preface learning physics aims at physics.pdf · 1 preface the study of physics helps to understand...

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Preface

The study of Physics helps to understand basic laws of nature and their manifestation in different

physical phenomena. It facilitates the development of experimental, observational, manipulative,

decision making and investigatory skills in the learners.

Learning Physics aims at:

Developing conceptual competence in the learners and make them realize and appreciate the

interface of Physics with other disciplines.

Exposing the learners to different processes used in Physics-related industrial and technological

applications.

The basic aim of Ready Reckoner is to provide the students with a ready to absorb material, which

can be very helpful at the time of revising the syllabus. Questions included at the end of each chapter

are in accordance with CBSE guidelines.

Each chapter has been divided into three sections:

Section A: Synopsis of the chapter emphasizing all the value points

Section B: Concept Based Exercises which includes Very Short, Short and Long Answer Questions

Section C: Enhancement exercises which includes HOTS and Value based questions

Practice sample papers

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INDEX

CHAPTER NO. CHAPTER NAME PAGE NO.

SYLLABUS AND MARKS DIVISION 3-5

DESIGN OF QUESTION PAPER 6-7

CH-1 PHYSICAL WORLD PHYSICS 8

CH-2 UNITS AND MEASUREMENT 9-18

CH-3& 4 MOTION IN ONE DIMENSION &MOTION IN A PLANE 19-33

CH-5 LAWS OF MOTION 34-49

CH-6 WORK, ENERGY AND POWER 50-58

CH-7 SYSTEM OF PARTICLES AND ROTATIONAL MOTION 59-70

CH-8 GRAVITATION 71-80

CH-9 &10 MECHANICAL PROPERTIES OF SOLIDS & FLUIDS 81-94

CH-11 &12 THERMAL PROPERTIES OF MATTER& THERMODYNAMICS 95--111

CH-13 KINETIC THEORY 112-119

CH-14 &15 OSCILLATIONS& WAVES 120-134

SAMPLE PAPERS 135-160

BIBLIOGRAPHY & STUDY TIPS 155

SYLLABUS & MARKS DIVISION PHYSICS

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COURSE STRUCTURE

CALSS XI (THEORY)

Unit I: Physical World and Measurement 10 Periods

Chapter–1: Physical World

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Physics-scope and excitement; nature of physical laws; Physics, technology and society.

Chapter–2: Units and Measurements

Need for measurement: Units of measurement; systems of units; SI units, fundamental and

derived units. Length, mass and time measurements; accuracy and precision of measuring

instruments; errors in measurement; significant figures. Dimensions of physical quantities,

dimensional analysis and its applications.

Unit II: Kinematics 24 Periods

Chapter–3: Motion in a Straight Line

Frame of reference, Motion in a straight line: Position-time graph, speed and velocity. Elementary

concepts of differentiation and integration for describing motion, uniform and non- uniform

motion, average speed and instantaneous velocity, uniformly accelerated motion, velocity - time

and position-time graphs. Relations for uniformly accelerated motion (graphical treatment).

Chapter–4: Motion in a Plane

Scalar and vector quantities; position and displacement vectors, general vectors and their

notations; equality of vectors, multiplication of vectors by a real number; addition and

subtraction of vectors, relative velocity, Unit vector; resolution of a vector in a plane, rectangular

components, Scalar and Vector product of vectors. Motion in a plane, cases of uniform velocity

and uniform acceleration projectile motion, uniform circular motion.

Unit III: Laws of Motion 14 Periods

Chapter–5: Laws of Motion

Intuitive concept of force, Inertia, Newton's first law of motion; momentum and Newton's second

law of motion; impulse; Newton's third law of motion. Law of conservation of linear momentum

and its applications. Equilibrium of concurrent forces, Static and kinetic friction, laws of friction,

rolling friction, lubrication. Dynamics of uniform circular motion: Centripetal force, examples of

circular motion (vehicle on a level circular road, vehicle on a banked road).

Unit IV: Work, Energy and Power 12 Periods

Chapter–6: Work, Energy and Power

Work done by a constant force and a variable force; kinetic energy, work energy theorem, power.

Notion of potential energy, potential energy of a spring, conservative forces: conservation of

mechanical energy (kinetic and potential energies); nonconservative forces: motion in a vertical

circle; elastic and inelastic collisions in one and two dimensions.

Unit V: Motion of System of Particles and Rigid Body 18 Periods

Chapter–7: System of Particles and Rotational Motion

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Centre of mass of a two-particle system, momentum conservation and centre of mass motion.

Centre of mass of a rigid body; centre of mass of a uniform rod. Moment of a force, torque,

angular momentum, law of conservation of angular momentum and its applications. Equilibrium

of rigid bodies, rigid body rotation and equations of rotational motion, comparison of linear and

rotational motions. Moment of inertia, radius of gyration, values of moments of inertia for simple

geometrical objects (no derivation). Statement of parallel and perpendicular axes theorems and

their applications.

Unit VI: Gravitation 12 Periods

Chapter–8: Gravitation

Kepler's laws of planetary motion, universal law of gravitation. Acceleration due to gravity and its

variation with altitude and depth. Gravitational potential energy and gravitational potential,

escape velocity, orbital velocity of a satellite, Geo-stationary satellites.

Unit VII: Properties of Bulk Matter 24 Periods

Chapter–9: Mechanical Properties of Solids

Elastic behaviour, Stress-strain relationship, Hooke's law, Young's modulus, bulk modulus, shear

modulus of rigidity, Poisson's ratio; elastic energy.

Chapter–10: Mechanical Properties of Fluids

Pressure due to a fluid column; Pascal's law and its applications (hydraulic lift and hydraulic

brakes), effect of gravity on fluid pressure. Viscosity, Stokes' law, terminal velocity, streamline

and turbulent flow, critical velocity, Bernoulli's theorem and its applications. Surface energy and

surface tension, angle of contact, excess of pressure across a curved surface, application of

surface tension ideas to drops, bubbles and capillary rise.

Chapter–11: Thermal Properties of Matter

Heat, temperature, thermal expansion; thermal expansion of solids, liquids and gases, anomalous

expansion of water; specific heat capacity; Cp, Cv -calorimetry; change of state - latent heat

capacity. Heat transfer-conduction, convection and radiation, thermal conductivity, qualitative

ideas of Blackbody radiation, Wein's displacement Law, Stefan's

law, Greenhouse effect.

Unit VIII: Thermodynamics 12 Periods

Chapter–12: Thermodynamics

Thermal equilibrium and definition of temperature ( zeroth law of thermodynamics), heat, work

and internal energy. First law of thermodynamics, isothermal and adiabatic processes. Second

law of thermodynamics: reversible and irreversible processes, Heat

Engine and refrigerator.

Unit IX: Behaviour of Perfect Gases and

Kinetic Theory of Gases 08 Periods

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Chapter–13: Kinetic Theory

Equation of state of a perfect gas, work done in compressing a gas, Kinetic theory of gases -

assumptions, concept of pressure. Kinetic interpretation of temperature; rms speed of gas

molecules; degrees of freedom, law of equi-partition of energy (statement only) and application

to specific heat capacities of gases; concept of mean free path, Avogadro's

number.

Unit X: Oscillations and Waves 26 Periods

Chapter–14: Oscillations

Periodic motion - time period, frequency, displacement as a function of time, periodic functions.

Simple harmonic motion (S.H.M) and its equation; phase; oscillations of a loaded spring- restoring

force and force constant; energy in S.H.M. Kinetic and potential energies; simple pendulum

derivation of expression for its time period. Free, forced and damped oscillations (qualitative

ideas only), resonance.

Chapter–15: Waves

Wave motion: Transverse and longitudinal waves, speed of travelling wave, displacement relation

for a progressive wave, principle of superposition of SSSSSSSSwaves, reflection of waves, standing

waves in strings and organ pipes, fundamental mode and harmonics, Beats, Doppler effect

QUESTION PAPER DESIGN:

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S.No Typology of Questions VSA-

Objective

Type

(1 mark)

SA

(2 marks)

LA-1

(3 marks)

LA-II

(5 marks)

Total

Marks

Percentage

1. Remembering: Exhibit memory

of previously learned material

by recalling facts, terms, basic

concept, and answers

2 2 1 - 9 12%

2. Understanding: Demonstrate

understanding of acts and ideas

by organizing, comparing,

translating, interpreting, giving

descriptions and stating main

ideas.

6 2 2 1 21 30%

3. Applying: Solve problems to

new situations by applying

acquired knowledge, facts,

techniques and rules in a

different way.

6 2 1 2 23 33%

4. Analysing and Evaluating:

Examine and break information

into parts by identifying motives

or causes. Make references and

find evidence to support

generalizations. Present and

defend opinions by making

judgements about information,

validity of ideas or quality of

work based on a set of criteria.

6 1 2 - 14 20%

5. Creating: Compile information

together in a different way by

combing elements in a new

pattern or proposing alternative

solutions.

- - 1 - 3 5%

Total 20x1=20 7x2=14 7x3=21 3x5=15 70 100

Practical 30 Marks

Note:

1. Internal Choice: There is no overall choice in the paper. However, there will be at least 33% internal choice.

2. The above template is only a sample. Suitable internal variations may be made for generating similar templates

keeping the overall weightage to different form of questions and typology of questions same.

PHYSICAL WORLD PHYSICS

SYNOPSIS

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1. Physics deals with the study of the basic laws of nature and their manifestation in different

phenomena. The basic laws of physics are universal and apply in widely different contexts and

conditions.

2. The scope of physics is wide, covering a tremendous range of magnitude of physical

quantities.

3. Physics and technology are related to each other. Sometimes technology gives raise to new

physics at other times physics generates new technology. Both have direct impact on society.

4. There are four fundamental forces in nature that govern the diverse phenomena of the

macroscopic and the microscopic world. These are the ‘gravitational force ‘, the

electromagnetic force’, ‘the strong nuclear force’, and the weak nuclear force’

5. The physical quantities that remain unchanged in a process are called conserved quantities.

Some of the general conservation laws in nature include the law of conservation of mass,

energy, linear momentum, angular momentum, charge, parity, etc.

6. Conservation laws have a deep connection with symmetries of nature. Symmetries of space

and time, and other types of symmetries play a central role in modern theories of

fundamental forces in nature.

7. Gravitational force is the force of mutual attraction between any two objects by virtue of their

masses. It is always attractive.

8. Electromagnetic Force is the force between charged particles .It acts over large distances and

does not need any intervening medium. Enormously strong compared to gravity. It can be

attractive or repulsive.

9. Strong nuclear force is the force that binds the nucleons together. It is the strongest of all the

fundamental forces. It is charge independent and very short range.

10. Weak nuclear force appears only in certain nuclear processes such as β-decay. Weak nuclear

force is not as weak as the gravitational force.

11. In a chemical reaction if the total binding energy of the reacting molecules is less than that of

the product molecules the difference appears as heat and the reaction is exothermic.

12. In a chemical reaction if the total binding energy of the reacting molecules is more than that

of the product molecules the difference amount of energy is absorbed and the reaction is

endothermic.

13. In a nuclear process mass gets converted into energy. This is the energy which gets released

in a nuclear power generation and nuclear explosions.

CHAPTER-2

UNITS AND MEASUREMENT

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SYNOPSIS

1. Physics – It is a quantitative science, based on measurement of physical quantities.

2. Physical quantities – Quantities which can be measured directly or indirectly and in terms of

which laws of physics can be expressed.

Fundamental or Base quantities – Independent of other physical quantities. Seven

fundamental physical quantities are Length, Mass, Time, Electric current, Thermodynamic

temperature, Amount of substance, and Luminous intensity.

Derived Quantities – Depends upon fundamental quantities. Eg. Acceleration, velocity etc.

3. Physical units

Fundamental or base units - They cannot be further resolved into more simpler units. Seven

fundamental units are metre, kilogram, second, Ampere, Kelvin, mole and Candela. Two

supplementary units in relation to quantities plane angle and solid angle are: - radian,

Steradian.

Derived units – Can be expressed as a combination of the base units.eg. m/s, m/s2.

4. The International System of units based on seven base units is at present internationally

accepted unit system and is widely used throughout the world

5. S.I. Unit – Suitable size, easily reproducible, accurately defined and do not change with time

and physical conditions.

6. Measurement of large distances by Parallax Method

Parallax- It is the apparent shift in the position of the object against the reference point in the

background. Distance(r) = where θ is parallax angle and x is basis the distance

between the two points of observation.

7. Error (Δa)– It is the difference between the true value and the measured value of the quantity.

Δa = True value – Measured value = a mean - a.

8. Accuracy - how close the measured value is to the true value of the quantity.

9. Precision - tells us to the limit to which quantity is measured.

10. The errors in measurement can be classified as

Systematic errors

Random errors and

Least count error

11. Systematic errors - Either positive or negative. Sources of systematic errors

are

Instrumental errors

Imperfection in experimental technique or procedure

Personal errors

12. Random errors - due to unpredictable fluctuations in experimental conditions

13. Least count error – due to the resolution of the instrument.

14. Absolute error - magnitude of the difference between the individual measurement and the

true value of the quantity measured. Ex: ∆a = I a-amean I

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15. Mean absolute error = Δamean=

16. Relative error - ratio of the mean absolute error to the mean value of the quantity measured.

Relative error = ∆amean/amean

17. Percentage error = ( ∆amean/amean) X 100

18. Combination of errors

a) ERROR OF A SUM OR A DIFFERENCE

When two quantities are added or subtracted, the absolute error in the final result is the sum of the absolute errors in the individual quantities. IF Z=A+ B then the max possible error in Z, ∆Z =∆A + ∆B IF Z=A- B then the max possible error in Z, ∆Z =∆A + ∆B

b) ERROR OF A PRODUCT OR A QUOTIENT

When two quantities are multiplied or divided the relative error is the sum of the relative errors in the multipliers Suppose Z = A x B or Z = A/B then the max relative error in ‘Z’ = ∆Z/Z= (∆A/A) + (∆B/B)

c) ERROR IN CASE OF A QUANTITY RAISED TO A

POWER

The relative error in a physical quantity raised to the

power k is the k times the relative error in the individual quantity.

Suppose Z = Ak then ∆Z/Z = K (∆A/A)

19. Significant figures - Reliable digits plus the first uncertain digit in a measurement.

20. Rules for finding the significant figures in a measurement

a) Always count nonzero digits

Example: 8.926 has four

b) Never count leading zeros

Example: 0.021 have two significant figures

c) Always count zeros which fall somewhere between two nonzero digits

Example: 20.8 has three significant figures, while 0.00104009 has six

d) Count trailing zeros if and only if the number contains a decimal point

Example: 210 and 210000 both have two significant figures, while 210 has three and 210.00

has five

21. For numbers expressed in scientific notation, ignore the exponent and apply Rules 1-4 to the

mantissa

Example: -4.2010 x 1028 has five significant figures

22. Rules for Arithmetic Operations with Significant Figures:

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In any mathematical operation involving addition, subtraction decimal places in the result will

correspond to lowest number of decimal places in any of the numbers involved.

Example: A= 334.5 kg; B= 23.45Kg then A + B =334.5 kg + 23.43 kg = 357.93 kg

The result with significant figures is 357.9 kg.

In a mathematical operation like multiplication or division, number of significant figures in the

product or in the quotient will correspond to the smallest number of significant figures in any

of the numbers involved.

Suppose F= 0.04 Kg X 0.452 m/sec2 =0.0108 kg-m/sec2

The final result is F = 0.01Kg-m/Sec 2

23. Rounding Off - While rounding off measurements the following rules are applied

a) Rule I: If the digit to be dropped is smaller than 5, then the preceding digit should be left

unchanged. For ex: 9.32 is rounded off to 9.3

b) Rule II: If the digit to be dropped is greater than 5, then the preceding digit should be raised

by 1.

Example: 8.27 is rounded off to 8.3 c) Rule III: If the digit to be dropped is 5 followed by digits other than zero, then the preceding

digit should be raised by 1

Example: 9.351 on being rounded off to first decimal, becomes 9.4 d) Rule IV: If the digit to be dropped is 5 or 5 followed by zeros, then the preceding digit is not

changed if it is even, is raised by 1 if it is odd.

Example: 5.45, on being rounded off, become 5.4 5.450 on being rounded off, becomes 5.4 7.35 , on being rounded off, becomes 7.4

24. Dimensions, Dimensional Formula and Dimensional Equation

a) Dimensions of a derived unit - Powers to which the fundamental units of mass, length and

time etc. must be raised to represent a derived unit.

Example: Density = Mass / Volume= M/L3= M1L-3 b) Dimensional formula - Expression which shows how and which of the fundamental units are

required to represent the unit of a physical quantity.

Example: M1 L 1T -2 is the dimensional formula of Force. 25. Categories of Physical Quantities

a) Dimensional Constants - Quantities which possess dimensions and have a fixed value. For ex

: Gravitational Constant

b) Dimensional Variables - Quantities which possess dimensions and do not have a fixed value.

For ex: velocity, acceleration etc.

c) Dimensionless Constants: Quantities which do not possess dimensions and have a fixed value.

For ex: π etc.

d) Dimensionless Variables: Quantities which are dimensionless and do not have a fixed value.

For ex: Strain, Specific Gravity etc.

26. Principle of Homogeneity of Dimensions

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a) A given physical relation is dimensionally correct if the dimensions of the various terms on

either side of the relation are the same.

27. Uses of Dimensional Equations

a) Conversion of one system of units into another.

b) Checking the accuracy of various formulae.

c) Derivation of formulae.

28. Limitations of Dimensional Analysis

a) It supplies no information about dimensionless constants. They have to be determined either

by experiment or by mathematical investigation.

b) It fails in case of exponential and trigonometric relations.

c) It fails when a physical quantity depends on more than three physical quantities.

d) It cannot identify all the factors on which the given physical quantity depends upon. The

method becomes more complicated when dimensional constants like G, h, etc. are involved.

CONCEPT BASED EXERCISE

VERY SHORT ANSWER QUESTIONS (1 MARK)

1. If f = x4, then relative error in f would be how many times the relative error in x?

2. Give two examples of non-dimensional variables.

3. If g is the acceleration due to gravity and is wavelength, then which physical quantity does

√(λg) represent?

4. If x = a +bt3, where x is in metres and t is in seconds, find the units of ‘a’ and ‘b’.

5. State the principle of homogeneity of dimensions.

SHORT ANSWER QUESTIONS (2 & 3 MARKS)

1. Describe the parallax method for the determination of the distance of a nearby star from the

earth.

2. Show that the maximum error in the sum of two quantities is equal to the sum of the absolute

errors in the individual quantities.

3. Show that the maximum fractional error in the product of two quantities is equal to the sum

of the fractional errors in the individual quantities.

4. Deduce the dimensional formula for the following quantities (i) Gravitational constant (ii)

Power (iii) Relative density (iv) Force (v) Pressure.

5. Mention various sources of errors while taking measurements.

6. Write differences between accuracy and precision.

NUMERICALS

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1. Solve the following to correct significant figures

a) (i) 5.1m + 13.235m (ii) 14.632kg / 5.52345kg

b) Find the area of a circle of radius 3.458 cm up to correct significant figures.

(i) 18.3m (ii) 2.6491kg (iii) 37.58cm2

2. The sides of a rectangle are (12.5 ± 0.2) cm and (8.3 ± 0.1) cm. Calculate its perimeter with

error limits. (31.4 + 0.6 cm)

3. The length of a rod as measured in an experiment was found to be 2.48m, 2.46m, 2.49m,

2.50m and 2.48m. Find the average length, absolute error and percentage error. Express the

result with error limit. (2.48 m ± 0.40%)

4. The error in the measurement of radius of a sphere is 2%. What would be the error in the

volume of the sphere? (6%)

5. A physical quantity X is given by . If the percentage error of measurement in A, B,

C and D are 4%, 2%, 3% and 1% respectively, then calculate the percentage error in X.(28%)

6. The orbital velocity of the satellite depends on mass of the satellite, radius of the orbit around

the earth and acceleration due to gravity. Derive the relation for the orbital velocity.

7. According to Vander Waal’s gas equation, P and V denotes pressure and

temperature and T is temperature. If a and b are constants. Find

dimensional formula for ‘a’ and ‘b’ assuming equation to be dimensionally consistent.([a] = [

M1 L5 T-2]; [b] = [L3] )

8. Check by the method of dimensions whether the following equation is correct: =

where = frequency of vibration, I = length of the string, T = Tension in the string

and m = mass per unit length.

9. Derive by method of dimensions, an expression for energy of the body executing SHM,

assuming that this energy depends on mass, frequency and the amplitude of vibration.

10. A physical quantity is measured as a = (2.1 ± 0.5) units. Calculate the percentage error in

i) Q2 ii) 2Q. i) 48% ii) 24%

ENRICHMENT EXERCISE

1) What are the dimensions of 1/µ 0ε 0 where symbols have their usual meaning?

2) The pairs of physical quantities that have the same dimensions are:

a) Reynolds’s number and coefficient of friction,

b) Curie and frequency of a light wave

c) Latent heat and gravitational potential

d) Planck’s constant and torque

Ans : (a), (b).

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3) If L,C,R represent inductance , capacitance and resistance respectively, the combinations

having dimensions of frequency are

(a)1/√CL (b) L/C (c) R/L (d ) R/C

Ans : (a) and (c).

4) If the error in radius is 3%, what is error in volume of sphere?

(a)3 % (b) 27 % (c) 9 % (d) 6 %

Ans : ( c ) 9%.

MCQ

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Answer Key :

1. (d) 2. (b) 3. (c) 4. (c) 5. (a) 6. (b) 7. (c) 8. (c) 9. (c) 10. (c) 11. (c) 12. (a) 13. (c) 14. (b) 15. (a) 16. (b)

17. (a) 18. (a) 19. (d) 20. (d)

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CHAPTER-3

MOTION IN ONE DIMENSION

SYNOPSIS

1. Rest – An object does not change its position w.r.t. its surroundings with the passage of time.

2. Motion - An object changes its position w.r.t. its surroundings with the passage of time.

3. One Dimensional Motion – The motion of an object is said to be one dimensional motion if

only one out of the three coordinates specifying the position of the object changes with time.

4. The position of the object can be specified with reference to a conveniently chosen origin. For

motion in a straight line, position to the right of the origin is taken as positive and to the left

as negative.

5. Path length or Distance(x) - The total length of the path traversed by an object.

6. Displacement - It is the change in position of an object in a fixed direction. ∆x=x2-x1, Path

length is greater than or equal to the magnitude of the displacement between the two

positions. Displacement is a vector Quantity. It can be positive, negative or zero.

7. An object is said to be in uniform motion in a straight line if its displacement is equal in equal

intervals of time. Otherwise the motion is said to be non-uniform.

8. Speed is the ratio of the total path length traversed to the corresponding time interval.

Speed=Distance travelled/time taken. It is a scalar quantity.

9. Uniform Speed – Object covers equal distances in equal intervals of time.

10. Average speed =

11. Instantaneous Speed – Speed of an object at any particular instant of time.

Instantaneous Speed =

12. Velocity is the rate of change of position of an object in a particular direction.

Velocity=Displacement/time taken. It is a Vector quantity.

13. Uniform Velocity – Object covers equal displacements in equal intervals of time.

14. Average velocity = 𝐷𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡

𝑇𝑜𝑡𝑎𝑙 𝑡𝑖𝑚𝑒 𝑡𝑎𝑘𝑒𝑛=

∆𝑥

∆𝑡

15. Instantaneous velocity – velocity of an object at any particular instant of time.

Instantaneous velocity = dx

dt

16. The average speed of an object is greater than or equal to the magnitude of the average

velocity over a given interval of time.

17. The velocity at a particular instant is equal to the slope of the tangent drawn on position –

time graph at that instant.

18. On a x-t graph, the average velocity over a time interval is the slope of the line connecting the

initial and final positions corresponding to that interval.

19. Acceleration – Rate of change of velocity of an object.

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Acceleration = Change in velocity/ Time taken. It is a vector quantity. 20. Uniform Acceleration – Velocity of an object changes by equal value in equal interval of time.

21. Average acceleration is the change in velocity divided by the time interval during which he

change occurs. a = ∆v/∆t

22. Instantaneous acceleration is defined as the limit of the average acceleration as the time

interval ∆t goes to zero. 𝑎 = 𝑑��

𝑑𝑡.

23. The acceleration of an object at a particular time is the slope of the velocity- time curve at

that instant of time.

24. The area under the velocity- time curve between times t1 and t2 is equal to the displacement

of the object during that interval of time.

25. For objects in uniformly accelerated rectilinear motion,

a) v= u+ at

b) s=ut + ½ a t2

c) v2- u2 = 2 a s

26. The steepness of the slope of position vs. time graph tells us the magnitude of the velocity &

its sign indicates the direction of the velocity.

27. Motion Under Gravity – In the absence of air resistance, all bodies fall with the same

acceleration ‘g’ near the surface of the earth( for h<<R, g is taken to be constant).

28. Relative Velocity – The relative velocity of an object B with respect to object A when both are

in motion is the rate of change of position of object B with respect to object A. Relative

velocity of A wrt B �� AB = �� A – �� B

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VERY SHORT ANSWER TYPE QUESTIONS (1 MARK)

1. Is the direction of acceleration same as the direction of velocity?

2. What is the nature of the displacement-time curve of a body moving with uniform

acceleration?

3. Two balls of different masses (one lighter and other heavier) are thrown vertically upwards

with the same speed. Which one will pass through the point of projection in their downward

direction with the greater speed?

4. Draw position-time graph for two objects having zero relative velocity.

5. Under what condition is the average velocity equal to the instantaneous velocity?

SHORT ANSWER TYPE QUESTIONS (2 & 3 MARKS)

1. Two straight lines A and B drawn on the same displacement-time graph make angles 30o and

60o with time-axis respectively. Which line represents greater velocity? What is the ratio of

two velocities?

2. If in case of a motion, displacement is directly proportional to the square of the time elapsed,

what do you think about its acceleration i.e. constant or variable? Explain why.

3. An object is in uniform motion along a straight line. What will be position-time graph for the

motion of the object if

a) xo = +ve, v = +ve

b) xo = +ve, v = -ve

c) xo = -ve, v = +ve

both xo and v are negative?

The letter xo and v represent position of the object at t = 0 and uniform velocity of the object

respectively.

4. Distinguish between speed and velocity.

5. Using calculus method prove that v2 – u2 = 2as.

6. Draw the following graphs for an object projected upward with a velocity vo, which comes to

the same point after some time:

a) Acceleration v/s time graph.

b) Speed v/s time graph.

c) Velocity v/s time graph.

LONG ANSWER TYPE QUESTIONS (5 MARKS)

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1. Derive the following equations of motion for uniformly accelerated motion from velocity

time graph:

a) v = u + at

b) s = ut + ½ at2

c) v2 – u2 = 2as

NUMERICALS

1. On a 60 km straight road, a bus travels the first 30 km with a uniform speed of 30 kmh–1. How

fast must the bus travel the next 30 km so as to have average speed of 40 kmh–1 for the entire

trip? (60km/h)

2. The displacement x of a particle varies with time as x = 3t2 – 10t + 15. Find the position,

velocity and acceleration of the particle at t = 0.(15m, -10m/s,6m/s2)

3. A driver take 0.20 second to apply the brakes (reaction time). If he is driving car at a speed of

54 kmh–1 and the brakes cause a deceleration of 6.0 ms–2. Find the distance travelled by car

after he applies the brakes.(21.75 m)

4. A ball thrown vertically upwards with a speed of 19.6 ms–1 from the top of a tower returns to

the earth in 6s. Find the height of the tower. (g = 9.8 m/s2) (58.8m)

5. A food packet is released from helicopter which is rising steadily at 2m/s. After two seconds.

What is the velocity of the packet? How far is it below the helicopter?

( g = 9.8 m/s2)( -17.6m/s, 19.6m)

6. Starting from rest a car accelerates uniformly with 3m/s2 for 5s and then moves with uniform

velocity. Draw the distance-time graph of the motion of the car upto t = 7s.

7. Two trains 120m and 80m in length are running in opposite directions with velocities 42km/h

and 30 km/h. In what time they will completely cross each other? (10s)

8. Two buses start simultaneously towards each other from towns A and B which are 480 km

apart. The first bus takes 8 hours to travel from A to B while the second bus takes 12 hours to

travel from B to A. Determine when and where the buses will meet. (4.8h, 288 km from A)

9. A motorboat covers the distance between the two spots on the river in 8 h and 12 h

downstream and upstream respectively. Find the time required by the boat to cover this

distance in still water. ( 9.6h )

10. A body covers 12 m in 2nd second and 20 m in 4th second. How much distance will it cover in

4 seconds after the 5th second?(136m)

ENRICHMENT EXERCISE

22

1. A ball is released from the top of the tower of height h metres. It takes T seconds to reach the

ground. What is the position of the ball in T/3 seconds? (8h/9 above the ground)

2. A jugglers maintains four balls in motion, making each in turn rise to a height of 20m from his

hand. With what velocity does he project them and where will the other three balls be at the

instant when the fourth one is just leaving the hand? Take g = 10m/s2.

3. A car, starting from rest, accelerates at the rate f through a distance s, then continues at

constant speed for some time t and then decelerates at the rate f/2 to come to rest. If the

total distance is 5s, then prove that s = ½ ft2.

HOTS

1. A boat is sent across a river with a velocity of 8km/h. if the resultant velocity of boat is 10

km/h , then calculate the velocity of the river.

Ans : 6 km/h.

2. A cricket ball is hit at 450to the horizontal with a kinetic energy E. calculate the kinetic energy

at the highest point.

Ans: E/2.(because the horizontal component uCos450is present on highest point.)

3. Speed of two identical cars are u and 4u at a specific instant. The ratio of the respective

distances at which the two cars stopped from that instant.

Ans : 1 : 16

4. A projectile can have the same range R for two angles of projection. If t1and t2 be the time of

flight in the two cases, then prove that t1t2 = 2R/g

23

CHAPTER-4

24

MOTION IN A PLANE

SYNOPSIS

1. Scalar quantities are quantities with magnitudes only. Examples are distance, speed mass and

temperature.

2. Vector quantities are quantities with magnitude and direction both. Examples are

displacement, velocity and acceleration. They obey special rules of vector algebra.

3. Representation of a vector

Represented by a straight line with an arrowhead over it.

Length gives magnitude

Arrowhead gives direction

4. Position vector - Gives position w.r.t. to the origin.

5. Displacement Vector – Tells how much and in which direction an object has changed its

position in a given interval of time.

6. Equal Vectors – Two vectors having same magnitude and same

direction.

7. Negative of a vector – Another vector having same magnitude but

opposite direction.

8. Parallel Vector – Two vectors having same direction.

𝐴

��

9. Anti – parallel Vector – Two vectors having opposite direction.

25

A

B

10. Modulus of a vector (IAI ) – Magnitude of a vector.

11. A unit vector associated with a vector A has magnitude one and is along the vector A . �� = 𝑟

|𝑟 |.

The unit vectors i, j ,k are vectors of unit magnitude and point in the direction of x, y, and z-

axes respectively in a coordinate system.

12. A vector ’A’ multiplied by a real number ‘λ’ is also a vector, whose magnitude is ‘λ’ times the

magnitude of the vector ‘ A ‘and whose direction is same or opposite depending upon

whether ‘λ’ is positive or negative.

13. A null or zero vector (O) - A vector with zero magnitude. Since the magnitude is zero, we don’t

have to specify its direction. Eg. Velocity vector of a stationary object is a null vector. It has

the properties:

A + O =A

λO =O

OA=O

14. Two vectors A and B may be added graphically using head to tail method or parallelogram

method.

a) Triangle law of vector addition – If two vectors can be represented both in magnitude and

direction by the two sides of a triangle taken in the same order, then the resultant is

represented completely both in magnitude and direction by the third side of the triangle.

26

b) Parallelogram law of vector addition – If two vectors can be represented both in magnitude

and direction by the two adjacent sides of a parallelogram drawn from a common point then

their resultant is completely represented both in magnitude and direction by the diagonal of

the parallelogram passing through that point.

R = (A2 + B2 + 2AB cosθ)1/2

tan β = Bsinθ

A + Bcosθ

Resolution of a Vector – It is the process of splitting of a vector into two or more vectors in

such a way that their combined effect is same as that of the given vector. If a vector is

resolved into two components along the two mutually perpendicular directions, they are

called ‘rectangular components’.

15. Rectangular components of a vector in a plane - If 𝐴 makes an angle 𝜃 with x-axis and A x and A y be

-axis and y-axis respectively, then A = Ax + Ay

= Ax

i+Ayj Here Ax = A cosθ and Ay = A sinθ and tan θ = Ay

Ax

16. The dot product of two vectors A andB , represented by A.B is a scalar, which is equal to the product

of the magnitudes of A and B and the cosine of the smaller angle between them. If θ is the smaller

angle between A and B, then A.B= AB cosθ.

17. The vector product or cross product of two vectors A and B is represented as A XB . If θ is the

smaller angle between A and B ,then A B C ABs in �� where n is a unit vector perpendicular to

the plane containing A and B .

18. Right handed screw rule. It states that if a right handed screw placed with its axis

perpendicular to the plane containing the two vectors A and B is rotated from the direction

of A to the direction of B through smaller angle, then the sense of the advancement of the tip

of the screw gives the direction of(A XB ).

19. Position Vector – r = rxi + ryj + rzk

20. Displacement from position r to position r’ is given by ∆r = r’-r = (x’-x)i + ( y’-y)j

27

21. Velocity Vector - V=vxi + vyj + vz k where Vx = dx/dt, Vy = dy/dt ,Vz= dz/dt . When position of

an object is plotted on a coordinate system v is always tangent to the curve representing the

path of the object.

22. Average acceleration is given by a = (v-v’)/∆t= ∆v/∆t

23. Instantaneous acceleration =lim ∆v/∆t = dv/dt ∆t→0

24. In component form, we have a = axi + ay j +az k Where ax = dvx/dt, ay = dvy/dt, az = dvz /dt

25. If an object is moving in a plane with constant acceleration a (IaI =√(ax2 + ay

2)) And its position

vector at time t = 0 is r0, then at any other time t, it will be at a point given by :r = r0 + V0t + ½

a t2 and its velocity is given by : V = V0t + at where V0 is the velocity at time t = 0

26. In component form

X = x0 +V0xt + ½ ax t2 Y= y0 + V0y t + ½ ay t2

Vx = V0x + ax t Vy = V0y + ay t

27. Motion in a plane can be treated as superposition of two separate simultaneous one

dimensional motions along two perpendicular directions.

28. Projectile. Projectile is the name given to a body which is thrown with some initial velocity

with the horizontal direction and then it is allowed to move under the effect of gravity alone.

29. Angular velocity (ω) : It is defined as the time rate of change of angular displacement of the

object i.e. ω= dθ/dt. Its S.I unit is rad/s.

30. Uniform circular motion: When a point object is moving on a circular path with a constant

speed, then the motion of the object is said to be a uniform circular motion.

28

31. Centripetal acceleration: It is defined as the acceleration of as object undergoing uniform

circular motion. It always acts along the radius towards the centre of the circular path. The

magnitude of centripetal acceleration is, a = v2/r = ω2r.

32. The rate of change of angular velocity (α) is called its angular acceleration (α) i.e. α=dw

dt.

33. The acceleration which changes the magnitude of the velocity is called tangential

acceleration. It is given by aT = rα, where α is the angular acceleration. The direction of

tangential acceleration is along the tangent to curved path.

34. When a body moves in a circular path with increasing angular velocity, it has two linear

accelerations.

a) Centripetal acceleration, ac = v2/r ;

b) tangential acceleration, aT = r α; Resultant acceleration of the body is a = √ac2 + aT

2 and tan β = aT

acwhere β is the angle between aT and ac



1. What does the slope of v-t graph indicate?

2. Under what condition the average velocity equal to instantaneous velocity.

3. The position coordinate of a moving particle is given by x=6+18t+9t2(x in meter, t in seconds)

what is it’s velocity at t=2s

4. Give an example when a body moving with uniform speed has acceleration.

5. Two balls of different masses are thrown vertically upward with same initial velocity. Height

attained by them are h1 and h2respectively what is h1/h2.

6. State the essential condition for the addition of the vector.

7. What is the angle between velocity and acceleration at the peak point of the projectile

motion?

8. What is the angular velocity of the hour hand of a clock?

9. What is the source of centripetal acceleration for earth to go round the sun?

10. What is the average value of acceleration vector in uniform circular motion?

29


1. The sum and difference of two vectors are perpendicular to each other. Prove that the vectors

are equal in magnitude.

2. The resultant of two vectors P and Q is perpendicular to P and its magnitude is half that of Q.

What is the angle between P and Q?

3. A skilled gunman always keeps his gun slightly tilted above the line of sight while shooting.

Why?

4. A person sitting in a moving train throws a ball vertically upwards. How will the ball appear to

move to an observer a) sitting inside the train b) standing outside the train? Give reason for

your answer.

5. Prove that the horizontal range is same when angle of projection is i) greater than 45o by

certain value and ii) less than 45o by the same value.

6. Do A+B and A-B lie in the same plane? Give reason.

7. A railway carriage moves over a straight track with acceleration a. A passenger in the carriage

drops a stone. What is the acceleration of the stone w.r.t. the carriage and the earth?

8. If ‘R’ is the horizontal range for Ɵ inclination and H is the height reached by the projectile,

show that R(max.) is given by Rmax =4H

9. A body is projected at an angle Θ with the horizontal. Derive an expression for its horizontal

range. Show that there are two angles Θ1 and Θ2 projections for the same horizontal range.

Such that (Θ1 +Θ2 ) = 900.

10. Prove that there are two values of time for which a projectile is at the same height . Also show

that the sum of these two times is equal to the time of flight.

11. Draw position –time graphs of two objects , A and B moving along straight line, when their

relative velocity is zero.

12. When the angle between two vectors of equal magnitudes is 2π/3, prove that the magnitude

of the resultant is equal to either.

13. A ball thrown vertically upwards with a speed of 19.6 m/s from the top of a tower returns to

the earth in 6s. find the height of the tower. ( g = 9.8 m/sec2)

14. Find the value of λ so that the vector = 2 + λĵ +k and = 4î – 2ĵ – 2k are perpendicular to each.

15. Show that a given gun will shoot three times as high when elevated at angle of 600 as when

fired at angle of 300 but will carry the same distance on a horizontal plane.

30


1. State the parallelogram law of vector addition and find the magnitude and direction of the

resultant of two vectors P and Q inclined at an angle with each other. What happens, when

= 0o and = 90o?

2. What is a projectile? Derive the expression for the trajectory, time of flight, maximum height

and horizontal range for a projectile thrown upwards, making an angle with the horizontal

direction.

3. Define centripetal acceleration. Derive an expression for the centripetal acceleration of a

body moving with uniform speed v along a circular path of radius r. Explain how it acts along

the radius towards the centre of the circular path.

NUMERICALS

1. A motorboat is racing towards north at 25 kmh–1 and the water current in that region is 10

kmh–1 in the direction of 60° east of south. Find the resultant velocity of the boat.

( 21.8 kmh–1, 23.4°)

2. If A = 3i + 4j and B = 7i + 24j, find a vector having the same magnitude as B and parallel to A.

(15i + 20j)

3. Find the angles between A = i +2 j - k and B = -i + j – 2k. (600)

4. A force F = 4i + j + 3k N acts on a particle and displaces it through displacement

S = 11i + 11j + 15k metre. Calculate the work done by the force. (100J)

5. Determine the area of the parallelogram whose adjacent sides are formed by the vectors

A = 2i + 3j - 4k and B = i + j – k.

6. Find a unit vector perpendicular to the vectors A = i –3 j + k and B = i + j + k.

(3√5 square units)

7. A hiker stands on the edge of a cliff 490 m above the ground and throws a stone horizontally

with an initial speed of 15 ms -1. Neglecting air resistance, find the time taken by the stone to

reach the ground and the speed with which it hits the ground (g = 9.8 ms–2)

(99.1m/s)

8. A stone tied to the end of a string 80 cm long is whirled in a horizontal circle with a constant

speed. If the stone makes 14 revolutions in 25 seconds, what is the magnitude and direction

of acceleration of the stone? (991.2 cms-2)

9. A cyclist is riding with a speed of 27 kmh–1. As he approaches a circular turn on the road of

radius 30 m, he applies brakes and reduces his speed at the constant rate 0.5 ms–2. What is

the magnitude and direction of the net acceleration of the cyclist on the circular turn?

(0.86m/s2, 54o 28 ')

10. To a driver going east in a car with a velocity of 40 kmh-1, a bus appears to move towards

north with a velocity of 40 kmh-1. What is the actual velocity and direction of motion of

the bus? ( 300, east of north)

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ENRICHMENT EXERCISE

1. The x- and y- components of A are 4m and 6m respectively. The x- and y- components of

vector A + B are 10m and 9m respectively. Calculate for the vector B a) its x- and y-

components b) its length and c) the angle it makes with the x-axis.

2. A ball is thrown from a point with a speed of vo at an angle of projection . From the same

point and at the same instant, a person starts running with a constant speed of vo/2 to catch

the ball. Will the person be able to catch the ball? If yes, what should be the angle of

projection? (60o)

HOTS

1. A boat is sent across a river with a velocity of 8km/h. if the resultant velocity of boat is 10

km/h , then calculate the velocity of the river.

Ans : 6 km/h.

2. A cricket ball is hit at 450to the horizontal with a kinetic energy E. calculate the kinetic energy

at the highest point.

Ans : E/2.(because the horizontal component uCos450is present on highest point.)

3. Speed of two identical cars are u and 4u at a specific instant. The ratio of the respective

distances at which the two cars stopped from that instant.

Ans : 1 : 16

4. A and B are two vectors and Θ is the angle between them, If calculate

the value of angle Θ .

5. A projectile can have the same range R for two angles of projection. If t1and t2 be the time of

flight in the two cases, then prove that t1t2 = 2R/g

KINEMATICS MCQ

33

CHAPTER-5

34

LAWS OF MOTION

SYNOPSIS

1. Newton’s first law of motion states that everybody continues to be in its state of rest or of

uniform motion unless it is acted by an external force.

2. If external force on a body is zero, its acceleration is zero.

3. Momentum (p) of a body is the product of mass (m) and velocity (v) : p = mv.

4. Newton’s Second law of motion: It states that the rate of change of momentum of a body is

proportional to the applied force and takes place in the direction in which force acts. Thus F =

k dp/dt = k ma. SI unit of force : 1N= 1kgms-2

5. Force is not always in the direction of motion .Depending on the situation F may be along v,

opposite to v, normal to v, or may make some other angle with v. In every case it is parallel to

acceleration.

6. If v = 0 at an instant, i.e., if a body is momentarily at rest, it does not mean that force or

acceleration are necessarily zero at that instant.

7. Impulse (J) – It is a large force acting for a short time to produce finite change in momentum. It

is equal to change in momentum. Impulse = Fav X ∆t = m (v- u). Unit of measurement of Impulse

is Ns.

8. Newton’s third law of motion: “To every action, there is an equal and opposite reaction.” Action

and reaction can act on different bodies and so they cannot be cancelled out. The internal action

and reaction forces between parts of a body do however sum to zero. FAB = - FBA

9. According to the principle of conservation of linear momentum, the vector sum of linear

momenta of all the bodies in an isolated system is conserved.

10. Apparent weight of a man in an elevator is given by R = m(g ± a).

a) When a lift moves upwards with uniform acceleration, apparent weight of a body in the lift

increases, R=m(g+a); decreases when lift moves downwards with acceleration a, R=m(g-a).

b) When a lift is at rest or moves with uniform velocity, the apparent weight of the body is equal to

its true weight, R=mg.

c) When a lift falls freely (a=g), the apparent weight of a body becomes zero. R=m(g-g)=0.

11. Equilibrium of concurrent forces – When a number of forces act on a body at the same point and

the net unbalanced force is zero, the body will continue in its state of rest or of uniform motion

along a straight line and is said to be in equilibrium.

F2

F1 F1 + F2 +F3 = 0

F3

12. Motion of connected bodies - When two bodies of masses m1 and m2 are tied at the ends of an

inextensible string passing over a light frictionless pulley, acceleration of the system & tension in

the string is given by

35

13. Friction is the opposing force that comes into play when one body is actually moving over the

surface of another body or tries to move over the surface of the other, a force comes into play

which acts parallel to the surface of contact and opposes the relative motion, called as friction.

Cause of friction - Force of adhesion between the molecules of the surfaces in contact and

roughness of surfaces in contact.

14. Types of friction

a) Static friction: It comes in to effect when object is at rest but external force is applied. Static

friction is a self-adjusting force.

b) Limiting friction: the maximum force of static friction which comes into play when a body just

starts moving over the surface of another body. Limiting friction is the maximum value of static

friction.

c) Kinetic friction: the force of friction which comes into play when a body is in steady motion over

the surface of another body. Kinetic friction is less than limiting friction.

d) Rolling friction- it comes in to play when object is in rolling.

15. Coefficient of friction: μ = f/R when f = Frictional force and R = Normal reaction

16. Static frictional force‘fs‘oppose impending relative motion: kinetic frictional force’ fk ‘opposes

actual relative motion.

36

17. They are independent of the area of contact.

18. They satisfy the following approximate laws:

a) fs ≤ (fs) max = μs R, fk = μk R, μk < μs

b) Where μs(coefficient of static friction ) and μk( Coefficient of kinetic friction) are constants

characteristic of the pair of surfaces in contact.

Rolling Frictional Force - Frictional force which opposes the rolling of bodies (like cylinder, sphere,

ring etc.) over any surface is called rolling frictional force. Rolling frictional force acting between

any rolling body and the surface is almost constant and is given by μrN. Where μr is coefficient of

rolling friction and N is the normal reaction between the rolling body and the surface. fr = μrN

Note – Rolling frictional force is much smaller than maximum value of static and kinetic frictional

force.

fr << fk < fs(max)

or, μrN << μkN < μsN

or, μr << μk < μs

Cause of Rolling Friction – When anybody rolls over any surface it causes a little depression and

a small hump is created just ahead of it. The hump offers resistance to the motion of the rolling

body, this resistance is rolling frictional force. Due to this reason only, hard surfaces like cemented

floor offers less resistance as compared to soft sandy floor because hump created on a hard floor

is much smaller as compared to the soft floor.

Need to Convert Kinetic Friction into Rolling Friction – Of all the frictional forces rolling frictional

force is minimum. Hence in order to avoid the wear and tear of machinery it is required to convert

kinetic frictional force into rolling frictional force and for this reason we make the use of ball-

bearings.

Friction: A Necessary Evil – Although frictional force is a non-conservative force and causes lots

of wastage of energy in the form of heat yet it is very useful to us in many ways. That is why it is

considered as a necessary evil.

Advantages of Friction -

a) Friction is necessary in walking. Without friction it would have been impossible for us to walk.

37

b) Friction is necessary for the movement of vehicles on the road. It is the static frictional force

which makes the acceleration and retardation of vehicles possible on the road.

c) Friction is helpful in tying knots in the ropes and strings.

d) We are able to hold anything with our hands by the help of friction only.

Disadvantages of Friction -

a) Friction causes wear and tear in the machinery parts.

b) Kinetic friction wastes energy in the form of heat, light and sound.

c) A part of fuel energy is consumed in overcoming the friction operating within the various parts of

machinery.

Methods to Reduce Friction –

a) By polishing – Polishing makes the surface smooth by filling the space between the depressions

and projections present in the surface of the bodies at microscopic level and there by reduces

friction.

b) By proper selection of material – Since friction depends upon the nature of material used hence

it can be largely reduced by proper selection of materials.

c) By lubricating – When oil or grease is placed between the two surfaces in contact, it prevents the

surface from coming in actual contact with each other. This converts solid friction into liquid

friction which is very small.

19. Angle of friction (θ) is the angle which resultant of F and R makes with the direction of R. The

relation between θ and μ is μ = tan θ.

20. Angle of Repose (α) is the minimum angle of inclination of a plane with the horizontal, such that

a body placed on the plane just begins to slide down. μ = tanα

38

21. Work done against friction

Work done in moving a body up an inclined plane W = mg( sin + k cos ) S

Work done in moving a body down an inclined plane W = mg( k cos - sin )

22. Why is it easier to pull a lawn roller than to push it?

fk = k ( W + F sin ) > fk = k ( W – F sin )

The force of friction is more in case of push than in case of pull.

23. Centripetal force is the name given to the force that provides inward radial acceleration to a body

in circular motion. We should always look for some material force like tension, gravitational force,

electrical force, friction etc. as the centripetal force. F = m 2r or F=mv2/r

24. During motion on level curved road, the necessary centripetal force is provided by the force of

friction between the tyres and the road. The maximum velocity with which a vehicle can go round

a level curve without skidding is v =√𝜇𝑟𝑔

25. To increase speed on turn, curved roads are usually banked i.e. outer edge of the curved road is

raised suitably above the inner edge. If θ is the angle of banking, then tan 𝜃=𝑣2

𝑟𝑔.

26. When frictional force is ignored, the optimum speed is v0 = (rg tan θ)1/2.

27. While rounding a banked curved road, maximum permissible speed is given by

vmax=√[𝑟𝑔(𝜇𝑠+𝑡𝑎𝑛𝜃)

1−𝜇𝑠𝑡𝑎𝑛𝜃] when friction is taken in to account.

39

28. When a cyclist takes a turn. He bends a little inwards from his vertical position, while turning.

Angle θ of bending from vertical position is given by tan θ = 𝑣2

𝑟𝑔

CENTRIFUGAL FORCE

It is a pseudo force experienced by a body which is a part of the circular motion. It is a non-realistic

force and comes into action only when the body is in a circular motion. Once the circular motion of

the body stops, this force ceases to act. Its magnitude is exactly same as that of centripetal force but

it acts opposite to the direction of the centripetal force that is in the radially outward direction.

Frame of reference attached to a body moving on a circular path is a non-inertial frame since it an

accelerated frame. So when ever anybody is observed from this frame a pseudo force F = ma = mv2/r

= mrω2 must be applied on the body opposite to the direction of acceleration along with the other

forces. Since the acceleration of the frame in circular motion is centripetal acceleration a = v2/r

directed towards the center of the circular path, the pseudo force applied on the bodies observed

from this frame is F = mv2/r directed away from the center of the circular path. This pseudo force is

termed as a centrifugal force.

CENTRIFUGE

It is an apparatus used to separate cream from milk. It works on the principal of centrifugal force. It

is a cylindrical vessel rotating with high angular velocity about its central axis. When this vessel

40

contains milk and rotates with high angular velocity all the particles of milk start moving with the

same angular velocity and start experiencing centrifugal force FCentrifugal= mrω2in radially outward

direction. Since centrifugal force is directly proportional to the mass of the particles, massive particles

of milk on experiencing greater centrifugal force starts depositing on the outer edge of the vessel and

lighter cream particles on experiencing smaller centrifugal force are collected near the axis from

where they are separated apart.

Inertial and Non-inertial Frame of Reference

Frame of reference is any frame with respect to which the body is analysed. All the frames which are

at rest or moving with a constant velocity are said to be inertial frame of reference. In such frame of

reference all the three laws of Newton are applicable.

Any accelerated frame of reference is said to be non-inertial frame of reference. In such frames all

the three laws of Newton are not applicable as such. In order to apply Newton’s laws of motion in a

non-inertial frame, along with all other forces a pseudo force F = ma must also be applied on the

body opposite to the direction of acceleration of the frame

41

MEMORY MAP

42


VERY SHORT TYPE ANSWER QUESTION (1 MARK)

1. Is net force needed to keep a body moving with uniform velocity?

2. Is Newton’s 2ndlaw (F=ma) always valid. Give an example in support of your answer?

3. Action and reaction forces do not balance each other. Why?

4. Can a body remain in state of rest if more than one force is acting upon it?

5. Is the centripetal force acting on a body performing uniform circular motion always constant?

6. The string is holding the maximum possible weight that it could withstand. What will happen

to the string if the body suspended by it starts moving on a horizontal circular path and the

string starts generating a cone?

7. What is the reaction force of the weight of a book placed on the table?

8. What is the maximum acceleration of a vehicle on the horizontal road? Given that coefficient

of static friction between the road and the tyres of the vehicle is μ.

9. Why guns are provided with the shoulder support?

43

10. While paddling a bicycle what are the types of friction acting on rear wheels and in which

direction?

Short Answer Questions (2 or 3 marks)

1. Explain why the water doesn’t fall even at the top of the circle when the bucket full of water

is upside down rotating in a vertical circle?

2. The displacement of a particle of mass 1kg is described by s = 2t + 3t2. Find the force acting

on particle?

3. A particle of mass 0.3 kg is subjected to a force of F = -kx with k = 15 Nm–1. What will be its

initial acceleration if it is released from a point 10 cm away from the origin?

4. Three forces F1, F2 and F3 are acting on the particle of mass m which is stationary. If F1 is

removed, what will be the acceleration of particle?

5. A spring balance is attached to the ceiling of a lift. When the lift is at rest spring balance reads

50 kg of a body hanging on it. What will be the reading of the balance if the lift moves :-

a) Vertically downward with an acceleration of 5 ms–2

b) Vertically upward with an acceleration of 5 ms–2

c) Vertically upward with a constant velocity. Take g = 10m/s2. [(i) 25kgf,(ii) 75kgf, (iii) 50kgf]

6. Is larger surface area break on a bicycle wheel more effective than small surface area brake?

Explain?

7. Calculate the impulse necessary to stop a 1500 kg car moving at a speed of 25ms–1?

8. Give the magnitude and directions of the net force acting on a rain drop falling freely with a

constant speed of 5 m/s?

9. A block of mass .5kg rests on a smooth horizontal table. What steady force is required to give

the block a velocity of 2 m/s in 4 s?

10. Calculate the force required to move a train of 200 quintal up on an incline plane of 1 in 50

with an acceleration of 2 ms–2. The force of friction per quintal is 0.5 N?

LONG TYPE ANSWER QUESTION (5 MARKS)

1. What is meant by banking of roads? What is the need for banking a road? Obtain an

expression for the maximum speed with which a vehicle can safely negotiate a curved road

banked at an angle θ. The coefficient of friction between the wheels and the road is µ.

2. Define the terms static friction, limiting friction and kinetic friction and represent them on the

graph between friction and applied force.

3. Using graph explain that static force is a self-adjusting force.

NUMERICALS

44

1. The velocity of a body of mass 2kg as a function of t is given by v(t) = (2t i + t2 j)m/s. Find the

momentum and the force acting on it, at time t=2s. ((8i+8j)kgm/s,(4i + 4j)N)

2. A bullet of mass 0.01 kg is fired horizontally into a 4 kg wooden block at rest on a horizontal

surface. The coefficient of kinetic friction between the block and surface is 0.25. The bullet remain

embedded in the block and combination moves 20 m before coming to rest. Find the speed of

the bullet strike the block? (4000m/s)

3. An elevator and its load weigh a total of 800kg. Find the tension T in the supporting cable when

the elevator, originally moving downwards at 20m/s is brought to rest with a constant retardation

in a distance of 50m. (1.014x104 N)

4. A mass of 6 kg is suspended by a rope of length 2 m from the ceiling. A force of 50 N horizontally

is applied at the mid – point. P of the rope. Calculate the angle of rope makes with the vertical.

Neglect the mass of rope. (g = 9.8 ms–2) (θ = 40°)

5. Three blocks of masses m1 = 10 kg, m2 = 20 kg and m3 = 30 kg are connected by strings on smooth

horizontal surface and pulled by a force of 60 N. Find the acceleration of the system and tensions

in the string. (1m/s2, T1 = 10N, T2 = 30N)

6. A body of mass 15kg is hung by a spring balance in a lift. What would be the reading of the balance

when (i) the lift is ascending with an acceleration of 2m/s2, (ii) descending with the same

acceleration, (iii) descending with a constant velocity of 2m/s. (18kg f, 12kg f, 15kg f)

7. The coefficient of friction between the ground and the wheels of a car moving on a horizontal

road is 0.5. If the car starts from rest, what is the minimum distance in which it can acquire a

speed of 72km/h. (40m)

45

8. A block of metal of mass 50g when placed over an inclined plane at an angle of 150 slides down

without acceleration. If the inclination is increased by 150, what would be the acceleration of the

block? (tan150= 0.267) (2.6m/s2)

9. A circular race

track of radius

300m is

banked at an

angle of 150. If the

coefficient of

friction

between the

wheels of a

car and the road

is 0.2, what is

the (i) the optimum speed of the car to avoid wear and tear of the tyres?,(ii) maximum permissible

speed to avoid slipping?(28.1m/s,38.1m/s)

10. A rectangular block lies on a rough inclined surface. The coefficient of friction between the

surface and the box is μ. Let the mass of the box be m

a) At what angle θ of the plane to the horizontal will the box just start to slide down the plane?

b) What is the force acting on the box down the plane, if the angle of inclination is increased to α >

θ?

c) What is the force needed to be applied upwards along the plane to make the box remain either

stationary or just move up with the uniform speed?

ENRICHMENT EXERCISE

1. A block of mass 0.1 kg is held against a wall by applying a horizontal force of 5 N on the block. If

the coefficient of friction between the block and the wall is 0.5, what is the magnitude of the

frictional force acting on the block? (0.98N)

2. An aeroplane requires for takeoff a speed of 80 km/h, the run on the ground being 100m. The

mass of the aeroplane is 104 kg and the coefficient of friction between the plane and the ground

is 0.2. Assume that the plane accelerates uniformly during the takeoff. What is the maximum

force required by the engine of the plane for the off? (4.43 x 104N)

MCQ (LAWS OF MOTION)

48

CHAPTER - 6

WORK, ENERGY

AND POWER

SYNOPSIS

49

1. Work done (W) by a constant force F in producing a displacement 𝑠 in a body is W = 𝐹 . 𝑠 = Fs

cosθ, where θ is smaller angle between𝐹 𝑎𝑛𝑑 𝑠 .

2. If the force is not constant, W =∫𝐹. 𝑑𝑠 = area under the force displacement graph.

3. Work done is a scalar quantity.

4. Work done = Positive when θ lies between 0 and 𝜋/2. Work done = negative when θ lies

between 𝜋

2 and 𝜋. Work done = zero, when𝜃 =

𝜋

2.

5. SI unit of work is joule and the cgs unit of work is erg.

6. Energy of a body is defined as the capacity of the body to do the work. Energy is a scalar

quantity. Energy has the same units and dimensions as those of work.

7. “Principle of conservation of energy”, according to which sum of total energy in this universe

remains constant. The amount of energy disappearing in one form is exactly equal to the

amount of energy appearing in any other form.

OR

The principle of conservation of mechanical energy states that the total mechanical energy of

a body remains constant if and only if the forces that act on the body are conservative.

8. Kinetic Energy (K) of a body is the energy possessed by the body due to its motion. K.E. of

translation = ½ mv2

9. Potential energy (U) - Energy stored in a body or system due to its configuration or shape.

10. Gravitational Potential energy = mgh. U may be positive or negative; when forces involved

are repulsive, U is positive. When forces involved are attractive, P.E. is negative.

11. Mechanical Energy - Mechanical energy of a particle or system is defined as the sum of K and

U of the system. K is always positive, but the mechanical energy may be zero, positive or

negative. Negative mechanical energy represents a bound state.

12. The Work-Energy theorem states that the change in kinetic energy of a body is the work done

by the net force on the body. K f - Ki =Wnet

13. Conservative and Non-conservative Forces

S.No. Conservative Forces Non Conservative Forces

50

1. Work done by such forces in

displacing a particle does not a

particle depends upon the path

along which particle is displaced.

Work done by such forces in

displacing depend upon the path

along which particle is displaced.

2. Work done by such forces in

displacing a particle around a

closed path is zero.

Work done by such forces in

displacing a particle around a

closed path is NOT zero.

3. K.E. of particle remains constant. K.E. of particle changes.

14. For a conservative force in one dimension, we may define potential energy Function U(x) such

that F(x) = -dU(x)/dx

15. In a conservative force, work done is independent of the path followed by the body.

16. The elastic potential energy of a spring of force constant k and extension x is PE= ½ Kx2.

17. Collisions when a body strikes against body or one body influences the other from a distance,

collision is said to be occur. Collisions are of two types : (a) Perfectly elastic collision - in which

there is no change in kinetic energy of the system, i.e., total K.E. before collision = total K.E.

after collision.(b) Perfectly inelastic collision - in which K.E. it NOT conserved. The bodies stick

together after impact.

18. If the initial and final velocities of colliding bodies lie along the same line then it is known as

head on collision.

51

a) Law of conservation of linear momentum. m1u1 + m2u2 = m1v1 + m2 v2

b) principle of conservation of K.E. 1

2𝑚1𝑢1

2 + 1

2𝑚2𝑢2

2 = 1

2𝑚1𝑣1

2 +1

2𝑚2𝑣2

2

𝑣1 = (𝑚1 − 𝑚2) + 2𝑚2𝑢2

𝑚1 + 𝑚2

𝑣2 = (𝑚2 − 𝑚1) + 2𝑚1𝑢1

𝑚1 + 𝑚2

1. Coefficient of restitution or resilience of the two bodies. e = -𝑣1 − 𝑣2

𝑢1− 𝑢2 For a perfectly elastic

collision, e = 1 and for a perfectly inelastic collision,e = 0. (0 < e <1)

S.No. Perfectly Elastic Collision Perfectly Inelastic Collision

1. Particles do not stick together after

collision.

Particles stick together after collision.

2. Rel. vel. of separation after collision

= rel. vel. of approach

before collision.

Rel. vel. of separation after collision is

zero.

3. Coeff. of restitution, e = 1. Coeff. of restitution, e = 0

4. Linear momentum is conserved. Linear momentum is conserved.

5. K.E. is conserved. K.E. is not conserved.

2. Power of a body is defined as the time rate of doing work by the body.

3. 𝑃 = 𝑑𝑊

𝑑𝑡= 𝐹 . 𝑣 . Here, θ is angle between 𝐹 and 𝑣 of the body.

4. The practical unit of power is horse power (h . p), where 1 h.p = 746 W

MEMORY MAP

52



1. Define the conservative and non-conservative forces? Give example of each?

2. A light body and a heavy body have same linear momentum. Which one has greater K.E?

3. If the momentum of the body is doubled by what percentage does its K.E changes?

4. A truck and a car are moving with the same K.E on a straight road. Their engines are

simultaneously switched off which one will stop at a lesser distance?

5. What happens to the P.E of a bubble when it rises upon water?

6. Define spring constant of a spring?

7. What happens when a sphere collides head on elastically with a sphere of same mass initially at

rest?

8. Derive an expression for K.E of a body of mass m moving with a velocity v by calculus method.

9. After bullet is fired, gun recoils. Compare the K.E. of bullet and the gun.

10. In which type of collision there is maximum loss of energy

SHORT TYPE ANSWER QUESTION (2 & 3 MARKS)

1. A bob is pulled sideway so that string becomes parallel to horizontal and released. Length of the

pendulum is 2 m. If due to air resistance loss of energy is 10% what is the speed with which the

bob arrives the lowest point?

53

2. Find the work done if a particle moves from position r1 = (4i + 3j + 6k) m to a position r = (14i =

13j = 16k) under the effect of force, F = (4i + 4j - 4k)N?

3. 20 J work is required to stretch a spring through 0.1 m. Find the force constant of the spring. If

the spring is stretched further through 0.1m calculate work done?

4. A pump on the ground floor of a building can pump up water to fill a tank of volume 30m3 in 15

min. If the tank is 40 m above the ground, how much electric power is consumed by the pump?

The efficiency of the pump is 30%.

5. Spring of a weighing machine is compressed by 1cm when a sand bag of mass 0.1 kg is dropped

on it from a height 0.25m. From what height should the sand bag be dropped to cause a

compression of 4cm?

6. Show that in an elastic one dimensional collision the velocity of approach before collision is equal

to velocity of separation after collision?

7. A spring is stretched by distance x by applying a force F. What will be the new force required to

stretch the spring by 3x? Calculate the work done in increasing the extension?

8. Write the characteristics of the force during the elongation of a spring. Derive the relation for the

P.E. stored when it is elongated by length. Draw the graphs to show the variation of potential

energy and force with elongation?

9. How does a perfectly inelastic collision differ from perfectly elastic collision? Two particles of

mass m1 and m2 having velocities u1 and u2 respectively make a head on collision. Derive the

relation for their final velocities?

2. In lifting a 10 kg weight to a height of 2m, 250 Joule of energy is spent. Calculate the acceleration

with which it was raised?(g=10m/s2)

LONG ANSWER TYPE QUESTION (5 MARKS)

1. Show that at any instant of time during the motion total mechanical energy of a freely falling

body remains constant. Show graphically the variation of K.E. and P.E. during the motion.

2. Define spring constant. Write the characteristics of the force during the elongation of a spring.

Derive the relation for the PE stored when it is elongated by X. Draw the graphs to show the

variation of P.E. and force with elongation.

3. How does a perfectly inelastic collision differ from perfectly elastic collision? Two particles of

mass m1 and m2 having velocities U1 and U2 respectively make a head on collision. Derive the

relation for their final velocities. Discuss the following special cases.

a) m1 = m2

b) m1 >> m2 and U2 = 0

c) m1 << m2 and U1 = 0

NUMERICALS

54

1. Calculate the power of a crane in watts, which lifts a mass of 100kg to height of 10m in 20s.

(500W)

2. A bob is pulled sideway so that the string becomes parallel to horizontal and released. Length of

the pendulum is 2m. If due to air resistance loss of energy is 10%, find the speed with which the

bob arrived at its lowest point. (6m/s)

3. A 16kg block moving on a frictionless horizontal surface with a velocity of 5m/s compresses an

ideal spring and comes to rest. If the force constant of the spring be 100N/m, then how much is

the spring compressed? (2m)

4. A particle of mass 1g moving with a velocity v1= (3i-2j) collides elastically with another particle of

mass 2g with velocity v2= (4i-6k). Find the velocity of the particles after collision. (4.6 m/s)

5. A force F = 2j N acts in a region, where a particle moves clock wise along the sides of a square of

length 2m. Find the total amount of work done. (8J)

6. If the linear momentum increases by 20%, what will be the percentage increase in the kinetic

energy of the body? (44%)

7. A railway carriage of mass 9000 kg moving with a speed of 36 km/h collides with a stationary

carriage of the same mass. After the collision, the carriages get coupled and move together. What

is their common speed after collision? What type of collision is this? (5m/s, Inelastic)

8. What percentage of kinetic energy of a moving particle is transferred to a stationary particle,

when a stationary particle of mass i) 9 times in mass ii) equal in mass and iii) 1/19th of its mass?

(36%, 100%, 19%)

9. A ball bounces to 80% of its original height. What fraction of its mechanical energy is lost in each

bounce? Where does this energy go? (0.20)

10. A raindrop of mass 1g falling from a cliff of height 1km hits the ground with a speed of 50m/s.

What is the work done by the unknown resistive force? (-0.875J)

ENRICHMENT EXERCISE

55

1. A mass less pan is placed on an elastic spring. Spring is compressed by 0.01 m when a sand bag

of mass 0.1 kg is dropped on it from a height 0.24 m. From what height should the sand bag be

dropped to cause a compression of 0.04 m? (3.96m)

2. A pendulum bob of mass 10-2 kg is raised to a height of 0.05 m and then released. At the bottom

of its swing, it picks up a mass of 10-3 kg. To What height will the combined mass rise? Take g =

10 m/s2. (0.0414m)

3. A ball whose kinetic energy is E, is projected at an angle of 45o to the horizontal. What will be

kinetic energy of the ball at the highest point of its flight? (1/2 E)

HOTS

1. A body of mass m is placed on a rough horizontal surface having coefficient of static friction μ

with the body. Find the minimum force that must be applied on the body so that it may start

moving? Find the work done by this force in the horizontal displacement s of the body?

2. Two blocks of same mass m are placed on a smooth horizontal surface with a spring of constant

k attached between them. If one of the block is imparted a horizontal velocity v by an impulsive

force, find the maximum compression of the spring?

2. A block of mass M is supported against a vertical wall by a spring of constant k. A bullet of mass

m moving with horizontal velocity v0 gets embedded in the block and pushes it against the wall.

Find the maximum compression of the spring?

3. Prove that in case of oblique elastic collision of a moving body with a similar body at rest, the two

bodies move off perpendicularly after collision?

4. A chain of length L and mass M rests over a sphere of radius R (L < R) with its one end fixed at the

top of the sphere. Find the gravitational potential energy of the chain considering the center of

the sphere as the zero level of the gravitational potential energy

58

CHAPTER-7

SYSTEM OF PARTICLES AND ROTATIONAL MOTION

SYNOPSIS

1. Particle – It is an object whose mass is finite but whose size and internal structure can be

neglected.

2. System – It is a collection of a very large number of particles which mutually interact with one

another.

3. A rigid body is one for which the distance between different particles of the body do not change,

even though there is a force acting on it.

4. In pure translation every particle of the body moves with the same velocity at any instant of time.

5. In rotation about a fixed axis, every particle of the rigid body moves in a circle which lies in a plane

perpendicular to the axis and has its centre on the axis. Every point in the rotating rigid body has

the same angular velocity at any instant of time.

6. A rigid body fixed at one point or along a line can have only rotational motion. A rigid body not

fixed in some way can have either pure rotation or a combination of translation and rotation.

7. Centre of mass of a body is a point where the entire mass of the body can be supposed to be

concentrated. Moves as if the external force acts on the entire mass concentrated at this point.

In the absence of any external force, the centre of mass moves with a constant velocity.

8. For a system of n-particles, the centre of mass is given by 𝑟 = 𝑚1𝑟1 +𝑚2𝑟2 +⋯+𝑚𝑛𝑟𝑛

𝑚1+𝑚2+⋯+𝑚𝑛

9. Torque (𝜏) - The turning effect of a force with respect to some axis, is called moment of force or

torque due to the force. Torque is measured as the product of the magnitude of the force and

the perpendicular distance of the line of action of the force from the axis of rotation. 𝜏 = 𝑟 𝑋 𝐹

; SI unit of torque is Nm.

momentum (�� )- 10. Angular

It is the rotational analogue of linear

momentum and is measured as the product of the linear momentum and the perpendicular

distance of its line action from the axis of rotation. If 𝑝 is linear momentum of the particle and 𝑟

its position vector, then angular momentum of the particle, �� = 𝑟 𝑋𝑝 . SI unit of angular

momentum is kg m2 s–1.

59

11. Relation between torque and angular momentum : 𝜏 =𝑑��

𝑑𝑡

12. Principle of moments of rotational equilibrium – Clockwise moment = Anticlockwise moment

13. Couple – A pair of equal and opposite forces acting on a body along two different lines of action

constitute a couple. Couple has a turning effect but cannot produce translational motion.

Moment of couple = τ = Force x Arm of the couple

14. Geometrical meaning of angular momentum – The angular momentum of a particle is equal to

twice the product of its mass and areal velocity. L = 2m ΔA

Δt

15. For translational equilibrium of a rigid body, 𝐹 = ∑ 𝐹 𝑖𝑖 = 0

16. For rotational equilibrium of a rigid body, 𝜏 = ∑ 𝜏 𝑖𝑖 = 0

17. Equations of translatory motion Equations of rotational motion

1. v = u +at 1. ω2 = ω1 + αt

2. s = ut + ½ at2 2. θ = ω1t + ½ αt2

3. v2 – u2 = 2as 3. ω22- ω1

2 =2αθ

18. Moment of inertia (I). The moment of inertia of a rigid body about a given axis is the sum of the

products of masses of the various particles with squares of their respective perpendicular

distances from the axis of rotation.I = m1r12 + m2r2

2 + …+ mnrn2 .SI unit of moment of inertia is kg

m2.

19. Radius of gyration (K). It is defined as the distance of a point from the axis of rotation at which, if

whole mass of the body were concentrated, then K=√𝑟12+𝑟22+ …+𝑟𝑛2

𝑛 𝑎𝑛𝑑 I = MK2. SI unit of radius

of gyration is m.

20. Theorem of perpendicular axes. It states that the moment of inertia of a 2-d object about an axis

perpendicular to its plane is equal to the sum of the moments of inertia of the lamina about any

two mutually perpendicular axes in its plane and intersecting each other at the point, where the

perpendicular axis passes through the plane. Iz = Ix + ly where X and Y-axes lie in the plane of the

object and Z-axis is perpendicular to its plane and passes through the point of intersection of X

and Y axes.

21. Theorem of parallel axes. It states that the moment of inertia of a rigid body about any axis is

equal to moment of inertia of the body about a parallel axis through its centre of mass plus the

product of mass of the body and the square of the perpendicular distance between the axes. I =

Ic + M h2, where Ic is moment of inertia of the body about an axis through its centre of mass and

h is the perpendicular distance between the two axes.

22. Law of conservation of angular momentum. If no external torque acts on a system, the total

angular momentum of the system remains unchanged. Constant vector or, I�� = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 or

I1ω1 = I2ω2 provided no external torque acts on the system.

60

23. Motion of a body rolling without slipping on an inclined plane acceleration 𝑎 = 𝑚𝑔 𝑠𝑖𝑛𝜃

𝑚+𝐼/𝑟2

24. Kinetic energy of a rolling body is EK = K.E of translation (KT) + K.E. of rotation (Ke)

Example = ½ Mv2 + ½ I ω2

64



1. Does centre of mass of a body necessarily lie inside the body? Give an example.

2. Which physical quantity is conserved when a planet revolves around the sun?

3. A particle moves on a circular path with decreasing speed. What happens to its angular

momentum?

4. Two solid spheres of the same mass are made of metals of different densities. Which of them

has a large moment of inertia about the diameter?

5. On what factors does radius of gyration of body depend?

6. Can a body be in equilibrium while in motion? If yes, give an example.

7. There is a stick half of which is wooden and half is of steel. (i) it is pivotedat the wooden end

and a force is applied at the steel end at right angle to its length (ii) it is pivoted at the steel

end and the same force is applied at the wooden end. In which case is the angular acceleration

more and why?

8. If earth contracts to half its radius what would be the length of the day at equator?

9. An internal force cannot change the state of motion of centre of mass of a body. How does

the internal force of the brakes bring a vehicle to rest?

10. Why do we prefer wrench of longer arm.

11. If one of the particles is heavier than the other, to which will their centre of mass shift?

12. Can centre of mass of a body coincide with geometrical centre of the body?

13. Which physical quantity is represented by a product of the moment of inertia and the angular

velocity?

14. Which component of linear momentum does not contribute to angular momentum?

15. A disc of metal is melted and recast in the form of solid sphere. What will happen to the

moment of inertia about a vertical axis passing through the centre ?

16. What is rotational analogue of mass of body?

17. Is the angular momentum of a system always conserved? If no, under what condition is it

conserved?

18. Can a body in translatory motion have angular momentum? Explain.

19. A person is sitting in the compartment of a train moving with uniform velocity on a smooth

track. How will the velocity of centre of mass of compartment change if the person begins to

run in the compartment?

20. A particle performs uniform circular motion with an angular momentum L. If the frequency of

particle’s motion is doubled and its K.E is halved, what happens to the angular momentum?

65


1. How will you distinguish between a hard-boiled egg and a raw egg by spinning it on a table

top?

2. When does a rigid body said to be in equilibrium? State the necessary condition for a body to

be in equilibrium.

3. Show that in the absence of any external force, the velocity of the centre of mass remains

constant.

4. Prove that the rate of change of angular momentum of a system of particles about a reference

point is equal to the net torque acting on the system.

5. Derive a relation between angular momentum, moment of inertia and angular velocity of a

rigid body.

6. Disc rotating about its axis with angular speed ω0 is placed on a perfectly frictionless table.

The radius of the disc is R. Find the linear velocities at points A, B & C on the disc.

7. Prove that the acceleration of a solid cylinder rolling without slipping down an inclined plane

is 2𝑔 𝑠𝑖𝑛𝜃

3

8. Show that moment of a couple does not depend on the point about which moment is

calculated.


1. Define the term angular momentum. Give its unit and dimensions

i. Prove that the angular momentum of a particle is twice the product of its mass and areal

velocity. How does it lead to the Kepler’s second law of planetary motion?

ii. State law of conservation of momentum. Derive a relation between angular momentum,

moment of inertia and angular velocity of rigid body.

2. A body whose moment of inertia is 3 kgm2, is at rest. It is rotated for 20s with a moment of

force 6 Nm. Find the angular displacement of the body. Also calculate the work done.

3. Define rotational motion of a body. Derive the three equation of rotational motion

a) ω2 = ω1 + αt

b) θ = ω1t + ½ αt2

c) ω22- ω1

2 =2αθ

66

NUMERICALS

1. Three masses 3kg, 4kg & 5kg are located at the corners of an equilateral triangle of side 1m.

Locate the centre of mass of the system. (0.54 m, 0.36 m)

2. Two particles of mass 100g and 300g at a given time have velocities (10i-7j-3k) & (7i-9j+6k)

respectively. Determine the velocity of COM. ((31i - 34 j+ 15k)/4 m/s)

3. The angular speed of a motor wheel is increased from 1200rpm to 3120 rpm in 16 seconds.

(i) What is its angular acceleration, assuming the acceleration to be uniform? (ii)How many

revolutions does the wheel make during this time?( 4π rad s–1 , 576)

4. A 3m long ladder weighing 20kg leans on a frictionless wall. Its feet rest on the floor 1m from

the wall as shown. Find the reaction forces F1 & F2 of the wall and the floor. (199 N, 800)

5. Calculate the ratio of radii of gyration of a circular ring and a disc of the same radius with

respect to the axis passing through their centres and perpendicular to their planes. (√2:1)

6. An automobile moves on a road with a speed of 54km/h. The radius of its wheels is 0.35m.

What is the average torque transmitted by its brakes to a wheel if the vehicle is brought to

rest in 15s? The moment of inertia of the wheel about the axis of rotation is 3kg/m2. (-8.57

kgm2s-2)

7. A solid cylinder rolls down an inclined plane. Its mass is 2kg and radius 0.1m. If the height of

the inclined plane is 4m, what is the rotational K.E. when it reaches the foot of the plane?

(26.13J)

8. Four particle masses 4kg, 2kg, 3kg and 5kg are respectively located at the four corners A, B, C

and D of a square of side 1 m. Calculate the moment of inertia of the system about

a) An axis passing through the point of intersection of the diagonals and perpendicular to the

plane of the square.

b) The side AB

c) The diagonal BD. (7 kgm2, 8 kgm2, 3.5 kgm2)

9. A flywheel of mass 25kg has a radius of 0.2m. It is making 240 r.p.m. What is the torque

necessary to bring it to rest in 20s? If the torque is due to a force applied tangentially on the

rim of the flywheel, what is the magnitude of the force? ( -π/5 Nm, π N)

10. Calculate the kinetic energy of rotation of a circular disc of mass 1kg and radius 0.2m rotating

about an axis passing through its centre and perpendicular to its plane. The disc is making

30/π rotations per minute .(0.01J)

67

ENRICHMENT EXERCISE

1. Determine the force exerted by the liquid at the other end. A light string is wound round a cylinder

and carries a mass tied to it at the free end. When the mass is released, calculate.

a) The linear acceleration of the descending mass

b) Angular acceleration of the cylinder

c) Tension in the string.

2. A rod of length L and mass M is hinged at point O. A small bullet of mass m hits the rod, as shown

in figure. The bullet get embedded in the rod. Find the angular velocity of the system just after

the impact

3. A uniform disc of radius R is put over another uniform

disc of radius 2R of the same thickness and density. The peripheries of the two discs touch each

other. Locate the centre of mass of the system.

4. Two blocks of masses 10 kg and 20 kg are placed on the x-axis. The first mass is moved on the axis

by a distance of 2 cm. By what distance should the second mass be moved to keep the position

of centre of mass unchanged?

5. A square plate of mass 120 g and edge 5.0 cm rotates about one of edges. If it has a uniform

angular acceleration of 0.2 rad/s2, what torque acts on the plate?

MCQ CENTRE OF MASS

70

Answer

(MCQ) Key :

58. (d) 59. (d) 60. (c) 61. (a) 62. (b) 63. (d) 64. (a) 65. (a) 66. 67. 68. (b) 69. (a)

70. (a) 71. (d) 72. (b) 73. (a) 74. (a) 75. (d) 76. (a)

72

CHAPTER-8

GRAVITATION

SYNOPSIS

1. Newton’s law of gravitation. It states that the gravitational force of attraction acting between two

bodies of the universe is directly proportional to the product of their masses and is inversely

proportional to the square of the distance between them, i.e.,

F = 𝐺𝑚1𝑚2

𝑟2 ; where G is the universal gravitational constant.

2. The value of G = 6.67 × 10–11 Nm2 kg–2

3. Garvitational Forces – Are independent of intervening medium, Obey’s Newton III Law of

motion,has spherical symmetry, obey principle of superposition.

4. Gravity - It is the force of attraction exerted by earth towards its centre on a body lying on or near

the surface of earth.

5. Acceleration due to gravity (g). It is defined as the acceleration set up in a body while falling freely

under the effect of gravity alone. It is a vector quantity.𝑔 = 𝐺𝑀

𝑟2

where M and R are the mass and radius of the earth.

6. Principle of superposition – The resultant force on a particle due to number of masses is the

vector sum of the gravitational forces exerted by the individual masses on the given particle.

7. Weight – Gravitational force with which a body is attracted towards the centre of earth.

W = mg. Unit – Newton

8. Variation of acceleration due to gravity.

a) (i) Effect of altitude, 𝑔ℎ = 𝑔𝑅2

(𝑅+ℎ)2 and gh = g[1 −

2ℎ

𝑅] The first relation is valid when h is

comparable with R and the second relation is valid when h < < R. The value of g decreases with

increase in h.

b) (ii) Effect of depth, gd = g[1 −𝑑

𝑅] The acceleration due to gravity decreases with increase in depth

d and becomes zero at the centre of earth.

c) (iii) Effect of non-sphericity of the earth – As Re>Rp , so the value of g is minimum at the equator

and minimum at the poles.

9. Gravitational field- It is the space around a material body in which its gravitational pull can be

experienced by other bodies. The intensity of gravitational field at a point at a distance r from the

centre of the body of mass M is given by I = GM/r2 = g (acceleration due to gravity).

73

10. Gravitational potential - The gravitational potential at a point in a gravitational field is defined as

the amount of work done in bringing a body of unit mass from infinity to that point without

acceleration. Gravitational potential at a point, V = 𝑊

𝑚0=

𝐺𝑀

𝑟.

11. Gravitational potential energy U = gravitational potential × mass of body = [−𝐺𝑀𝑚

𝑟]

12. Gravitational intensity (I) is related to gravitational potential (V) at a point by the relation,

I = 𝑑𝑣

𝑑𝑟.

13. Satellite - A satellite is a body which is revolving continuously in an orbit around a comparatively

much larger body.

a. Orbital speed of a satellite when it is revolving around earth at height h is given by

b) 𝑣𝑜 = 𝑅√𝑔

𝑅+ℎ

c) When the satellite is orbiting close to the surface of earth, i.e., h < < R, then v0=√𝑔𝑅

a. Time period of satellite (T). It is the time taken by the satellite to complete one revolution around

the earth. 𝑇 = 2𝜋

𝑅√

(𝑅+ℎ)3

𝑔

b. Height of satellite above the earth’s surface : h = [𝑇2𝑅2𝑔

4 𝜋2 ] – R

c. Total energy of satellite, E = P.E. + K.E. = - 𝐺𝑀𝑚

2(𝑅+ℎ) . If the satellite is orbiting close to earth, then r

= R. Now total energy of satellite. E = [−𝐺𝑀𝑚

2𝑅]

d. Binding energy of satellite = -E = 𝐺𝑀𝑚

2𝑅

e. Angular Momentum of a satellite – L = m

14. Escape speed. The escape speed on earth is defined as the minimum speed with which a body

has to be projected vertically upwards from the surface of earth so that it just crosses the

gravitational field of earth. Escape velocity ve is given by, ve = √2𝐺𝑀

𝑅= √2𝑔𝑅 . For earth, the value

of escape speed is 11.2 kms–1.

15. Geostationary Satellite – It revolves around the earth with the same angular speed and in the

same direction as the earth rotates about its own axis. Height of geostationary satellite above the

earth’s surface is 35800km.

16. Polar satellite: It is a satellite which revolves in a polar orbit, passes over North & south poles of

the earth. Its height above the earth is around 500-800km and time period is around 100minutes.

17. Kepler’s laws of Planetary motion

a. All planets move in elliptical orbits with the sun at one of its focal points.

b. The radius vector drawn from the sun to a planet sweeps out equal areas in equal time intervals.

This follows from the fact that the force of gravitation on the planet is central and hence angular

momentum is conserved.

c. The square of the elliptical period of a planet is proportional to the cube of the semi-major axis

of the elliptical orbit of the planet.

74


VERY SHORT TYPE ANSWER QUESTIONS (1 MARK)

1. How is the gravitational force between two masses affected when they are dipped in water

keeping the separation between them the same? ( Remains same)

2. What is the angle between the equatorial plane and the orbital plane of

a) Polar satellite and

b) Geostationary satellite? (900)

3. The gravitational force between two bodies is 1 N if the distance between them is doubled, what

will be the force between them? (one – fourth)

4. Identify the position of sun in the following diagram if the linear speed of the planet is greater at

C than at D.

5. The time period of the satellite of the earth is 5 hr. If the separation between earth and satellite

is increased to 4 times the previous value, then what will be the new time period of satellite?

6. The mass of moon is nearly 10% of the mass of the earth. What will be the gravitational force of

the earth on the moon, in comparison to the gravitational force of the moon on the earth?

7. The force of gravity due to earth on a body is proportional to its mass, then why does a heavy

body not fall faster than a lighter body?

8. The force of attraction due to a hollow spherical shell of uniform density on a point mass situated

inside is zero, so can a body be shielded from gravitational influence?

9. The gravitational force between two bodies in 1 N if the distance between them is doubled, what

will be the force between them?

10. A body of mass 5 kg is taken to the centre of the earth. What will be its

a) mass

b) (ii) weight there?

11. A satellite does not require any fuel to orbit the earth. Why?

12. A satellite of small mass burns during its descent and not during ascent. Why?

13. Is it possible to place an artificial satellite in an orbit so that it is always visible over New Delhi?

14. The time period of the satellite of the earth is 5 hr. If the separation between earth and satellite

is increased to 4 times the previous value, then what will be the new time period of satellite.

15. If the density of a planet is doubled without any change in its radius, how does ‘g’ change on the

planet?

75

16. Mark the direction of gravitational intensity at (i) centre of a hemispherical shell of uniform mass

density (ii) any arbitrary point on the upper surface of hemisphere.

17. Why an astronaut in an orbiting space craft is not in zero gravity although weight less?

18. Write one important use of (i) geostationary satellite (ii) polar satellite.

19. The distance of Pluto from the sun is 40 times the distance of earth if the masses of earth and

Pluto he equal, what will be ratio of gravitational forces of sun on these planets.

20. If suddenly the gravitational force of attraction between earth and satellite become zero, what

would happen to the satellite?


The figure shows elliptical orbit of a planet m about the sun S. The shaded area of SCD is twice

the shaded area SAB. If t1 is the time for the planet to move from D to C and t2 is time to move

from A to B, what is the relation between t1 and t2? ( t1 = 2t2 )

1.

2. Derive expression for the orbital velocity of a satellite.

3. Derive an expression for the gravitational potential energy at a point in the gravitational field of

earth.

4. Why does the earth impart the same acceleration to all bodies?

5. State and explain Kepler’s laws of planetary motion. Name the physical quantities which remain

constant during the planetary motion.

6. Derive expression for the variation of ‘g’ with height from the surface of the earth.


1. Obtain an expression for the acceleration due to gravity on the surface of the earth in terms of

mass of the earth and its radius. Discuss the variation of acceleration due to gravity with altitude,

depth and rotation of the earth.

2. Define orbital velocity of a satellite. Derive expressions for orbital velocity, time period, height,

and angular momentum of a satellite.

3. What is meant by gravitational potential energy of a body? Derive an expression for the

gravitational potential energy of a body of mass m located at distance r from the centre of earth.

4. Three mass points each of mass m are placed at the vertices of an equilateral triangle of

5. Side L. What is the gravitational field and potential due to three masses at the centroid of the

triangle? (Ans E=0 ;V= -331/2Gm/L )

NUMERICALS

76

1. At what height from the surface of the earth will the value of ‘g’ be reduced by 36% of its value

at the surface of earth? (1600 km)

2. At what depth is the value of ‘g’ same as at a height of 40km from the surface of earth? (80km)

3. The gravitational field intensity at a point 10,000km form the earth is 4.8N/kg. Calculate the

gravitational potential at that point. (-4.8X107J/kg)

4. Calculate the energy required to move an earth satellite of mass 103kg from a circular orbit of

radius 2R to that of radius 3R. Given mass of the earth = 6 X 1024kg, R=6400km.

(approx. 5X10-9J)

5. An artificial satellite of mass 100kg is in a circular orbit of 500km above the earth’s surface. Radius

of earth = 6400km. (i) Find the acceleration due to gravity at any point along the satellite path,(ii)

what is the centripetal acceleration of the satellite? (8.45m/s2, 8.45m/s2)

6. A remote sensing satellite of the earth revolves in a circular orbit at a height of 250km above the

earth’s surface. What is the (i) orbital speed and (ii) time period of revolution of the satellite?

Radius of the earth=3600km. (7.76 km/s, 5370 s)

7. Two satellites are at different heights from the surface of earth which would have greater

velocity. Compare the speeds of two satellites of masses m and 4m and radii 2R and R

respectively. (1:√2)

8. Find the potential energy of a system of four particles, each of mass m, placed at the vertices of

a square of side. Also obtain the potential at the centre of the square.

(-5.41Gm2/l, -4√2Gm/l)

9. Three equal masses of m kg each are fixed at the vertices of an equilateral triangle ABC.

(a) What is the force acting on a mass 2m placed at the centroid G of the triangle?

(b) What is the force if the mass at vertex A is doubled? ( 0, 2Gm2 j)

10. If the radius of the earth were to decrease by 1%, keeping its mass same, how will the acceleration

due to gravity change?

11. Draw graphs showing the variation of acceleration due to gravity with

a) Height above the earth’s surface

b) Depth below the earth’s surface.

12. The gravitational field intensity at a point 10,000 km from the centre of the earth is 4.8 N kg–1.

Calculate gravitational potential at that point.

77

ENRICHMENT EXERCISE

1. A mass ‘M’ is broken into two parts of masses m1 and m2. How are m1 and m2 related so

that force of gravitational attraction between the two parts is maximum?

2. If the radius of earth shrinks by 2%, mass remaining constant. How would the value of

acceleration due to gravity change?

3. A body hanging from a spring stretches it by 1 cm at the earth’s surface. How much will the

same body stretch at a place 1600 k/m above the earth’s surface? Radius of earth 6400 km.

4. Imagine a tunnel dug along a diameter of the earth. Show that a particle dropped from one

end of the tunnel executes simple harmonic motion. What is the time period of this motion?

5. A black hole is a body from whose surface nothing can escape. What is the condition for a

uniform spherical body of mass M to be a black hole? What should be the radius of such a

black hole if its mass is nine times the mass of earth?

6. A satellite is moving round the earth with velocity V0 what should be the minimum percentage

increase in its velocity so that the satellite escapes

MCQ ON GRAVITATION

79

77

NUMBER

MISSING

81

CHAPTER-9

82

MECHANICALPROPERTIES OF SOLIDS

SYNOPSIS

1. Elasticity: It is the property of the body by virtue of which the body regains its original

configuration (length, volume or shape) when the deforming forces are removed.

2. Plasticity: The property by virtue of which a body does not regain its original size and shape even

after the removal of the deforming forces.

3. Stress: The internal restoring force acting per unit area of a deformed body is called stress,i.e.,

Stress = restoring force/area.

4. Strain: It is defined as the ratio of change in configuration to the original configuration of the

body. Strain can be of three types: (i) Longitudinal strain (ii) Volumetric strain (iii) Shearing strain.

Strain = change in configuration

original configuration

5. Hooke’s law: It states that the stress is directly proportional to strain within the elastic limit.

6. Modulus of Elasticity or Coefficient of elasticity of a body is defined as the ratio of the stress to

the corresponding strain produced, within the elastic limit.

a) Modulus of elasticity is of three types:

7. Young’s Modulus of elasticity (Y): It is defined as the ratio of normal stress to the longitudinal

strain within the elastic limit, i.e., Y =normal stress

longitudnal strain=

F

a X

l

∆l

8. Bulk Modulus of elasticity : It is defined as the ratio of normal stress to the volumetric strain

within the elastic limit,

B = normal stress

compressional strain= −p

V

∆V

compressibility = 1

Bulk modulus

83

9. Rigid modulus of elasticity(η): It is defined as the ratio of tangential stress to the shearing strain,

within the elastic limit. η = tangential stress

Shear strain=

F

A θ =

F

AX

∆x

h

10. Stress-strain curve for a metallic wire:

11. Elastic potential energy stored per unit

volume of a strained body U = 𝑌 𝑥 𝑠𝑡𝑟𝑎𝑖𝑛2

2 where Y is the Young’s modulus of elasticity of a solid

body.

12. Work done in a stretched wire, W = ½ Load x Extension

13. cations of elasticity

a) Metallic part of machinery is never subjected to a stress beyond the elastic limit of material.

b) Metallic rope used in cranes to lift heavy weight are decided on the elastic limit of material

c) In designing beam to support load (in construction of roofs and bridges)

d) Preference of hollow shaft than solid shaft

e) Calculating the maximum height of a mountain

84

CHAPTER- 10

MECHANICAL PROPERTIES OF FLUIDS

SYNOPSIS

1. Total pressure at a point inside the liquid of density ρ at depth h is P = h ρ g + P0 where P0 is the

atmospheric pressure.

2. Pascal’s law: It states that the increase in pressure at one point of the enclosed liquid in

equilibrium at rest is transmitted equally to all other points of the liquid and also to the walls of

the container, provided the effect of gravity is neglected.

3. Viscosity: Viscosity is the property of a fluid by virtue of which an internal frictional force comes

into play when the fluid is in motion and opposes the relative motion of its different layers.

Viscous drag F acting between two layers of liquid each of area A, moving with velocity gradient

dv/dx is given by F = ŋ A dv/dx, SI unit of ŋ is poiseuille of Nsm–2 or Pascal-second

4. Stoke’s law : It states that the backward dragging

force F acting on a small spherical body of radius r, moving through a medium of viscosity ŋ, with

velocity v is given by F = 6 πŋrv.

5. Terminal velocity: It is the maximum constant velocity acquired by the body while falling freely in

a viscous medium. Terminal velocity v of a spherical body of radius r, density ρ while falling freely

in a viscous medium of viscosity ŋ, density ς is given by v = 2r2(ρ−σ)g

9η

85

6. Stream line flow of a liquid is that flow in which every particle of the liquid follows exactly the

path of its preceding particle and has the same velocity in magnitude and direction as that of its

preceding particle while crossing through that point.

7. Turbulent flow: It is that flow of liquid in which the motion of the particles of liquid becomes

disordered or irregular.

8. Critical velocity: It is that velocity of liquid

flow, up to which the flow of liquid is a streamlined and above which its flow becomes turbulent.

Critical velocity of a liquid (vc) flowing through a tube is given by vc = Nη

ρD

9. Equation of continuity, a v = constant where a = area of cross section v = velocity of flow of liquid.

10. Bernoulli’s Theorem : It states that for the stream line flow of an ideal liquid, the total energy

(the sum of pressure energy, the potential energy and kinetic energy) per unit volume remains

constant at every cross-section throughout the tube, i.e., P + ρgh + ½ ρv2 = constant

86

11. Torricelli’s Theorem: It states that the velocity of efflux, i.e., the velocity with which the liquid

flows out of an orifice is equal to that which a freely falling body would acquire in falling through

a vertical distance equal to the depth of orifice below the free surface of liquid. Quantitatively

velocity of efflux, v= 2gh, where h is the depth of orifice below the free surface of liquid.

12. Surface tension: It is the property of the liquid by virtue of which the free surface of liquid at rest

tends to have minimum surface area and as such it behaves as a stretched membrane. Surface

tension, S=F/l, where F is the force acting on the imaginary line of length l, drawn tangentially to

the liquid surface at rest. SI unit of surface tension is Nm–1 .

13. Surface energy: The additional potential energy

per unit area of the surface film as compared to the molecules in the interior is called surface

energy. surface energy = work done in increasing the surface area per unit increase in surface

area.

14. Angle of contact: Angle between tangents drawn at point of contact to liquid surface and to solid

surface drawn into liquid.

15. The liquids for which angle of contact is acute (concave meniscus) show a rise in capillary tube

while those for which angle of contact is obtuse (convex meniscus) show a fall in capillary tube.

16. Capillarity: The phenomenon of rise or fall of liquid in a capillary tube is called capillarity. The

height (h) through which a liquid will rise in a capillary tube of radius r which wets the sides of

the tube will be given by h=2Scosθ/rρg, where S is the surface tension of liquid, θ is the angle of

contact, ρ is the density of liquid and g is the acceleration due to gravity.

87

17. Excess pressure : For a curved surface in equilibrium; the concave side will have more pressure

than convex side (i) Excess of pressure inside a liquid drop, P = 2 S/R (ii) Excess of pressure inside

a soap bubble, P = 4 S/R where S is the surface tension and R is the radius of the drop or bubble.


VERY SHORT TYPE ANSWER QUESTION (1 MARK))

1. The Young’s modulus for steel is much more than that for rubber. For the same longitudinal

strain, which one will have greater tensile stress? (steel)

2. Identical springs of steel and copper are equally stretched. On which, more work will have to be

done?(one having higher value of Y ,more work)

3. What is (i) Young’s modulus, (ii) Bulk modulus for a perfectly rigid body?

4. What is the value of bulk modulus for an incompressible liquid?

5. Stress and pressure are both forces per unit area. Then in what respect does stress differ from

pressure?

6. Why is it easier to swim in sea water than in the river water?

7. Railway tracks are laid on large sized wooden sleepers. Why?

8. The dams of water reservoir are made thick near the bottom. Why?

9. Why is it difficult to stop bleeding from a cut in human body at high altitudes?

10. The blood pressure in human is greater at the feet than at the brain. Why?

11. Define coefficient of viscosity and write its SI unit.

12. Why machine parts get jammed in winter?

13. Why are rain drops spherical?

14. Why do paints and lubricants have low surface tension?

15. What will be the effect of increasing temperature on

a) Angle of contact

b) Surface tension.

16. For solids with elastic modulus of rigidity, the shearing force is proportional to shear strain. On

what factor does it depend in case of fluids?

17. How does rise in temperature effect

a) Viscosity of gases

b) Viscosity of liquids.

18. Hotter liquids move faster than the colder ones. Why?

19. Why two boats moving in parallel directions close to each other get attracted?

20. Mercury does not cling to glass. Why?

21. Oil spreads over the surface of water whereas water does not spread over the surface of oil. Why?

22. Water rises in a capillary tube whereas mercury falls in the same tube?

23. Why birds are often seen to swell their feather in winter?

24. A brass disc fits snugly in a hole in a steel plate. Should we heat or cool the system to loosen the

disc from the hole.

88

25. A wire is stretched by a force such that its length becomes double. How will the Young’s modulus

of the wire be affected?

26. How does the Young’s modulus change with rise in temperature?

27. Which of the three modulus of elasticity –Y, K and η is possible in all the three states of matter

28. The Young’s modulus of steel is much more than that for rubber. For the same longitudinal strain,

which one will have greater stress?

29. Which of the two forces –deforming or restoring is responsible for elastic behavior of substance?

30. Which mode of transfer of heat is the quickest?

31. A boat carrying a number of large stones is floating in a water tank. What will happen to the level

of water if the stones are unloaded into the water?

32. A rain drop of radius r falls in air with a terminal velocity v. What is the terminal velocity of a rain

drop of radius 3r?

33. When air is blown in between two balls suspended close to each other, they are attracted towards

each other. Why?

34. Why does air bubble in water goes up?

SHORT TYPE ANSWER QUESTION (2 & 3 MARKS)

1. State Hooke’s law. Write expression for young’s modulus of material of a wire of length ‘l’, radius

of cross - section ‘r’ loaded with a body of mass M producing an extension l in it. Mention its

unit and dimension.

2. Prove that the elastic potential energy per unit volume is equal to ½ Stress X Strain.

3. Define the term bulk modulus. Give its SI unit. Give the relation between bulk modulus and

compressibility.

4. Two wires of same length and material but of different radii are suspended from a rigid support.

Both carry the same load. Will the stress, strain and extension be same or different?

5. Stress strain curve for two wires of material A and B are as shown in Fig.

a) Which material in more ductile?

b) Which material has greater value of young modulus?

c) Which of the two is stronger material?

d) Which material is more brittle?

89

6. State Pascal’s law for fluids with the help of a neat labelled diagram explain the principle and

working of hydraulic brakes.

7. Define Capillarity and angle of contact. Derive an expression for the ascent of a liquid in a capillary

tube.

8. Define surface tension. Derive a relation between surface tension and surface energy. What is

the unit of surface tension?

9. What is laminar flow of a liquid? Draw velocity profiles for the laminar flow of viscous and non-

viscous liquids.

10. Explain what happens when the length of a capillary tube is less than the height upto which the

liquid may rise in it?

11. What is Reynold’s number? Write its significance. On what factors does it depend?

12. State stokes’ law. Given the numerical constant in stokes’ law as 6 , derive this law by method

of dimensions

LONG ANSWER QUESTION (5 MARKS)

1. a) Draw and discuss stress versus strain graph, explaining clearly the terms elastic limit,

permanent set, proportionality limit, elastic hysteresis, tensile strength.

b) An aluminum wire 1m in length and radius 1mm is loaded with a mass of 40 kg hanging vertically.

c) Young’s modulus of Al is 7.0 × 10–10 N/m2 Calculate (a) tensile stress (b) change in length (c) tensile

strain and (d) the force constant of such a wire.

2. State and prove Bernoullis theorem. Give its limitation. Name any two applications of the

principle.

3. Define terminal velocity. Obtain an expression for terminal velocity of a sphere falling through a

viscous liquid. Use the formula to explain the observed rise of air bubbles in a liquid.

4. a) State Pascal’s law of fluid pressure. With a suitable diagram, explain how is Pascal’s law applied

in a hydraulic lift?

b) In a hydraulic lift air exerts a force F on a small piston of radius 5cm. The pressure is transmitted

to the second piston of radius 15cm. If a car of mass 1350kg is to be lifted, calculate force F that

is to be applied.

NUMERICALS

1. The average depth of an ocean is 2500m. Calculate the fractional compression (ΔV/V) of water at

the bottom of ocean, given the bulk modulus of water is 2.3X109N/m2. (Ans .1.08%)

2. A force of 5 X 103N is applied tangentially to the upper face of a cubical block of steel of side

30cm. Find the displacement of the upper face relative to the lower one, and the angle of shear.

The shear modulus of steel is 8.3 X 1010Pa. (Ans.2 x 10-7 units, tan-1(0.67 x10-6))

3. Calculate the pressure at a depth of 10m in an ocean. The density of sea water is 1030kg/m3.

The atmospheric pressure is 1.01 X 105Pa. (Ans.2.04 × 105 pa).

90

4. What is the percentage increase in the length of a wire of diameter 2.5mm stretched by a force

of 100 kg wt? Young’s modulus of elasticity of the wire is 12.5 x 1011 dyne/cm2. (0.16%)

5. A structural rod has a radius of 10mm and a length of 1m. A 100kN force F stretches it along its

length. Calculate a) the stress b) elongation and c) strain on the rod. Given that the Young’s

modulus, Y, of the structural steel is 2.0 x 1011N/m2. (3.18 x 108N/m2,1.59mm,0.16%)

6. A metallic cube whose each side is 10cmm is subjected to a shearing force of 100kg f. The top

face is displaced through 0.25cm with respect to bottom. Calculate the shearing stress, strain and

shear modulus. (9.8 x 104N/m2,0.025 rad,3.92 x 106N/m2)

7. A rubber block 1cm x 3cm x10cm is clamped at one end with its 10 cm side vertical. A horizontal

force of 30N is applied to the free surface. What is the horizontal displacement of the top face?

Modulus of rigidity of rubber = 1.4 x 105 N/m2. (7.14 cm)

8. The velocity of water in a river is 180km/h near the surface. If the river is 5m deep, find the

shearing stress between horizontal layers of water. Coefficient of viscosity of water=10-2 poise.

(10-3N/m2)

9. Find the terminal velocity of a steel ball 2mm in diameter falling through glycerine. Relative

density of steel = 8, relative density of glycerine = 1.3 and viscosity of glycerine=8.3 poise.

(1.78cm/s)

10. Water is flowing through two horizontal pipes of different diameters which are connected

together. In the first pipe the speed of water is 4m/s and the pressure is 2X104N/m2. Calculate

the speed and pressure in the second pipe. The diameters of the pipes are 3cm and 6cm

respectively. (1m/s, 2.75 x 104N/m2)

11. At what velocity does water emerge from an orifice in a tank in which gauge pressure is

3X105N/m2 before the flow starts? Density of water=103kg/m3(24.495 m/s)

12. What would be the gauge pressure inside an air bubble of 0.2mm radius situated just below the

surface of water? Surface tension of water is 0.07N/m. (700 Pa)

13. Water rises to a height of 9cm in a certain capillary tube. If in the same tube, level of mercury id

depressed by 3cm, compare the surface tension of water and mercury. Relative density of

mercury is 13.6, angle of contact for water is zero and for mercury is 1350. (0.152)

14. A soap bubble of radius 1 cm expands into a bubble of radius 2cm. Calculate the increase in

surface energy if the surface tension for soap is 25 dyne/cm. (1.02 × 103 erg)

15. A glass plate of 0.20 m2 in area is pulled with a velocity of 0.1 m/s over a larger glass plate that is

at rest. What force is necessary to pull the upper plate if the space between them is 0.003m and

is filled with oil of co-efficient of viscosity = 0.01 Ns/m2. (66.7 x 10-3N)

16. The area of cross-section of a water pipe entering the basement of a house is 4 × 10–4 m2. The

pressure of water at this point is 3 × 105 N/m2, and speed of water is 2 m/s. The pipe tapers to an

area of cross section of 2 × 10–4 m2, when it reaches the second floor 8 m above the basement.

Calculate the speed and pressure of water flow at the second.( 2.16 × 105 N/m2)

17. A liquid rises to a height of 7cm in a capillary tube of radius 0.1 mm. The density of the liquid is

0.8 x 103 kgm-3. If the angle of contact between the liquid and the surface of the tube is zero,

calculate the surface tension of the liquid. Given g=10m/s2.

91

ENRICHMENT EXERCISE

1. A metallic sphere of radius 1 x 10-3m and density 1 x 104kgm-3 enters a tank of water, after a free

fall through a distance of h in the earth’s gravitational field. If its velocity remains unchanged after

entering water, determine the value h. Given coefficient of viscosity of water = 1 x 10-3Nsm-2, g =

10m/s2 and density of water = 1 x 103kg/m3. ( 20m )

2. Water stands at a height H in a tank whose side walls are vertical. A hole is made in one of the

walls at a depth h below the water surface. A) Find at what distance from the foot of the wall

does the emerging stream of water strike the floor? B) For what value of h, this range is

maximum? C) Can a hole be made at another depth so that the second stream has the same

range? (2 , H/2, h or (H-h))

3. Water from a tap emerges vertically downward with an initial speed of1.0m/s. The cross-sectional

area of the tap is 10-4 m2. Assume that the pressure is constant throughout the stream of water,

and that the flow is steady. What is the cross-sectional area of the stream 0.15m below the tap?

(5 X 10-5m2)

MCQ

95

CHAPTER-11

THERMAL PROPERTIES OF MATTER

96

SYNOPSIS

1. The ratio of work done (W) to the amount of heat produced (Q) is always a constant, represented

by J. i.e., where J is called Joule's mechanical equivalent of heat. The value of J = 4.186

joule/calorie.

2. Thermal Expansion: Almost all solids, liquids and gases expand on heating. The increase in the

size of a body when it is heated is called thermal expansion. The coefficient of linear expansion

(α), coefficient of superficial expansion (β) and coefficient of cubical expansion (γ) are related as

α = β/2 = γ/3.

3. Relation Between Different Temperature Scales

4. If temperature of a body of mass m rises by ΔT, then Q = mc Δ T where c is specific heat of the

material of the body. Specific heat of a body is defined as the amount of heat required to raise

the temperature of unit mass of the substance by one degree. Principal specific heat of a gas :

a) Specific heat at constant volume (cv)

b) Specific heat at constant pressure (cp)

Cv -molar heat capacity at constant volume

Cp - molar heat capacity at constant pressure

Cp – CV = R/J

5. The amount of heat required to change the state of unit mass of a substance at a constant

temperature is called its latent heat. It is denoted by L. When the state of body of mass m changes

at its melting point/boiling point, then heat required is Q = m L, where L is latent heat of the body.

6. Principle of calorimetry: When two substances at different temperature are mixed together, they

exchange heat. If we assume that no heat is lost to the surroundings, then according to principle

of calorimetry, Heat lost by one substance = Heat gained by another substance.

7. Modes of heat transfer:

a) Conduction: It is a process in which heat is transmitted from one part of a body to another

through molecular collisions without actual flow of matter.

97

b) Convection: It is process by which heat is transmitted from one point to another due to actual

motion of the heated particles.

c) Radiation: It is the process in which heat is transmitted from one place to another without heating

the intervening medium.

8. Coefficient of thermal conductivity (K) of a solid conductor is calculated from the relation ΔQ

Δt= −KA (

ΔT

ΔX)

where A is area of cross-section, Δx is distance between the hot and cold faces, ΔQ is the small

amount of heat conducted in a small time (Δt), ΔT is difference in temperatures of hot and cold

faces. Here (ΔT/Δx) temperature gradient, i.e., rate of fall of temperature with distance in the

direction of flow of heat.

9. Stefan’s law: When the temperature difference between body and surroundings is large, then

Stefan's law for cooling of body is obeyed. According to it, where E is the amount of thermal

energy emitted per second per unit area of a black body. T is the temperature of black body and

T0 is the temperature of surroundings, is the Stefan's constant. E = σ ( T4 − T04)

10. Wien's Displacement Law : The wavelength λmax at which the maximum amount of energy is

radiated decreases with the increase of temperature such that λmaxT =b, b is a constant & T is the

temperature of black body in Kelvin.

11. Newton’s Law of Cooling : It states that the rate of loss of heat of a liquid is directly proportional

to difference in temperature of the liquid and the surroundings provided the temperature

difference is small (around 40°C).

−dQ

dtα (T − To)



1. Why is water preferred to any other liquid in the hot water bottles?

2. A gas is free to expand what will be its specific heat?

3. At what temperature does a body stop radiating?

4. If Kelvin temperature of an ideal black body is doubled, what will be the effect on energy radiated

by it?

5. In which method of heat transfer gravity does not play any part?


1. Draw a graph to show the anomalous behaviour of water. Explain its importance for sustaining

life under water.

98

2. State & explain the three modes of heat transfer. Also write example for each.

3. Derive the relation between coefficient of linear and cubical expansion.

4. Plot a graph of temperature v/s time showing the change in the state of ice on heating and hence

explain the process with reference to latent heat.

5. How does the density of a solid or liquid change with temperature? Show that the variation is

given by ρ’=ρ (1-ϒΔT), where ϒ is the coefficient of cubical expansion.

6. A sphere, a cube and a disc made of same material and of equal masses heated to same

temperature of 200°C. These bodies are then kept at same lower temperature in the surrounding,

which of these will cool (i) fastest, (ii) slowest, explain.


1. State Newton’s law of cooling. Deduce the relations: log (T – T0) = –kt + c and T – T0 = C e–kt where

the symbols have their usual meanings. Represent Newton’s law of cooling graphically by using

each of the above equation.

2. On what factors does the rate of heat conduction in a metallic rod in the steady state depend?

Write the necessary expression and hence define the coefficient of thermal conductivity. Write

its unit and dimensions.

3. Discuss energy distribution of a black body radiation spectrum and explain Wein’s displacement

law of radiation and Stefan’s law of heat radiation.

NUMERICALS

1. If the volume of a block of metal changes by 0.12%, when it is heated through 200C, what is the

coefficient of linear expansion of metal? 2.0 x 10-5 ⁰C-1)

2. Calculate the force required to prevent a steel wire of 1mm2 cross-section from contracting when

it cools from 600C to 150C, if young’s modulus for steel is 2 X 1011N/m2 and its coefficient of linear

expansion is 0.000011/0C. (99N)

3. An iron sphere has a radius of 10cm at a temperature of 00C. Calculate the change in the volume

of the sphere, if it is heated to 1000C. Coefficient of linear expansion of iron is 11 X 10- 6/0C.

(13.8cm3)

4. Calculate the heat required to convert 3kg of ice at -120C kept in a calorimeter to steam at 1000C

at atmospheric pressure. Given: specific heat capacity of ice= 2100J/kg/K, specific heat capacity

of water=4186J/kg/K, latent heat of fusion of ice=3.35 X 105 J/kg, latent heat of steam=2.256 X

106J/kg. (9.1X106J)

5. When 0.45kg of ice of 00C is mixed with 0.9kg of water at 550C in a container, the resulting

temperature is 100C. Calculate the heat of fusion of ice. Specific heat of water= 4186J/kg/K.

(334400J/kg)

99

6. Steam at 1000C is passed into a copper cylinder 10mm thick and of 200cm2 area. Water at 1000C

collects at the rate of 150g/min. Find the temperature of the outer surface, if the conductivity of

copper is 0.8cal/s/cm/0C and latent heat of steam is 540cal/g. (91.56⁰C)

7. A body cools in 7 minutes from 600C to 400C. What will be its temperature after the next 7

minutes? The temperature of the surroundings is 100C. (28⁰C)

8. One end of a copper rod of uniform cross-section and of length 1.5m is kept in contact with ice

and the other end with water at 100oC. At what point along its length should a temperature of

2000C be maintained so that in steady state, the mass of ice melted be equal to that of the steam

produced in the same interval of time? Assume that the whole system is insulated from the

surroundings. Latent heat of fusion of ice = 80 cal/g. Latent heat of vaporization of water = 540

cal/g. (1.396m)

9. Calculate the temperature (in K) at which a perfect black body radiates energy at the rate of

5.67W/cm2. Given 5.67x10-8Wm-2K-4. (1000K)

10. The surface temperature of a hot body is 12270C. Find the wavelength at which it radiates

maximum energy. Given Wien’s constant = 0.2898 cm K. (19320 Å)

ENRICHMENT EXERCISE

1. A sphere of diameter 7 cm and mass 266.5g floats in a bath of a liquid. As the temperature is

raised, the sphere just begins to sink at a temperature of 35oC. If the density of the liquid at 0oC

is 1.527 g/cm3, find the coefficient of cubical expansion of the liquid. Neglect the expansions of

the sphere. (0.000084/⁰C)

2. Hot oil is circulated through an insulated container with wooden lid at the top whose conductivity

K = 0.149 J/(m0Csec), thickness t = 5mm, emissivity = 0.6. Temperature of the top of the lid is

maintained at Tl =127oC. If the ambient temperature Ta=27oC, calculate

a) Rate of heat loss per unit area due to radiation from the lid.

b) Temperature of the oil. (Given = 17/3 x10-8) (595W/m2,420K)

3. 2 kg of ice at -200C is mixed with 5kg of water at 20oC in an

insulating vessel having a negligible heat capacity. Calculate

the final mass of water remaining in the container. It is given that the specific heats of water and

ice are 1 kcal/kg/0C and 0.5 kcal/kg/0C, while the latent heat of fusion of ice is 80 kcal/kg. ( 6kg)

100

CHAPTER -12

THERMODYNAMICS

SYNOPSIS

101

1. Thermodynamics is the branch of physics that deals with the concepts of heat and temperature

and the inter-conversion of heat and other forms of energy.

a) It is concerned with macroscopic (large-scale) changes and observations.

2. Thermodynamic system- It is an assembly of a very large number of particles confined within

certain boundaries such that it has certain values of pressure, volume and temperature.

3. Surroundings- Everything outside the system which can have a direct effect on the system.

4. Thermodynamic variable – Variables needed to describe the physical state of a thermodynamic

system in equilibrium e.g. – Pressure, Volume, Temperature and Mass.

a) Extensive variables – These variables depend on the size of the system. Eg: volume, energy etc.

b) Intensive variables – These variables are independent of the size of the system. Eg: Temperature,

Pressure etc.

5. Equation of state of an ideal gas – PV = nRT.

6. Thermodynamic Equilibrium – If the variables describing the state of the system do not change

with time system is said to be in thermodynamic equilibrium.

a) Thermal equilibrium: All parts of the system and the surroundings are at the same temperature.

b) Mechanical equilibrium: Net external force and torque on a system are zero.

c) Chemical equilibrium: The chemical composition of a system does not change with time, i.e., no

chemical reactions occur.

7. Zeroth law of Thermodynamics – Two systems A and C which are separately in thermal

equilibrium with a third system B are also in thermal equilibrium with each other.

8. Temperature of a system is a physical quantity, equality of which is the only condition for the

thermal equilibrium of two systems or bodies in contact

9. Internal energy of a system (U) is defined as the sum of the kinetic energies and the potential

energies of the constituent particles of the system.

a) The kinetic energy is due to the motion of the system's particles (e.g., translations, rotations,

vibrations). It depends only on the temperature of the system.

b) Potential energy is due to the intermolecular force of attraction.

c) In ideal gases, there is no inter-particle interaction. Therefore, molecular potential energy of a

system is zero. So only kinetic energy contributes to the internal energy of thermodynamic

system.

d) Internal energy of a system is a thermodynamic state variable i.e. it depends only on the state of

existence of the system and not on the path along which that state has been brought.

10. Energy which is transferred from one system to another system due to temperature difference is

called heat.

102

11. Work is said to be done if a body or a system moves through a certain distance on the application

force.

12. First law of thermodynamics – The energy entering the system in the form of heat is equal to the

sum of the increase in the internal energy of the system and the energy leaving the system in the

form of work done by the system on its surrounding. Q = ΔU + W

13. Specific Heat Capacity.

a) The specific heat capacity (c) of a substance is the amount of energy needed to raise the

temperature of 1g of the substance by 1K (or 1 ) .

b) Q = m c ΔT ;Unit – J kg -1 ºC-1 Dimensions – [ M0 L2 T-2 K-1] Specific heat of water is the highest

among all the substances.

c) It is equal to 4186 J kg-1 C-1

d) Two principal specific heats

e) Molar Specific heat at constant pressure (C p) -It is defined as the quantity of heat required to

raise the temperature of 1mole of a gas through 1°C at constant pressure.

f) Molar Specific heat at constant volume ( C v) -It is defined as the quantity of heat required to

raise the temperature of 1mole of a gas through 1°C at constant volume. Cp – Cv = R ; = γ ;

Cp > Cv; Cv = Mcv

14. Indicator diagrams

a) Indicator diagram is the graphical representation of the state of a system with the help of two

thermodynamic variables.

b) Work done by the thermodynamic system is equal to the area under P-V diagram.

15. Quasi-static process - Arbitrarily slow process such that system always stays arbitrarily close to

thermodynamic equilibrium.

16. Cyclic Process - System returns to same thermodynamic state periodically.

a) Work done in a cyclic process is equal to the area of the loop representing the cycle.

b) Clockwise loop – Work done is positive. Work is done by the system.

c) Anticlockwise loop – Work done is negative. Work is done on the system.

17. Reversible process - It is a process where the effects of following a thermodynamic path can be

undone be exactly reversing the path.

18. Irreversible process- It is a process which cannot be retraced in the reverse direction exactly.

19. Thermo dynamical process

103

20. Conditions for isothermal process

a) Perfectly conducting walls

b) Compression and expansion should be extremely slow.

21. Conditions for adiabatic process

a) Perfectly insulating walls

b) Compression and expansion should be very fast.

22. Heat engine – It is a device which converts heat energy into mechanical energy in a cyclic process.

a) Efficiency is always less than 100%.

104

23. Second law of thermodynamics( Kelvin – Planck statement) – It is impossible to construct an

engine which will produce no effect other than extracting heat from a reservoir and performing

an equivalent amount of work.

24. Carnot’s theorem – It states that i) no engine working between two given temperatures can have

efficiency greater than that of the carnot engine ii) efficiency of the carnot engine is independent

of the nature of working substance.

25. Refrigerator or heat pump

coefficient of performance(β) = =

26. Sign conventions used in numericals

1. Heat absorbed by the system (Q) Positive

2. Heat given out by the system Negative

3. Work done by the system (W) Positive

4. Work done on the system Negative

5. Internal energy of a system increases Positive

6. Internal energy of a system decreases Negative

105



1. Can water be boiled without heating? If yes, Give reason for your answer.

2. The efficiency of heat engine cannot be 100%. Explain why.

3. Why is the efficiency of a heat engine more in hilly areas than in the plains?

4. How can a refrigerator be used as a room heater in winters?

5. What is the specific heat of a gas in an (i) isothermal process, (ii) adiabatic process?

6. On what factors, does the efficiency of Carnot engine depend?

7. Plot a graph between internal energy U and Temperature (T) of an ideal gas.

8. Refrigerator transfers heat from cold body to a hot body. Does this violate the second law of

thermo dynamics.

9. Why a gas is cooled when expanded?

10. Can the temperature of an isolated system change?

11. Which one a solid, a liquid or a gas of the same mass and at the same temperature has the

greatest internal energy.

12. Under what ideal condition the efficiency of a Carnot engine be 100%.

13. Which thermodynamic variable is defined by the first law of thermodynamics?

14. Give an example where heat be added to a system without increasing its temperature.

15. What is the efficiency of Carnot engine working between ice point and steam point?

16. In an effort to cool a kitchen during summer, the refrigerator door is left open and the kitchen

door and windows are closed. Will it make the room cooler

SHORT ANSWER TYPE QUESTIONS ( 2 & 3 MARKS)

1. Explain what is meant by isothermal and adiabatic operations.

2. Explain briefly the principle of a heat pump. What is meant by coefficient of performance?

3. State first law of thermodynamics on its basis establish the relation between two molar specific

heat for a gas.

4. Explain briefly the working principle of a refrigerator and obtain an expression for its coefficient

of performance.

5. What is a cyclic process? Show that the net work done during a cyclic process is numerically

equal to the area of the loop representing the cycle.

6. The initial state of a certain gas is (P1, V1, T1). It undergoes expansion till its volume becomes V2.

Consider the following two cases:

a) The expansion takes place at constant temperature.

b) The expansion takes place at constant pressure.

Plot the P-V diagram for each case. In which of the two cases, is the work done by the gas more?

106


1. Describe briefly Carnot engine and obtain an expression for its efficiency.

2. a) Derive an expression for work done during an adiabatic expansion.

b) State the two statements of second law of thermodynamics.

3. a) Derive the relation between specific heats of a gas at constant pressure and at constant

volume.

4. b) A Carnot engine takes 3X106cal of heat from a reservoir at 6270C and gives it to sink at 270C.

Find the work done by the engine.

NUMERICALS

1. One of the most efficient engines ever developed operated between 2100K and 700K. Its

actual efficiency is 40%. What percentage of its maximum possible efficiency is this? (60%)

2. Calculate the fall in temperature when a gas initially at 720C is expanded suddenly to eight

time’s original volume. (γ=5/3) (86.25 K)

3. Two samples of gas initially at the same temperature and pressure are compressed from

volume V to V/2. One sample is compressed isothermally and the other adiabatically in which

case the pressure will be higher?

4. A volume of 10m3 of a liquid is supplied with 100kcal of heat and expands at a constant

pressure of 10atm to a final volume 10.2m3. Calculate the work done and change in internal

energy. (48kcal, 52kcal).

5. A perfect Carnot engine utilizes an ideal gas. The source temperature is 500K and sink

temperature is 375K if the engine takes 600Kcal per cycle from the source, calculate (i) the

efficiency of the engine, (ii) work done per cycle, (iii) heat rejected to sink per cycle.

(25%, 150 kcal, 450 kcal)

6. If the coefficient of performance of a refrigerator is 5 and operates at the room temperature

(270C). Find the temperature inside the refrigerator. (250K)

7. Find the change in internal energy of a gas when i) it absorbs 80 cal of heat and performs work

equal to 170 J ii) it absorbs 20 cal of heat and work equal to 55 J is performed on it. (139 J)

8. An ideal Carnot engine takes heat from a source at 317 , does some external work, and

delivers the remaining energy to a heat sink at 117 . If 500 kcal of heat is taken from the

source, how much work is done? How much heat is delivered to the sink? (710 kJ,331kcal)

9. A refrigerator freezes 5kg of water at 0 into ice at 0 in a time interval of 20min. Assume

that room temperature is 20 . Calculate the minimum power needed to accomplish it.

(102.5W)

10. ½ mole of helium is contained in a container at S.T.P. How much heat energy is needed to

double the pressure of the gas, keeping the volume constant? Heat capacity of gas is 3 J g–1

K–1.

107

ENRICHMENT EXERCISE

1. When a system is taken from state A to state B along the path ACB, 80 k cal of heat flows into the system

and 30 kcal of work is done.

a) How much heat flows into the system along path ADB if the workdone is 10 k cal?

b) When the system is returned from B to A along the curved path the work done is 20 k cal. Does the system

absorb or liberate heat.

c) If UA = 0 and UD = 40 k cal, find the heat absorbed in the process AD

2. Two Carnot engines A and B are operated in series. The first one A

receives heat at 900 K and reject to a reservoir at temperature T K. The second engine B receives the heat

rejected by the first engine and in turn rejects to a heat reservoir at 400 K calculate the temperature T

when

a) The efficiencies of the two engines are equal

b) The work output of the two engines are equal

HOTS

1. If hot air rises, why is it cooler at the top of mountain than near the sea level?

2. Can water be boiled without heating?

3. Assuming a domestic refrigerator as a reversible heat engine working between melting point of

ice and the room temperature at 27oC, calculate the energy in joule that must be supplied to

freeze 1Kg of water at 0oC.

4. Can we increase the temperature of gas without supplying heat to it?

5. In an effort to cool a kitchen during summer, the refrigerator door is left open and the kitchen

door and windows are closed. Will it make the room cooler?

112

CHAPTER-13

KINETIC THEORY OF GASES

113

SYNOPSIS

1. Kinetic theory of gases relates the macroscopic properties of gases like pressure, temperature

etc. to the microscopic properties of its gas molecules example speed, kinetic energy etc.

2. Ideal gas is one for which the pressure P, volume V and temperature T are related by PV = nRT

where R is called the gas constant.

3. Pressure exerted by a gas : It is due to continuous collision of gas molecules against the walls of

the container and is given by the relation P= 1/3 ρ vrms2 =1/3(nm v2

rms)

4. Average K.E. per molecule of a gas: E= ½ m v2= 3/2 KBT

5. This is kinetic interpretation of temperature. The temperature of a gas is a measure of the average

kinetic energy of a molecule.

6. The temperature of a gas is a measure of the average kinetic energy of molecules, independent

of the nature of the gas or molecule. In a mixture of gases at a fixed temperature the heavier

molecule has the lower average speed.

7. Absolute zero: It is that temperature at which the root mean square velocity of the gas molecules

reduces to zero.

8. The number of degrees of freedom is total number of independent co-ordinates required to

describe completely the position and configuration of a system. f = 3N – k. For monoatomic gases,

f = 3; For diatomic gases, f = 5; For linear triatomic gas molecules, f = 7.

9. According to the law of equipartition of energy, for any dynamical system in thermal equilibrium,

the total energy is distributed equally amongst all the degrees of freedom. The average energy

associated with each molecule per degree of freedom = 1/2 kBT where kB is Boltzmann constant

and T is temperature of the system.

10. Mean free path of gas molecules is the average distance travelled by a molecule between two

successive collisions. It is represented by λ.

11. Behaviour of a real gas approaches the ideal gas behaviour for low pressures and high

temperatures.

117


118


1. What is the ratio of rms speed of oxygen and hydrogen molecules at the same temperature?

2. Draw the graph between P & 1/V for a perfect gas at constant temperature.

3. Isothermal curves for a given mass of gas are shown at two different temperatures T1 & T2. State

whether T1>T2 or T2>T1, justify your answer.

4. If the number of molecule in a container is doubled. What will be the effect on the rms speed of the

molecules?

5. The absolute temperature of a gas is increased 3 times, what is the effect on the root mean square velocity

of the molecules?

6. Why gases at high pressure and low temperature show large deviation from ideal gas behaviour.

7. A gas is filled in a cylinder fitted with a piston at a definite temperature and Pressure. Why the pressure of

the gas decreases when the piston is pulled out.

8. On what parameters does the mean free path depends.

9. Equal masses of oxygen and helium gases are supplied equal amount of heat. Which gas will undergo a

greater temperature rise and why?

10. Why evaporation causes cooling?


1. Write fundamental postulates of kinetic theory of gases.

2. State Avogadro’s law. Deduce it on the basis of kinetic theory of gases.

3. Two vessels of the same volume are filled with the same gas at the same temperature. If the

pressure of the gas in these vessels be in the ratio 1 : 2 then state

a) The ratio of the rms speeds of the molecules.

b) The ratio of the number of molecules.

4. Equal masses of oxygen and helium gases are supplied equal amount of heat. Which gas will

undergo a greater temperature rise and why?

5. State Graham’s law of diffusion. How do you obtain this from Kinetic Theory of gases.

6. Prove that the pressure exerted by a gas is given by where ρ is density and c is root mean square

velocity.

119

7. Prove that for a perfect gas having n degree of freedom where Cp and Cv have their usual meaning.


1. Given that P = 1/3 ρc2where P is the pressure, ρ is the density and c is the rms velocity of gas

molecules. Deduce Boyle’s law and Charles law of gases from it.

2. What is law of equipartition of energy? Find the value of γ= Cp/Cv for monoatomic and diatomic,

triatomic, polyatomic gas where symbol have usual meaning.

NUMERICALS

1. An air bubble of volume 1cm3 rises from the bottom of a lake 40m deep at a temperature of 120C.

To what volume does it grow when it reaches the surface which is at a temperature of 350C? (5.3

X 10-6m3)

2. Calculate the temperature at which the rms velocity of gas molecules becomes double its value

at 270C, pressure of the gas remaining same. (927oC)

3. At what temperature the rms speed of oxygen atom equals to r.m.s. speed of helium gas atom at

–10°C? Atomic mass of helium = 4 Atomic mass of oxygen = 32.(2104 K)

4. The density of Carbon dioxide gas at 0°C and at a pressure of 1.0 × 105 N/m2 is 1.98 kg/m3. Find

the root mean square velocity of its molecules at 0°C and 30°C. Pressure is kept constant.

(410m/s)

5. 0.014 kg of nitrogen is enclosed in a vessel at a temperature of 27°C. How much heat has to be

transferred to the gas to double the rms speed of its molecules.(2250 cal)

6. Hydrogen is heated in a vessel to a temperature of 10,000K. Let each molecule possess an average

energy E1. A few molecules escape into the atmosphere at 300K. Due to collisions, their energy

changes to E2. Calculate E1/E2.(140:3)

7. Calculate the kinetic energy per molecule and also rms velocity of a gas at 1270C. Given

kB=1.38x10-23Jmolecule-1K-1 and mass per molecule of the gas = 6.4 x 10-27kg.

(8.28 x 10-21J, 1.608 x 103m/s)

8. The rms velocity of hydrogen at STP is u m/s. If the gas is heated at constant pressure till its

volume is three fold, what will be its final temperature and the rms velocity? (√3 u m/s)

9. At what temperature the rms speed of oxygen atom equal to r.m.s. speed of heliums gas atom at –

10°CAtomic mass of helium = 4 Atomic mass of oxygen = 32.

10. The density of Carbon dioxide gas at 0°C and at a pressure of 1.0 × 105 N/m2 is 1.98 kg/m3. Find the root

mean square velocity of its molecules at 0°C and 30°C. Pressure is kept constant

120

ENRICHMENT EXERCISE

1. A vessel is filled with a mixture of two different gases. However, the number of molecules per

unit volume of the two gases in the mixture are the same.

a) Will the mean KE per molecule of both the gases be equal?

b) Will the rms velocities of the molecules be equal?

c) Will the pressure be equal? Give reasons.

2. An insulated container containing monoatomic gas of molar mass m is moving with a velocity vo.

If the container is suddenly stopped, find the change in temperature. (mvo2/3R)

3. Two vessels of the same size are at the same temperature. One of them contains 1g of H2 gas,

and the other contains 1g of N2 gas.

a) Which of the vessels contains more molecules?

b) Which of the vessels is under greater pressure and why?

c) In which vessel is the average molecular speed greater? How many times greater?

a) 14 times, b) 3.74 )

4. An ideal gas has a specific heat at constant pressure (Cp = 5 R/2). The gas is kept in a closed vessel

of volume 0.0083 m3 at a temperature of 300 k and a pressure of 1.6 × 106 Nm–2. An amount of

2.49 × 104 J of heat energy is supplied to the gas. Calculate the final temperature and pressure of

the gas

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CHAPTER-14

OSCILLATIONS

SYNOPSIS

1. Periodic Motion – It is the motion of a body that repeats itself after regular intervals of time.

2. Oscillatory Motion- It is the motion of a body in which it moves to and fro about a fixed point

after regular intervals of time.

3. Every oscillatory motion is a periodic motion, but every periodic motion is not oscillatory.

4. Time period (T) – It is the time required to complete one oscillation.

5. Frequency – It is the number of oscillations in one second.

a) T =

b) SI unit – Hertz (Hz) or s-1

6. Displacement (x)– It is the distance travelled by an oscillating particle at any instant from its

equilibrium or mean position.

a) Amplitude is the maximum displacement of the particle.

b) SI Unit – metre (m)

7. Periodic Function – It is the function which repeats itself after regular interval of time. Eg –sine

and cosine functions

8. Simple harmonic motion(SHM) – In this a body moves to and fro about a mean position under

the action of a restoring force which is directly proportional to its displacement from the mean

position and is always directed towards the mean position.

Characteristics of SHM

a) Motion of the particle is periodic.

b) Restoring Force F α -x.

c) Constant amplitude and fixed frequency.

d) The force always acts toward the equilibrium position.

9. The minus sign in the above equation indicates that the restoring force is directed so that the

mass restores its equilibrium position.

10. k is the restoring force produced per unit displacement.

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11. In SHM, the displacement x(t) of a particle from its equilibrium position is given by, x(t) = A cos(

in which A is amplitude of the displacement, the quantity ( is the phase of

motion, is the phase constant or initial phase and is the angular frequency.

12. Phase – The phase of an oscillating particle at any instant is its state as regards to its position and

direction of motion with respect to mean position.

13. Initial phase or Epoch – It is the phase when the particle starts oscillating.

14. Phase is either measured in terms of an angle or the time that has elapsed since the particle last

passed through its mean position in the positive direction

15. Angular frequency - Angular frequency gives the frequency with which phase changes.

16. Simple harmonic motion and uniform circular motion

a) Simple harmonic motion is the projection of uniform circular motion on the diameter of the circle

in which the circular motion occurs

17. Velocity and Acceleration

T

2

124

18. Phase relationship between displacement, velocity and acceleration

T 0

T

0

2

x(t) +A 0 -A 0 +A

v(t) 0 -ωA 0 +Ωa 0

a(t) - 2A 0 +ω2A 0 -ω2A

19. Energy in SHM

Energy Formula Maximum Value Minimum Value

Kinetic Energy(K) ½ k(A2-x2) ½ kA2 0 ( at mean position)

Potential Energy(U) ½ kx2 ½ kA2 0 (at extreme

position)

Total Energy K+U ½ kA2 ( remains same at all the positions of

the

Particle)

20. Oscillations due to a spring (Horizontal or Vertical)

125

21. Simple Pendulum - To = 2π√l

g

22. Undamped oscillations : When a simple harmonic system oscillates with a constant amplitude

(which does not change) with time, its oscillations are called undamped oscillations.

23. Damped oscillations : When a simple harmonic system oscillates with a decreasing amplitude

with time, its oscillations are called damped oscillations.

24. Free, forced and resonant oscillations

a) Free oscillations : When a system oscillates with its own natural frequency

b) Without the help of an external periodic force, its oscillations are called free oscillations.

c) Forced oscillations : When a system oscillates with the help of an external

d) Periodic force of frequency, other than its own natural frequency, its oscillations are called forced

oscillations.

e) Resonant oscillations: When a body oscillates with its own natural frequency, with the help of an

external periodic force whose frequency is the same as that of the natural frequency of the

oscillating body, then the oscillations of the body are called resonant oscillations.

127

WAVES

SYNOPSIS

1. A Wave Motion is a form of disturbance which travels through a medium on account of

repeated periodic vibrations of the particles of the medium about their mean position,

the motion being handed on from one particle to the adjoining particle. A material

medium is a must for propagation of waves. It should possess the properties of inertia,

and elasticity. The two types of wave motion are: (i) Transverse wave motion that travels

in the form of crests and troughs. (ii) Longitudinal wave motion that travels in the form of

compressions and rarefactions.

2. Speed of longitudinal waves in a long solid rod is where, Y is Young's modulus

of the material of solid rod and P is density of the material. The speed of longitudinal waves

in a liquid is given by where K is bulk modulus of elasticity of the liquid.

128

3. The expression for speed of longitudinal waves in a gas, as suggested by Newton and

modified later by Laplace is where γ= Cp/Cv and P is the pressure exerted by

the gas.

4. Factors affecting speed of sound in gas – a) Density b) Humidity c) Temperature d) Wind

e) Frequency f) Amplitude

5. The speed of transverse waves over a string is given by where, T is tension in

the string and m is mass of unit length of the string.

6. Equation of plane progressive waves travelling along positive direction of X-axis is

y= a sin (ωt-kx)

7. Superposition principle enables us to find the resultant of any number of waves meeting

at a point. The resultant displacement at any point in the medium is given by vector sum

of the displacements of the individual waves. 𝑦1 = 𝑦1 + 𝑦2 + ⋯+𝑦𝑛

8. On a string, transverse stationary waves are formed due to superimposition of direct and

the reflected transverse waves of waves meeting at a point.

9. A wave is reflected from a rigid boundary with a phase reversal whereas reflection from

an open boundary takes place without any phase change.

10. Stationary waves: When two progressive waves of equal amplitude and frequency,

travelling in opposite directions along a straight line superpose each other, the resultant

wave does not travel in either direction and is called stationary or standing wave.

11. The wavelength of nth mode of vibration of a stretched string is λn = 2L

n and its frequency,

vn = n v1. This note is called nth harmonic or (n – 1)th overtone.

12. Nodes are the points, where amplitude of vibration is zero, In the nth mode of vibration,

there are (n + 1) nodes located at distances (from one end) x = 0, L

n ,

2L

n , ….,L

13. Antinodes are the points, where amplitude of vibration is maximum. In the nth mode of vibration,

there are n antinodes, located at distances (from one end) x = L

2n,3L

2n,5L

2n , … . ,

(2n−1)L

2n

14. Separation between two successive nodes or antinodes = λ/2

15. Separation between a nearest node and a nearest antinode = λ/4.

16. On a stretched string, transverse waves are formed due to superposition of direct and the

reflected transverse waves. The frequency of vibration of a string is νn = n/2L

for nth mode. The various frequencies are in the ratio 1:2:3:4.

17. Organ pipe: It is a musical instrument in which sound is produced by setting an air column

into vibrations. Longitudinal stationary waves are formed on account of superposition of

incident and reflected longitudinal waves.

18. Closed organ pipe - Longitudinal stationary waves are formed in a closed organ pipe (closed

at one end). Antinode is formed at one end and node at the other end. The frequency of

vibration for nth mode is :

129

. The frequencies produced in closed organ pipe are in the ratio

1:3:5:7.

19. Open organ pipe: Longitudinal stationary waves are formed in open organ pipe. Antinodes

are formed at both the ends, separated by one node in between. The frequency of

vibration for the nth mode is νn = n/2L . The frequencies are in the ratio 1:2:3:4.

20. Beats: the periodic vibrations in the intensity of sound due to superposition of two sound

waves of slightly different frequencies are called beats. One rise and one fall of intensity

constitute one beat. The number of beats produced per second is called beat frequency,

νbeat=ν1-ν2.

21. Doppler effect in sound: the phenomenon of apparent change in the frequency of sound

due to the relative motion between the source of sound and the observer is called Doppler

effect. If v, vo, vs and vm are the velocities of sound, observer, source and the medium

respectively, then the apparent frequency is given by ν’= [(v+vm-vo)/(v+vm-vs)] * ν .

130



1. How is the time period effected, if the amplitude of simple pendulum is increased?

2. What are the two basic characteristics of a simple harmonic motion?

3. At what distance from the mean position, is the kinetic energy of simple harmonic oscillator equal

to potential energy?

4. Define angular frequency. Give its S.I. unit.

5. A girl is swinging in the sitting position. How will the period of the swing change if she stands up?

6. Why does sound travel faster in iron than in water or air?

7. Frequency is the most fundamental property of wave, why?

8. How do wave velocity and particle velocity differ from each other?

9. Why the pitch of an organ pipe on a hot summer day is higher?

10. If any liquid of density higher than the density of water is used in a resonance tube, how

will the frequency change?

11. When is the swinging of simple pendulum considered approximately SHM?

12. Can the motion of an artificial satellite around the earth be taken as SHM?

13. What is the phase relationship between displacement, velocity and acceleration in SHM?

14. Why does sound travel faster in iron than in water or air?

15. The speed of sound does not depend upon its frequency. Give an example in support of

this statement.

16. If an explosion takes place at the bottom of lake or sea, will the shock waves in water be

longitudinal or transverse?

17. Frequency is the most fundamental property of wave, why?

18. 18.Which of the following relationships between the acceleration ‘a’ and the

displacement ‘x’ of a particle involve simple harmonic motion

19. Can a motion be periodic and not oscillatory?

20. Can a motion be periodic and not simple harmonic? If your answer is yes, give an example

and if not, explain why?

21. How does the frequency of a tuning fork change, when the temperature is increased?

22. On blowing into it more strongly it produces the first overtone of the frequency 384Hz.

What is the type of pipe –Closed or Open?

23. All harmonic are overtones but all overtones are not harmonic. How?

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SHORT TYPE ANSWER QUESTIONS (2 & 3 MARKS)

1. Show that in a S.H.M the phase difference between displacement and velocity is /2, and

between displacement and acceleration is .

2. Deduce an expression for the velocity of a particle executing S.H.M. When is the particle’s velocity

(i) Maximum (ii) minimum?

3. A tunnel is dug through the centre of the Earth. Show that a body of mass ‘m’ when dropped from

rest from one end of the tunnel will execute SHM.

4. A cylindrical log of wood of height h and area of cross-section A floats in water. It is pressed and

then released. Show that the log would execute SHM with a time period. T=2π √(m/Aρg)

5. Distinguish with an illustration among free, forced and resonant oscillations.

6. Draw (a) displacement time graph (b) velocity time graph and (c) acceleration time

graph of a particle executing SHM with phase angle equal to zero.

7. Use the formula to explain, why the speed of sound in air

a) Is independent of pressure?

b) Increase with temperature?

8. Differentiate between closed pipe and open pipe at both ends of same length for frequency

of fundamental note and harmonics.

9. Distinguish between the condition of stationary waves and beats.

10. Define wave number and angular wave number and give their S.I. units.

11. Give a qualitative discussion of the different modes of vibration of an open organ pipe.

12. Give the differences between progressive and stationary waves.

13. What are beats? How are they produced? Briefly discuss one application for this

phenomenon.

LONG TYPE ANSWER QUESTION (5 MARKS)

1. Find the total energy of the particle executing S.H.M. and show graphically the variation of

potential energy and kinetic energy with time in S.H.M.

2. Define the terms harmonic oscillator, displacement, amplitude, cycle, time period, frequency,

angular frequency, phase and epoch with reference to an oscillatory system.

3. Show that S.H.M. may be regarded as the projection of uniform circular motion along a diameter

of the circle.

4. Show that for small oscillations the motion of a simple pendulum is simple harmonic. Derive an

expression for its time period. Does it depend on the mass of the bob? What will be the time

period of second’s pendulum if its length is doubled?

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NUMERICALS

1. A mass of 2kg is attached to the spring of spring constant 50 N/m. The block is pulled to a distance

of 5cm from its equilibrium position at x = 0 on a horizontal frictionless surface from rest at t=0.

Write the expression for its displacement at any instant t. ( 0.05cos(5t)m)

2. The time period of a body executing SHM is 1s. After how much time will its displacement be 1/√2

of its amplitude? ( 1/8 s)

3. The displacement of a particle having SHM is x= 10sin (10πt + π/4)m. Find amplitude, angular

frequency, epoch, time period, frequency, maximum velocity. (10m, 10π rad/s, π/4, 0.2s, 5Hz, 20π)

4. A particle executing linear SHM has a maximum velocity of 40cm/s and a maximum acceleration

of 50cm/s2. Find its amplitude and the period of oscillation. (5.03 s)

5. A particle executing SHM completes 1200 oscillations per minute and passes through the mean

position with a velocity of 31.4m/s. determine the maximum displacement of the particle from

the mean position. Also obtain the displacement equation of the particle if its displacement be

zero at the instant t=0. (0.025m, 0.025 sin(40πt)m)

6. If the length of a simple pendulum is increased by 45%, what is the percentage increase in its time

period? (22.5 %)

7. A point describes SHM in a line 6 cm long. Its velocity, when passing through the centre of line is

18 cm s–1. Find the time period. (1.047 s)

8. Find the period of vibrating particle (SHM), which has acceleration of 45 cms–2, when displacement

from mean position is 5 cm. (2.095 s)

9. Two identical springs, each of force constant k are connected in a) series b) parallel, and they

support a mass m. Calculate the ratio of the time periods of the mass in the two systems. (2)

10. Show that if a particle is moving in SHM, its velocity at a distance /2 of its amplitude from the

central position is half its velocity in central position.

11. Find the ratio of velocity of sound in hydrogen gas ( to that in helium gas ( at

the same temperature. Given that molecular weights of hydrogen and helium are 2 and 4

respectively. (1.68)

12. Audible frequencies have a range 20 Hz to 20,000 Hz. Express this range in terms of i) period T ii)

wavelength in air and iii) angular frequency. Given velocity of sound in air is 330 m/s. ( i) 5x10-2s

to 5x10-5s ii) 16.5m to 0.0165m iii) 40π rad/s to 40000π rad/s)

13. A steel wire has a length of 12m and a mass of 2.10kg. What will be the speed of a transverse wave

on the wire when a tension of 2.06 X 104N is applied? (343 s)

At what temperature will the speed of sound be double its value at 273°K? (1092 K).

14. A string of mass 2.5 kg is under a tension of 200N. The length of the stretched string is 20m.

If a transverse jerk is struck at one end of the string, how long does the disturbance take

to reach the other end? ( 0.5s )

15. The equation of a plane progressive wave is given by the equation: y = 10 sin 2

(t – 0.005x) where y and x are in cm and t in seconds. Calculate the amplitude, frequency,

wave length and velocity of the wave. (10cm ,1 Hz ,200cm ,200m/s)

133

16. A pipe 20cm long is closed at one end. Which harmonic mode of the pipe is resonantly

excited by a source of 1237.5Hz? (sound of velocity in air=330m/s) ( First mode)

17. A tuning fork of unknown frequency gives 4 beats with a tuning fork of frequency 310 Hz.

It gives the same number of beats on filing. Find the unknown frequency. (306 Hz)

18. A railway engine and a car are moving on parallel tracks in opposite directions with speed

of 144km/h and 72km/h, respectively. The engine is continuously sounding a whistle of

frequency 500Hz. The velocity of sound is 340m/s. Calculate the frequency of sound heard

in the car when (i) the car and the engine are approaching each other, (ii) the two are

moving away from each other. (600 Hz,421 Hz)

19. An observer moves towards a stationary source of sound with a velocity one-fifth of the

velocity of sound. What is the percentage increase in the apparent frequency? (20%)


1. Discuss the Newton's formula for velocity of sound in air. What correction was applied to

it by Laplace and why?

2. Discuss the formation of harmonics in a stretched string. Show that in case of a stretched

string the first four harmonics are in the ratio 1:2:3:4. Give the differences between

progressive and stationary waves.

(i) What are beats? Prove that the number of beats per second is equal to the difference

between the frequencies of the two superimposing waves. (ii)A wave travelling along a

string is given by y(x,t)= 0.005sin(50x-5t), all quantities are in SI units. Calculate (a) velocity

of the wave, (b) amplitude of particle velocity

ENRICHMENT EXERCISE

1. An open pipe is in second harmonic with frequency f1. Now one end of the tube is closed

and frequency is increased to f2 such that the resonance again occurs in nth harmonic. Find

the value of n. How are f1 and f2 related to each other? ( f2 = 5/4f)

2. Tube A has both ends open, while B has one end closed. Otherwise the two tubes are

identical. What is the ratio of fundamental frequency of the tubes A and B?(2)

3. A whistle producing sound waves of frequencies 9500 Hz and above is approaching a

stationary person with speed vs m/s. The velocity of sound in air is 300m/s. If the person

can hear frequencies upto a maximum of 10 kHz, What is the maximum value of vs upto

which he can hear the whistle? (15m/s)

4. Two simple harmonic motions are represented by the equations :

y1 = 0.1 sin(100 and y2 = 0.1 cos What is the phase difference of the velocity of

the particle 1 with respect to the velocity of particle 2? (-π/6)

5. A simple pendulum has time period T1. The point of suspension is now moved upward according

to the relation y = Kt2 (K = 1m/s2), where y is the vertical displacement. The time period now

becomes T2. What is the ratio T12/T2

2? Given g = 10m/s2. (6/5)

134

6. A particle executes S.H.M. between x= -A and x=+A. The time taken for it to go from 0 to A/2 is T1

and to go from A/2 to A is T2. Then how are T1 and T2 related? (2T1)

146

PRACTICE SAMPLE PAPER -1

CLASS – XI

PHYSICS

Time allowed: 3 hours Maximum Marks: 70

General Instructions:

7. All questions are compulsory. There are 26 questions in all.

8. This question paper has five sections: Section A, Section B, Section C, Section D and Section E

9. Section A contains five questions of one mark each, Section B contains five questions of two marks

each, Section C contains twelve questions of three marks each, Section D contains one value

based question of four marks and Section E contains three questions of five marks each.

10. There is no overall choice. However, an internal choice has been provided in one question of two

marks, one question of three marks and all the three questions of five marks weightage. You

have to attempt only one of the choices in such questions.

SECTION –A

1. If two sound waves of frequencies 500 Hz and 550 Hz superpose, can the beats be observed? Give

reason for your answer.

2. What is the phase relationship between particle displacement and velocity in SHM?

3. Draw the graph showing cooling of hot water with time.

4. State second law of thermodynamics.

5. Draw the graph showing the variation of acceleration due to gravity with height above the earth’s

surface.

SECTION – B

6. What happens to the change in internal energy of a gas during so thermal expansion?

7. Derive Boyle’s law on the basis of kinetic theory of gases.

OR

Using the law of equipartition of energy, determine the specific heat ratio of a monoatomic gas.

8. a) Define elastomers. Give its example.

b) Draw the stress-strain curve for the elastomers.

9. What is meant by banking of roads? What is the need for banking of a road?

10. a) Define torque. Write its S.I. unit.

b) Why are handles provided at the edges of the door?

11. At a time when the displacement is half the amplitude, what fraction of the total energy is kineti

and what fraction is potential in S.H.M.?

147

12. Show that motion executed by the bob of the simple pendulum is S.H.M. Derive an expression for

its time period.

13. State the law of equipartition of energy. Determine the specific heat ratio of a monoatomic gas.

14. Calculate the heat required to convert 3 kg of ice at –12 °C kept in a calorimeter to steam at 100

°C at atmospheric pressure. Given specific heat capacity of ice = 2100 J kg–1 K–1, specific heat

capacity of water = 4186 J kg– 1 K–1, latent heat of fusion of ice = 3.35 × 105 J kg–1 and latent heat

of steam = 2.256 ×106 J kg–1.

15. Name the three modes of transfer of heat from one object to other. Also cite one example for

each one of them.

16. Derive an expression for rate of flow of fluid as measured by venture meter using well labelled

diagram.

17. Two wires made of the same material are subjected to forces in the ratio of 1:4. Their lengths are

in the ratio 8:1 and diameter in the ratio 2:1. Find the ratio of their extensions.

18. In the figure given below, find the acceleration ‘a’ of the system and the tensions T1 and T2 in the

strings. Assume that the table and the pulleys are frictionless and the strings are massless. Take

g = 10 m/s2.

19. State Theorem of parallel and parallel axis theorem axes. The moment of inertia of a solid sphere

about a tangent is 5/3 MR2, where M is mass and R is radius of the sphere. Find the moment of

inertia of the sphere about its diameter.

20. a) Define orbital velocity.

b) Derive expression for the orbital velocity of a satellite revolving at distance ‘r’ from the earth.

21. Calculate the energy required to move an earth satellite of mass 103kg from a circular orbit of

radius 2R to that of radius 3R. Given mass of the earth = 6 X 1024kg, R=6400km.

22. Prove that in case of close organ pipe of length L, the frequencies of vibrating air column are

given by ν= (2n+1)(v/4L).

23. One day Arvind went to Super Bazar to purchase some groceries. There he saw an old lady

struggling with her shopping. He immediately showed her the lift and explained to her how she

can carry her goods from one floor to the other. Even then the old lady showed hesitation to use

the lift. On seeing this, Arvind took the lady into the lift and showed her how to operate the lift.

The old lady was very happy and easily finished her shopping.

148

What are the values shown by Arvind?

An elevator which can carry a maximum load of 1800 kg is moving up with a constant speed of 2

m/s. The frictional force opposing the motion is 4000N. Determine the maximum power delivered

by the motor to the elevator.

A spring balance is attached to the ceiling of a lift. A man hangs bag of mass 5 kg on the spring

balance, when the lift is stationary. What will be the reading of the spring balance, if the lift moves

downward with an acceleration of 5 m/s2.

SECTION – E

24. a) Derive Newton’s formula for the speed of sound in a gas. b) Why and what correction was applied by Laplace in this formula? c) Using the Laplace correction, deduce the formula for the speed of sound. (2+1+2)

OR

Discuss the effect of temperature and pressure on the velocity of sound.

What is Doppler’s effect of sound? Obtain an expression for apparent frequency of sound when

source and observer are moving away from each other.(2+3)

25. a) Define surface tension.

Derive a relation between surface tension and surface energy.

How is the surface tension of a liquid explained on the basis of intermolecular forces. (1+2+2)

OR

Write any two properties of streamlines.

A liquid is in streamlined flow through a pipe of non-uniform cross-section. Prove the sum of its

kinetic energy, pressure energy and potential energy per unit volume remains constant. (2+3)

26. a) Explain the construction and various operations for Carnot’s heat engine working between two

temperatures.

b) Hence derive from it the efficiency of the engine. (3+2)

OR

a) What is an adiabatic process?

b) State two essential conditions for such a process to take place.

c) Show analytically that work done by one mole of an ideal gas during a diabatic expansion

from V1 to volume V2 is given by 𝑾 =𝑹[𝑻𝟏−𝑻𝟐]

𝜸−𝟏 . (1+1+3)

149

PRACTICE SAMPLE PAPER-2

CLASS – XI

PHYSICS

Time allowed: 3 hours Maximum Marks: 70

SECTION - A

1. What is the significance of the area of closed curve on a P-V diagram?

2. How will the time period of a loaded spring change when taken to moon?

3. Give reason, why does the hair of a shaving brush cling together when taken out of the water?

4. If the temperature of a gas at constant pressure is increased four times, how will the velocity of

sound in the gas will be affected?

5. The weight of a body is less inside the earth than on the surface. Justify.

6. State Boyle’s law and deduce it on the basis of kinetic theory of gases.

OR

Deduce Graham’s law of diffusion from kinetic theory of gases using expression of pressure.

7. Diagrammatically show first two modes of vibrations in each case of an open organ pipe and find

the ratio of their frequencies. 8. Calculate the velocity of the bob of a simple pendulum at its mean position if it is able to rise to

a vertical height of 10 cm. ( g= 10 m/s2).

9. Obtain an expression for the orbital velocity of a satellite.

10. Represent graphically the variation of extension with load in an elastic body. On the graph mark:

a) Hooke’s law region.

b) Breaking point.

SECTION - C

11. a) Derive an expression for the time-period of the horizontal oscillations of a massless loaded

spring.

b) Army troops are not allowed to march in steps while crossing a bridge. Why?

12. Obtain the expression for the linear acceleration of a cylinder rolling down an inclined plane.

13. State the law of equipartition of energy and find the specific heat ratio for Helium.

14. State the law of conservation of angular momentum. A solid cylinder of mass 20 kg rotates about

its axis with angular speed 100 rad/s. The radius of the cylinder is 0.25m. What is the kinetic

energy associated with the rotation of the cylinder? What is the magnitude of the angular

momentum of the cylinder about its axis?

150

15. Show that the total mechanical energy of a body falling freely under gravity is conserved. Draw a

graph showing variation of potential energy and kinetic energy with respect to height of a free

fall under gravitational force.

16. Define free, forced and resonant oscillations. Give an example of each.

17. a) In a refrigerator, heat from inside at 277 K is transferred to a room at 300 K. How many joules

of heat shall be delivered to the room for each joule of electrical energy consumed ideally?

b) Can a kitchen be cooled by leaving the door of an electric refrigerator open? Give reason for

your answer.

18. A copper block of mass 2.5 kg is heated in a furnace to a temperature of 5000 C and then placed

on a large ice block. What is the maximum amount of ice that can melt? (Specific heat of copper

= 0.39 J/g/oC, and heat of fusion of water = 335 J/g).

OR

A large steel wheel is to be fitted on to a shaft of the same material. At 27oC, the outer diameter

of the shaft is 8.70 cm and the diameter of the central hole in the wheel is 8.69 cm. The shaft is

cooled using 'dry ice'. At what temperature of the shaft does the wheel slip on the shaft? Assume

coefficient of linear expansion of the steel to be constant over the required temperature range.

Αsteel = 1.20 x 10-5 /K.

19. Draw energy distribution curves for a black body at two different temperatures T1 and T2 (T1 > T2). Write any two conclusions that can be drawn from these curves.

20. a) State Pascal's law. b) To lift an automobile of 2000 kg, a hydraulic lift with a larger piston 900 cm2 in area is used.

Calculate the force that must be applied to the smaller piston of area 10 cm2to accomplish this

task.

21. a) What force would be required to stretch a wire of 4 × 10-4 m2 cross – section, so that its length

becomes 3 times of its original length? Given that Young’s modulus of the material of the wire is

3.6 × 10 11 Nm-2.

b) What is the modulus of rigidity of a fluid? (2+1)

22. Distinguish between stationary and progressive waves. (Three points)

SECTION – D

23. Mythili was a student of class IX. She was sitting in a garden along with her grandmother, who

was a retired Physics teacher. Suddenly she saw an orange falling from the tree. Immediately she

asked her grandmother that both the orange and earth experience equal and opposite forces of

gravitation, then why it is the orange that falls towards the earth and not the earth towards the

orange. Her grandmother explained her the reason in a simple way.

a) What are the values being displayed by Mythili?

b) What in your opinion may be the reason for this observation?

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c) What is the maximum value of potential energy that can be possessed by a heavenly body? Give

the general expression for potential energy of an object near the surface of earth.

SECTION – E

24.

a) Define coefficient of viscosity.

b) State Stroke’s law.

c) Explain, how a body attains a terminal velocity when it is dropped from rest in a viscous medium.

d) Derive an expression for the terminal velocity of a small spherical body falling through a viscous

medium. (1+1+1+2)

OR

e) State and prove Bernoulli’s theorem. (ii) A cylindrical vessel of uniform cross-section contains

liquid upto the height ‘H’. At a depth h=H/2 below the free surface of the liquid there is an orifice.

Using Bernoulli’s theorem, find the velocity of efflux of the liquid.

25. a) Draw a neat P-V diagram showing cycle of operations for an ideal heat engine. b) Briefly explain the four stages of operations in proper order.

c) Derive the expression of efficiency of a Carnot’s heat engine.

OR

a) State first law of thermodynamics. On its basis establish the relation between two molar specific

heats for a gas.

b) Derive an expression for work done during an isothermal process.(3+2)

25. a) Why does sound travel faster in moist air than in dry air?

b) The equation of a plane progressive wave is y = 10 sin2π (t – 0.005x) where y and x are in cm and t in seconds. Calculate the amplitude,

frequency, wavelength and velocity of the wave.

c) Derive an expression for the apparent frequency of sound when a source moves towards the

stationary observer. (1+2+2)

OR

a) Derive Newton’s formula for the speed of sound in a gas.

b) Why was the correction needed in Newton’s formula?

c) Show mathematically the correction applied by the Laplace in Newton’s formula. (2+1+2)

152

PRACTICE SAMPLE PAPER -3

CLASS: XI

SUBJECT: PHYSICS

TIME: 3 HOURS M M: 70

General Instruction:

1. All questions are compulsory. There are 26 questions in all.

2. This question paper has five sections: Section A, Section B, Section C, Section D and Section E.

3. Section A contains five questions of one mark each, Section B contains five questions of two marks

each, Section C contains twelve questions of three marks each, Section D contains one value

based question of four marks and Section E contains three questions five marks each.

4. There is no overall choice. However, an internal choice has been provided in one question of two

marks, one question of three marks and all the three questions of five marks.

5. Use of calculators is not permitted. Use log tables, if necessary.

Section A

1. What is the angle of friction between two surfaces in contact if the coefficient of friction is 13?

2. Why springs are made of steel and not of copper?

3. Write the SI unit of specific heat?

4. Write the values of amplitude and frequency from the equationY = Asinωtof SHM.

5. In an open organ pipe, third harmonic is 450 Hz. What is the frequency of fifth harmonic?

Section B

6. A ball is thrown vertically upwards with a velocity of 20 meter sec-1from the top of a multi-storey

building. The height of the point from where the ball is thrown is 25 meter from the ground.

a) How high will the ball rise? (b)How long will it be before the ball hits the ground?

7. Show that the impulse of a force is equal to the change in momentum Produced by the force.

8. Define gravitational potential .Write its S.I unit. Write an expression for the Gravitational

potential at a point in the gravitational field of the earth.

9. Find the temperature at which the r.m.s speed of the molecules of a given gas becomes n times

their r.m.s speed under STP conditions. Assume the pressure to remain constant?

OR

At what temperature is the root mean square speed of an atom in argon gas cylinder equal to the

r.m.s speed of helium gas atoms at –20°C? (Atomic mass of argon = 39.9 u, and that of helium=

4.0 u.

153

10. What is the change in internal energy of a gas during (i) isothermal expansion and (ii) adiabatic

expansion?

Section C

11. We measure the period of oscillation of simple pendulum in successive Measurements, the

reading turn out to be 2.63 s,2.56s, 2.42 s, 2.71 s and 2.80 s. calculate the mean absolute error,

relative error and percentage error.3

12. Draw the position-time graph of a particle3(a)moving with uniformly accelerated (a > 0)

rectilinear motion.(b)moving with uniformly retarded motion(a < 0).(c)moving with zero

acceleration (a=0)

13. Show that Newton’s second law of motion is the real law of motion.

14. State the law of conservation of mechanical energy. Show that the total mechanical energy of a

body falling freely under gravity is conserved.

15. The kinetic energy of a body is increases by 300%. What is the percentage increase in the linear

momentum of the body?

16. (a)Define Centre of mass of a system?

(b)A particle performing uniform circular motion has angular momentum what will be the new

angular momentum, if its angular frequency is doubled and its kinetic energy halved?

17. Define moment of inertia of a body. Mention two factors on which the moment of inertia of a

body depends. Write the expression for the moment of inertia of (a) a thin uniform circular ring

about an axis passing through its Centre and Perpendicular to its plane. And

(b)a disc about an axis passing through its Centre and Perpendicular to its plane.

18. Define acceleration due to gravity. Show that value of ‘g’ decrease with depth.

OR

Define acceleration due to gravity. Show that value of ‘g’ decrease with height.

19. State and Derive equation of continuity.

20. (a)Using the law of equipartition of energy, calculate the value of the specific heats of solids.

(b)Write the value of degree of freedom for monoatomic gas.

21. The speed of longitudinal wave ‘V’ in a given medium of density ρ is given by the formula

Use this formula to explain why the speed of sound in air (a) increase with Temperature (b)

independent of Pressure(c) increases with Humidity

22. Draw labelled diagram of heat pump. Define coefficient of performance and write it’s an

expression?

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Section D

23. Mrs.Raj royal estate had many sprawling lawn. Her grandson Saurabh was visiting her in his

summer holydays. One day, just for fun, he started pushing and pulling a lawn roller. He felt that

it easier to pull a lawn roller than to pushit. He asked Mr. Thomas, the estate officer, the reason

of easier pull and difficult Push.Mr.Thomas was surprised at this observation of Saurabh. He

talked to the Gardeners but they knew nothing. Finally, he approached the physics teacher of

science school run by Mrs.Raj. The Physics teacher explained to Thomas that there is more friction

at the time of pushing as compared to the friction at the time of pulling. Thomas explained this

fact to Saurabh. Saurabh was overjoyed and thanked Tomas.

(a)What according to you, are the values displayed by Thomas? (b)Why is it easier to pull a body

than to push it?

Section E

24. Derive an expression for the kinetic energy and potential energy of a harmonic Oscillator. Hence

Show that total energy is conserved in SHM. Draw graph for (a) Energy versus time and (b) Energy

versus displacement.

OR

Define stationary wave and derive an expression for a stationary wave formed by two sinusoidal

waves and obtain the position of nodes and antinodes.

25. Define terminal velocity. Show that the terminal velocity V of a sphere of Radius r, density ρ falling

vertically through a viscous fluid of density σ and Coefficient of viscosity ɳis given by

Using this formula to explain the observed rise of air bubble in a 5 liquid.

OR

State newton’s law of cooling .Derive mathematical expression for it. A body cools from 80°Cto

50°Cin 5 minutes. Calculate the time it takes to cool from 60°Cto 30°C. The temperature of the

surrounding is 20°C.

26. What is meant by banking of road? What is need for banking a road? Obtain an expression for

the maximum speed with which a vehicle can safely negotiate a curved road banked at an angle

θ.

OR

a) Define centripetal acceleration. Derive an expression for the centripetal acceleration of a particle

moving with uniform speed v along a circular path of radius r.

b) Show that there are two angles of projection for which the horizontal range is Same .Also show

that the sum of the Maximum heights for these two angles are independent of the angle of

projection.

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MOCK PAPER

Ist Term Exam, SESSION: 2019-20

CLASS – XI, SUBJECT: PHYSICS

TIME: 3 Hrs MM MARKS: 70

Q.No 1 to 20 are Multiple choice questions and carry 1 mark each.

1. Measure of two quantities along with the precision of respective measuring instrument is

A = 2.5 m s-1 ± 0.5 ms-1 B = 0.10 s ± 0.01 s

The value of A B will be

(a) (0.25 ± 0.08) m (b) (0.25 ± 0.5) m (c) (0.25 ± 0.05) m (d) (0.25 ± 0.135) m

2. If momentum (P ), area (A) and time ( T ) are taken to be fundamental quantities, then energy has

the dimensional formula

(a) (P1 A-1 T1) (b) (P2 A1 T1) (c) (P2 A-1/2 T1) (d) (P1 A1/2 T-1)

3. If P, Q, R are physical quantities, having different dimensions, which of the following combinations

can never be a meaningful quantity?

(a) (P – Q)/R (b) PQ – R c) PQ/R (d) (PR – Q2)/R (e) (R + Q)/P

4. On the basis of dimensions, decide which of the following relations for the displacement of a

particle undergoing simple harmonic motion is not correct:

5. The displacement of a particle is given by x = (t – 2)2 where x is in metres and t in seconds. The

distance covered by the particle in first 4 seconds is

(a) 4 m (b) 8 m (c) 12 m (d) 16 m

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6. A vehicle travels half the distance L with speed V1and the other half with speed V2, then its

average speed is

7. For the one-dimensional motion, described by x = t–sint

(a) x (t) > 0 for all t > 0. (b) v (t) > 0 for all t > 0. (c) a (t) > 0 for all t > 0.

(d) v (t) lies between 0 and 2.

8. A spring with one end attached to a mass and the other to a rigid support is stretched and

released.

(a) Magnitude of acceleration, when just released is maximum.

(b) Magnitude of acceleration, when at equilibrium position, is maximum.

(c) Speed is maximum when mass is at equilibrium position.

(d) Magnitude of displacement is always maximum whenever speed is minimum.

9.

(a) 45° (b) 90° (c) –45° (d) 180°

10. Which one of the following statements is true?

(a) A scalar quantity is the one that is conserved in a process.

(b) A scalar quantity is the one that can never take negative values.

(c) A scalar quantity is the one that does not vary from one point to another in space.

(d) A scalar quantity has the same value for observers with different orientations of the axes.

11. The horizontal range of a projectile fired at an angle of 15° is 50 m. If it is fired with the same

speed at an angle of 45°, its range will be

(a) 60 m (b) 71 m (c) 100 m (d) 141 m

12. Three vectors A, B and C add up to zero. Find which is false.

(a) (A * B) * C is not zero unless B,C are parallel

(b) (A * B).C is not zero unless B,C are parallel

(c) If A,B,C define a plane, (A * B) *C is in that plane

(d) (A * B).C=|A||B||C| → C2 = A2 + B2

13. A hockey player is moving northward and suddenly turns westward with the same speed to avoid

an opponent. The force that acts on the player is

(a) Frictional force along westward. (b) Muscle force along southward.

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(c) frictional force along south-west (d) muscle force along south-west.

14. In Fig. 5.1, the co-efficient of friction between the floor and the body B is 0.1. The co-efficient of

friction between the bodies B and A is 0.2. A force F is applied as shown on B. The mass of A is

m/2 and of B is m. Which of the following statements are true?

(a) The bodies will move together if F = 0.25 mg (b) The body A will slip with respect to B if F = 0.5

mg.

(b) The bodies will move together if F = 0.5 mg.(d) The bodies will be at rest if F = 0.1 mg.

(c) The maximum value of F for which the two bodies will move together is 0.45 mg.

15. A body of mass 10 kg is acted upon by two perpendicular forces, 6N and 8N. The resultant

acceleration of the body is

(a) 1 m s–2 at an angle of tan-1 (4 / 3) w.r.t. 6N force.

(b) 0.2 m s–2 at an angle of tan-1 (4 / 3) w.r.t. 6N force.

(c) 1 m s–2 at an angle of tan-1 (3 / 4) w.r.t. 8N force.

(d) 0.2 m s–2 at an angle of tan-1 (3 / 4) w.r.t. 8N force.

16. Two billiard balls A and B, each of mass 50g and moving in opposite directions with speed of 5m

s-1 each, collide and rebound with the same speed. If the collision lasts for 10-3 s, which of the

following statements are true?

(a) The impulse imparted to each ball is 0.25 kg m s-1 and the force on each ball is 250 N.

(b) The impulse imparted to each ball is 0.25 kg m s–1 and the force exerted on each ball is 25 × 10–5

N.

(c) The impulse imparted to each ball is 0.5 Ns.

(d) The impulse and the force on each ball are equal in magnitude and opposite in direction.

17. A body is falling freely under the action of gravity alone in vacuum. Which of the following

quantities remain constant during the fall?

(a) Kinetic energy (b) Potential energy (c) Total mechanical energy (d) Total linear momentum.

18. The potential energy function for a particle executing linear SHM is given by (1/2)kx2 where k is

the force constant of the oscillator (Fig. 6.2). For k = 0.5N/m, the graph of V(x) versus x is shown

in the figure. A particle of total energy E turns back when it reaches x = ± xm . If V and K indicate

the P.E. and K.E., respectively of the particle at x = +xm, then which of the following is correct?

158

(a) V = O, K = E (b) V = E, K = O (c) V < E, K = O (d) V = O, K < E.

19. During inelastic collision between two bodies, which of the following quantities always remain

conserved?

20. A mass of 5 kg is moving along a circular path of radius 1 m. If the mass moves with 300

revolutions per minute, its kinetic energy would be

(a) 250 π2 (b) 100 π2 (c) 5 π2 (d) 0

(a) Total kinetic energy (b) Total mechanical energy (c) Total linear momentum (d) Speed of each

body.

21. A physical quantity X is related to four measurable quantities a, b, c and d as follows:

X=a2b3c5/2d–2.

The percentage error in the measurement of a, b, c and d are 1%, 2%, 3% and 4%, respectively.

What is the percentage error in quantity X ? If the value of X calculated on the basis of the above

relation is 2.763, to what value should you round off the result.

Q-21 to Q-27 carry 2 marks

22. A fighter plane is flying horizontally at an altitude of 1.5 km with speed 720 km/h. At what angle

of sight (w.r.t. horizontal) when the target is seen, should the pilot drop the bomb in order to

attack the target?

23. A boy travelling in an open car moving on a levelled road with constant speed tosses a ball

vertically up in the air and catches it back. Sketch the motion of the ball as observed by a boy

standing on the footpath. Give explanation to support your diagram.

24. The average work done by a human heart while it beats once is 0.5J. Calculate the power used by

heart if it beats 72 times in a minute.

25. Derive work –energy theorem

26. Block A of weight 100 N rests on a frictionless inclined plane of slope angle 30° . A flexible cord

attached to A passes over a frictionless pulley and is connected to block B of weight W. Find the

weight W for which the system is in equilibrium.

27. Derive v2-u2 = 2as using calculus method

Q-28 to 34 carry 3 marks

28. If velocity of light c, Planck’s constant h and gravitational constant G are taken as fundamental

quantities then express mass, length and time in terms of dimensions of these quantities.

29. A girl riding a bicycle with a speed of 5 m/s towards north direction, observes rain falling vertically

down. If she increases her speed to 10 m/s, rain appears to meet her at 45° to the vertical. What

is the speed of the rain? In what direction does rain fall as observed by a ground based observer?

30. An adult weighing 600N raises the centre of gravity of his body by 0.25 m while taking each step

of 1 m length in jogging. If he jogs for 6 km, calculate the energy utilized by him in jogging

assuming that there is no energy loss due to friction of ground and air.

159

Assuming that the body of the adult is capable of converting 10% of energy intake in the form of

food, calculate the energy equivalents of food that would be required to compensate energy

utilised for jogging.

31. A helicopter of mass 2000 kg rises with a vertical acceleration of 15 m s–2. The total mass of the

crew and passengers is 500 kg. Give the magnitude and direction of the (g = 10 m s–2)

(a) Force on the floor of the helicopter by the crew and passengers.

(b) Action of the rotor of the helicopter on the surrounding air.

(c) Force on the helicopter due to the surrounding air.

32. Derive an expression for centripetal acceleration. Also discuss what happens when body moves

in non- uniform circular motion

33. A mass of 6 kg is suspended by a rope of length 2 m from the ceiling. A force of 50 N horizontally

is applied at the mid – point. P of the rope. Calculate the angle of rope makes with the vertical.

Neglect the mass of rope. (g = 9.8 ms–2)

34. (a) A light body and a heavy body have same linear momentum. Which one has greater K.E?

(b) A block of mass M is supported against a vertical wall by a spring of constant k. A bullet of

mass m moving with horizontal velocity v0 gets embedded in the block and pushes it against the

wall. Find the maximum compression of the spring?

35 to 37 carry 5 marks

35. If a body is projected with some initial velocity making an angle θ with the horizontal, show that

its path is a parabola. Then find,

a) The maximum height attained b) Time for maximum height c) Horizontal range

d) Maximum horizontal range e) The time of flight.

36. (a) A block of mass 0.1 kg is held against a wall by applying a horizontal force of 5 N on the block.

If the coefficient of friction between the block and the wall is 0.5, what is the magnitude of the

frictional force acting on the block?

(b) Discuss the motion of a vehicle on a banked road

37. a) Two town A and B are connected by a regular bus service with a bus leaving in either direction

every T min. A man cycling with a speed of 20 kmh–1 in the direction A to B notices that a bus

goes past him every 18 min in the direction of his motion, and every 6 min in the opposite

160

direction. What is the period T of the bus service and with what speed do the buses ply of the

road?

b) An object is in uniform motion along a straight line, what will be position time graph for the

motion of the object if

a) x0 = positive, v = negative v is constant b) both x0 and v are negative v is constant

where x0 is position at t = 0

161

BIBLIOGRAPHY

Ncert Physics- Part –I & II

Concepts Of Physics By H.C. Verma

Exemplar Problems In Physics By Ncert

Support Material By Doe

Websites For Diagrams

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