class xi - regular notes unit 1 chapter 1 physical …...fundamental and derived quantities...

Unit # 1 - Part # 1 Physical World of Measurement(Physical Quantities, Prefixes & Dimensional Analysis)

Prepared By: Er. Veenus Girdhar

www.physicswithveenus.com 2

Part #1 – Physical Quantities and its Types Physical Quantity ♦A quantity which can be measured and by which various physical happenings can be explained and expressed in form of laws is called a physical quantity. For example length, mass, time, force etc. ♦On the other hand various happenings in life e.g., happiness, sorrow etc. are not physical quantities because these cannot be measured. ♦Measurement is necessary to determine magnitude of a physical quantity, to compare two similar physical quantities and to prove physical laws or equations. ♦Any physical quantity is represented completely by its magnitude and unit. For example, your mass is say 60 kg. Here

Ø 60 represents the numerical value of the quantity called mass Ø kg represents the unit of quantity under consideration.

Thus in expressing a physical quantity we choose a unit and then find that how many times that unit is contained in the given physical quantity, ♦Therefore, Physical Quantity (Q) = nu Here, n = the numerical value and

u = the unit. ♦Also, as the unit (u) changes, the magnitude (n) will also change but product ‘nu’ will remain same. Thus, n1u1 = n2u2 And nu = constant

⇒ n∝ 1u

Types of Physical Quantities ♦Physical quantities are of three types

Ø Ratio Ø Scalar Ø Vector

♦Ratio: These are those physical quantities that are ratio of two similar quantities. It has no unit. e.g.

Ø Relative density = Density of object/Density of water at 4oC Ø Refractive index = Velocity of light in air/Velocity of light in medium Ø Strain = Change in dimension/Original dimension

♦ Angle is exceptional physical quantity, which though is a ratio of two similar physical quantities (angle = arc / radius) but still requires a unit (degrees or radians) to specify it along with its numerical value. ♦Scalar: These are those physical quantities that have magnitude only and they do not have any direction e.g. Length, time, work, energy etc. Scalar quantities can be added or subtracted with the help of following ordinary laws of addition or subtraction. ♦Vector: These are those physical quantities that have both magnitude and direction e.g. displacement, velocity, acceleration, force etc. Vector physical quantities can be added or subtracted according to vector laws of addition. These laws are different from laws of ordinary addition. ♦There are certain physical quantities, which behave neither as scalar nor as vector.


For example, Ø Moment of inertia: It is not a vector, as by changing the sense of rotation its

value is not changed. It is also not a pure scalar as it has different values in different directions

Ø Strain: It is not a vector, as it does not have any direction. It is also not a pure scalar because it has different values in different directions

Such physical quantities are called Tensors. Fundamental and Derived Quantities ♦Fundamental quantities: Out of large number of physical quantities, which exist in nature, there are only few quantities, which are

Ø Independent of all other quantities and Ø Do not require the help of any other physical quantity for their definition,

These are called absolute quantities or fundamental or base quantities. ♦All other quantities are based upon and can be expressed in terms of these quantities. ♦Derived quantities: All other physical quantities can be derived by suitable multiplication or division of different powers of fundamental quantities. These are therefore called derived quantities. ♦For example, if length is defined as a fundamental quantity then area and volume are derived from length and are expressed in term of length with power 2 and 3 over the term of length. ♦In mechanics

Ø Length, Ø Mass and Ø Time

are arbitrarily chosen as fundamental quantities. ♦However this set of fundamental quantities is not a unique choice. In fact any three quantities in mechanics can be termed as fundamental as all other quantities in mechanics can be expressed in terms of these. e.g.

Ø If speed and time are taken as fundamental quantities, length will become a derived quantity because then length will be expressed as speed multiplied by time.

Ø If force and acceleration are taken as fundamental quantities, then mass will be defined as Force divided by acceleration and will be termed as a derived quantity.

Fundamental and Derived and Supplementary Units ♦In physics, we choose mass, length and time are arbitrary chosen as fundamental quantities. Their units are called fundamental units. ♦Other units, which can be expressed in terms of fundamental units, are called derived units. ♦In addition to these two, we use two supplementary units, which are:

Ø Radian: It is unit of plane angle. Ø Steradian: It is unit of solid angle.

♦ Apart from fundamental and derived units we also use very frequently practical units. These may be fundamental or derived units e.g., light-year is a practical unit (fundamental) of distance while horse-power is a practical unit (derived) of power. Characteristics of a Standard Unit


♦In early days, people used to measure a distance or length of an object in terms of distance between tip of index finger and thumb in stretched mode, length of human foot etc. Such units of length measurement cannot be considered as standard and hence were discarded with the passage of time. ♦A unit selected, as standard unit must possess the following characteristics

Ø It should be well defined Ø It should be of suitable size i.e. neither too large nor too small in

comparison to quantity being measured. Ø It should be easily reproducible at all places. Ø It should not change with time or place to place. Ø It should be easily accessible. Ø It should not be adhoc or ambiguous in definition.

System of units ♦A complete set of units, both fundamental and derived for all kinds of physical quantities is called system of units. The common systems are:

Ø CGS system: The system is also called Gaussian system of units. In it length, mass and time have been chosen as the fundamental quantities and corresponding fundamental units are centimeter (cm), gram (g) and second (s) respectively.

Ø MKS system: The system is also called Giorgi system. In this system also length, mass and time have been taken as fundamental quantities, and the corresponding fundamental units are metre, kilogram and second.

Ø FPS system: In this system foot, pound and second are used respectively for measurements of length, mass and time. In this system force is a derived quantity with unit poundal or pound

Ø SI system: It is known as International system of units, and is infact extended system of units applied to whole physics. There are seven fundamental quantities and two supplementary quantities in this system. These quantities and their units are given in the following table

The two supplementary quantities and their units are

Ø ♦ Practical units may or may not belong to a system but can be expressed in any system of units e.g., 1 mile = 1.6 km = 1.6 x 103 m. Advantage of SI ♦Following are the main advantages of S.I. system over other systems of units i.e. (C.G.S, FPS and MKS):

Quantity Name of Unit Symbol Length metre m Mass kilogram kg Time second s Electric Current ampere A Temperature kelvin K Amount of Substance mole mol Luminous Intensity candela cd


(a) It is a coherent system of units i.e. a system based on a certain set of fundamental units from which all derived units are obtained by multiplication or division without introducing numerical factors

(b) SI is a rational system of units as it assigns only one unit to a particular physical quantity e.g. Joule is the S.I. unit for all types of energies while MKS units of mechanical energy, heat energy and electrical energy are Joule, calorie and watt hour respectively.

(c) It is an absolute system of units : There are no gravitational units on the system. The use of factor 'g' is thus eliminated.

(d) It is a metric system i.e. the multiples and submultiples of units are expressed as powers of 10.

(e) SI is an extended or broader system of units as it has seven fundamental and two supplementary units.

(f) The symbols fixed by SI are not subject to change whatever may be the language, e.g., unit of mass kilogram having symbol 'kg' has to be written like this only in French or German or Hindi or Punjabi etc.

SI Prefixes ♦The magnitude of physical quantities varies over a wide range e.g.

• The range of distance is from 10–15m (which is separation between two protons) to 1026m (which is distance of a quasar from the earth).

• The range of mass is from 10– 31 kg (which is mass of electron) to 1041kg (which is mass of our galaxy).

♦In physics, certain recommended standard prefixes are: Power of 10 Prefix Symbol

24 yotta Y 21 zetta Z 18 exa E 15 peta P 12 tera T 9 giga G 6 mega M 3 kilo K 2 hecto h 1 deka da

– 1 deci d – 2 centi c – 3 milli m – 6 micro µ – 9 nano n – 12 pico p – 15 femto f – 18 atto a – 21 zepto z – 24 yocto y

Various important Practical Units ♦1 Angstrom (Å) = 10 – 10 m and is approximately the size of an atom.


♦1 fermi (1fm) = 10 – 15 m and is approximately the size of a nucleus. It is also the smallest unit of length. ♦1 micron = 10– 6 m and is approximately the wavelength of visible light. ♦Parallactic Second (Parsec): It is the largest practical unit of distance. It is the distance of an arc of length one astronomical unit (1.496x1011m) subtends an angle of one second of arc. Also;

1 parsec = 3.08x1016m = 3.26 light year ♦Light Year: It is also a practical unit of distance used in astronomy to measure the distance of distant stars. One light year is the distance traveled by light in one year in vacuum or free space i.e.

1 Light Year = (3x108) x (365 x 24 x 60 x 60) m = 9.5 x 1015m ♦Astronomical Unit: It is denoted by A.U. and is defined as the average distance between sun and earth, where

1A.U. = 1.496 x 1011m ≈ 1.5 x 1011m ♦Chandera Shekhar Limit (C.S.L.): It is the largest practical unit of mass, where

1 C.S.L. = 1.4 times the mass of sun ♦1 Metric Ton = 1000kg.♦Atomic Mass Unit (amu): It is not the atomic standard of mass but a practical unit of mass used in atomic and nuclear physics and is the smallest unit. It is defined at present as 1/12th of the mass of one C-12 atom, where

1 amu = 1.67 x 10-27kg♦1 Shake = smallest unit of time = 10-8 sec, it is now obsolete unit of time. Rules For Writing Units ♦The units named to commemorate a scientist are not written with capital initial letter.

Ø The unit of force is written as newton and not as Newton. Ø The unit of energy is written as joule and not as Joule.

♦The symbols for the units named after scientists are usually the first initial letter of their names in capital.

Ø Unit of force is denoted as N, named after Newton. Ø Unit of energy is denoted as J, named after Joule. Ø Unit of pressure is denoted as Pa, named after Pascal.

♦The units do not have plural forms. Ø It is wrong to write 10 newton force as 10 newtons. Ø It is wrong to write 10 joule of energy as 10 joules.

♦No full stops is put between the symbols for the units. ♦The use of double prefixes is conventionally prohibited.

Ø The distance 3x109m will not be written as 3Mkm. Ø However, in some books, 1 pm has been expressed as 1µµm, but such use

should be avoided. ♦More than one parenthesis is not allowed.

Ø Unit of energy per second per unit area cannot be written as J/s/m2, but should be written as Jm-2s-1.


Assignment # 1 Q1. Arrange AU, light year and parsec in ascending order. Ans.: AU < light year < parsec Q2. How many light years make 1 parsec? Q3. How many meter are there in

(a) 1 parsec (b) 1 light year (c) 1 angstrom (d) 1 astronomical units

Q4. Calculate the number of astronomical units in one meter?

Ans.: 1AU =1.49x1011m , therefore 1m = 11.49x1011

AU = 6.67x10−12AU

Q5. How many parsec are there in one meter?

Ans.: 1parsec = 3.08x1016m , therefore 1m = 13.08x1016

parsec = 3.25x10−17 parsec

Q6. How many light years are there in one meter? Q7. How many parsec are there in one AU? Q8. How many light years are there in one AU? Q9. Do Å and AU represent the same units of length? Ans.: No, they are different representations. In fact

(a) 1Å = 10 – 10 m (b) 1AU = 1.5 x 1011m

Q10. Convert the followings: (a) 1nm into km. (b) 1 pico second into microseconds. (c) 1dm into 1Å. (d) 1µF into pF.

Q11. The normal duration of a physics practical period in a school is about 100 minutes. Express this period in micro-centuries, if 1 century = 100 years.

[1.9 micro-centuries.]

Q12. Which of the following is not the unit of length? (a) Light year (b) AU (c) Year (d) Parsec.

Q13. 0.1 cm is also equal to (a) 107 Å (b) 108 nm (c) 10 – 4 km (d) 1012 fermi

Q14. 1 nanometer/1 attometer is equal to (a) 106 (b) 108 (c) 107 (d) 109

Q15. Which of the following is not a unit of time? (a) Parallactic second


(b) Solar day (c) Lunar month (d) Leap year

Q16. Which of the following is not the unit of distance? (a) Angstrom (b) Barn (c) Fermi (d) Parsec

Q17. What is the order size of our galaxy? Ans.: Size of our galaxy is of the order of 1020 meter. Q18. What is the order of mass of universe? Ans.: Mass of universe is of the order of 1055 kg. Q19. Suggest a distance corresponding to each of the following order of length

(a) 107 m (b) 104 m (c) 102 m (d) 10 – 3 m (e) 10 – 6 m (f) 10 – 14 m

Ans.: Suggestion to distance corresponding to each of the following order of length (a) 107 m = Radius of earth (b) 104 m = Height of mount Everest (c) 102 m = Length of hockey field = length of race track = length of cricket field (d) 10 – 3 m = thickness of cardboard (e) 10 – 6 m = mean free path of molecules = wavelength of visible light (f) 10 – 14 m = size of nucleus.

Q20. What is the common difference between bar and torr? Ans.: Both bar and torr represent the unit of pressure, where

(a) 1 bar = 1 atmospheric pressure = 760mm of Hg column (b) 1 torr = 1 mm of Hg column

Therefore, 1 bar = 760 torr Q21. In British engineering system the unit of mass is ‘slug’ while the unit of force is pound. Define 1 slug. How many kg are there in a slug? Ans.: 1 slug is defined as the mass of a body which moves with an acceleration of 1 ft/s2 under the action of force of 1 pound. Also 1 slug = 14.59 kg. Q22. Fill in the blanks:

(a) The volume of a cube of side 1cm is equal to _________ m3. (b) The surface area of a solid cylinder of radius 2.0cm and height 10.0cm is equal

to _____________ (mm)2. (c) A vehicle moving with a speed of 18 km/h covers ________ m in one second. (d) The relative density of lead is 11.3. Its density is __________ g/cm3 or __________

kg/m3. [10– 6 m3; 1.26 x 104 mm2; 5m; 11.3 g/cc or 11.2 x 103 kg/m3]

Q23. Fill in the blanks by suitable conversion of units: (a) 1 kg m2 s– 2 = _______________ g cm2 s– 2 [107] (b) 1m = _______________ light year (c) 3 m s– 2 =______________ km h– 2 [3.888 x 104] (d) G = 6.67 x 10– 11 N m2 kg – 2 = ______________ cm3 s – 2 g – 1. [6.67 x 10– 8]

Q25. Why do we have different units for same physical quantity?


Ans.: The value of a given physical quantity may vary over a wide range. To express the quantity in proper format, we may need different units. Further, a given quantity may be expressed in terms of different quantities. For example:

(a) Kinetic energy of bus, car or athletes are expressed in Joules (b) Kinetic energy of elementary particles such as neutrons or electrons are (c) expressed in MeV or keV or eV etc. (d) Electrical energy is expressed in kWh

This will also lead to different equivalent units of the same quantity. Q36. Express unified atomic mass unit in kg. Q37. How many electrons will make 1 kg? Q38. State True or false:

(a) Radian is the unit of plane angle and it is a supplementary unit. (b) The unit is always written in singular form, e.g. foot not feet. (c) If a unit is named after a person, the unit is not written with capital initial letter. (d) The unit of solid angle is steradian and it is a supplementary unit. (e) If A + B = C − D, then units of A, B, C and D must be same.

Q39. Why Length, mass and time are chosen as base quantities in mechanics? Q40. We know that length vary with temperature. How can then we call it a fundamental quantity? Q41. What are the advantages of defining standard meter in terms of wavelength of light? Q42. We know that length vary with time. How can then we call it a fundamental quantity? Q43. Is the definition of a physical quantity for which no method of measurement is known, has some meaning or not? Q44. What is difference between Nm, mN and nm?


Part # 2 – Dimensional Analysis Introduction ♦The term “dimensions of a physical quantity” refers to the power of fundamental quantities by which they should be raised so as to express given physical quantity. ♦The dimensions of mass, length and time are [M], [L] and [T] respectively. They can also be written as [M1L0T0], [M0L1T0] and [M0L0T1] respectively. ♦There are seven fundamental quantities in SI system of units. These are:

♦The derived units of all the physical quantities can be suitably expressed in terms of the fundamental units of mass, length and time raised to some power. ♦The dimensions of velocities can be obtained from dimensions of fundamental quantities. For example, dimensions of velocity are derived as:

velocity = displacement

time = [L][T]

= [M0LT-1]

Thus, dimensions of velocity are zero in mass, one in length and minus one in time. ♦The expression of [M0LT-1] is called dimensional formula of velocity. ♦The equation obtained, when a physical quantity is equated with its dimensional formula, is known as dimensional equation e.g. dimensional equation of velocity is [velocity] = [M0LT-1] ♦The dimensional formula of any quantity gives following information:

• On what fundamental quantities does the given quantity depends e.g. unit of velocity depends upon the units of length and time and is independent of the unit of mass.

• How a given physical quantity depends upon fundamental quantities e.g. velocity is directly proportional to length but is inversely proportional to time.

♦The dimensional formula is always written in square brackets. ♦The dimensions of a base quantity in terms of other base quantities are always zero. Dimensional formula of various Physical Quantities ♦Few Important dimensions are:

S.No. Physical Quantity Formula used

Dimensional Formula

1. Mass ----- [ML0T0] 2. Length ----- [M0LT0] 3. Time ----- [M0L0T1] 4. Area A = lb [M0L2T0]

Fundamental Quantity

Fundamental Units

Symbols Dimensions

Mass killigram kg [M] Length meter m [L] Time second s [T]

Electric current ampere A [A] or [I] Temperature kelvin K [K]

Luminous Intensity

candela Cd [Cd]

Amount of substance

mole mole [mol]


5. Volume V = lbh [M0L3T0] 6 Speed or velocity V = x/t [M0LT-1] 7. Acceleration A = v/t [M0LT-2] 8. Force* F = ma [MLT-2] 9. Linear Momentum* P = mv [MLT-1] 10. Impulse* I = change in momentum [MLT-1] 11. Work* W = (Force)(distance) [ML2T-2] 12. Energy* E = ½ mv2 [ML2T-2] 13. Power* P = work/time [ML2T-3] 12. Pressure* P = F/area [ML-1T-2] 13. Acceleration due to

gravity* Acceleration [M0LT-2]

14. Moment of force or torque*

τ = Fd [ML2T-2]

15. Couple* -do- [ML2T-2] 16. Angle θ = l/r Dimensionless 17. Stress* Stress = force /area [ML-1T-2] 18. Strain* Strain = ΔL/L Dimensionless 19. Coefficient of Elasticity E = stress/strain [ML-1T-2] 20. Surface Tension* T = Force/length [ML0T-2] 21. Surface energy* Energy/area [ML0T-2] 22. Velocity Gradient** Velocity per unit length [M0L0T-1] 23. Mass Gradient** Mass per unit length [ML-1T0] 24. Pressure Gradient** Pressure per unit length [ML-2T-2] 25. Force Gradient** Force per unit length [ML0T-2] 26. Coefficient of Viscosity*

(η) F = 6πηrv [ML-1T-1]

27. Radius of Gyration Distance [M0LT0] 28. Moment of Inertia* I = mr2 [ML2T0] 29. Angular velocity* (ω) ω = θ/t [M0L0T-1] 30. Angular Acceleration* (α) α = ω/t [M0LT-2] 31. Angular Momentum* (L) L = mvr [ML2T-1] 32. Time Period Time [M0L0T] 33. Frequency ν = 1/T [M0L0T-1] 34. Rate of Flow* Volume per second [M0L3T-1] 35. Wavelength Length of one wave [M0LT0] 36. Wave number Number of waves per meter [M0L-1T0] 37. Force Constant* Force/length [ML0T-2] 38. Reynolds Number It is a pure number Dimensionless 39. Trigonometric relations It is a pure number Dimensionless 40. Exponential functions It is a pure number Dimensionless 41. Logarithmic function It is a pure number Dimensionless 42. Refractive Index It is a pure number Dimensionless 43. Amplitude Maximum displacement [M0LT0] 44. Velocity amplitude** Maximum value of velocity [M0LT-1] 45. Acceleration amplitude** Maximum value of

acceleration [M0LT-2]

46. Density* Density = mass/volume [ML-3T0]


47. Linear mass density** Mass per unit length [ML-1T0] 48. Surface mass density** Mass per unit area [ML-2T0] 49 Heat* It is a form of energy [ML2T-2] 50 Latent heat* (L) Heat = mL [M0L2T-2] 51 Specific Heat* Heat = (mass)(s.heat)(temp) [M0L2T-2K-1]

Some Important Dimensions (Very Important) ♦Distance covered in nth seconds: It refers to the distance covered in one particular second. Therefore;

Sn =distancetime

= [LT −1]

It should be remembered that whenever we there is a quantity of unit magnitude is involved than that final answer is always obtained by dividing by that quantity ♦Electric Potential: Electric potential at a point is defined as work done in bringing unit charge from infinity to that point. Once gain, there is a stress on unit positive charge; therefore, electric potential is not work but it is work divided by charge i.e.

Potential = Workcharge

=ML2T −2⎡⎣ ⎤⎦AT[ ] = ML2T −3A−1⎡⎣ ⎤⎦

♦Surface energy: It is defined as work done in increasing area of surface film by unity against the force of surface tension. Once gain, there is a stress on increasing area by unity (unity means one), therefore;

Surface Energy = WorkArea

=ML2T −2⎡⎣ ⎤⎦L2⎡⎣ ⎤⎦

= ML0T −2⎡⎣ ⎤⎦

♦Avogadro’s constant: It is defined as number of molecules present in one mole. Once gain, there is a stress on one mole; therefore;

Avagadro 's Constant = Number of moleculesnumber of moles

= mol−1⎡⎣ ⎤⎦

♦Planck’s Constant (h): The energy is emitted by a source in the form of packets of energy called photons. The energy of one photo is given by E = hυ where h is Planck’s constant and υ is frequency. Therefore;

Planck 's Constant = EnergyFrequency

=ML2T −2⎡⎣ ⎤⎦T −1⎡⎣ ⎤⎦

= ML2T −1⎡⎣ ⎤⎦

♦Gravitational Constant (G): It is defined using Universal law of gravitation i.e.

F =G Mmr2

⇒ G = Fr2

Mm=

MLT −2⎡⎣ ⎤⎦ L[ ]2M[ ] M[ ] = M −1L3T −2⎡⎣ ⎤⎦

♦Universal Gas Constant (R): It is defined using ideal gas equation, according to which;

PV = nRT ⇒ R = PVnT

=ML−1T −2⎡⎣ ⎤⎦ L3⎡⎣ ⎤⎦mol[ ] K[ ] = ML2T −2K −1mol−1⎡⎣ ⎤⎦

♦Boltzmann Constant (k): t is ratio of universal gas constant (R) and Avogadro’s constant (N) i.e.

k = RN

=ML2T −2K −1mol−1⎡⎣ ⎤⎦

mol−1⎡⎣ ⎤⎦= ML2T −2K −1⎡⎣ ⎤⎦


♦Stefen’s law and Stefen’s constant (σ): Stefen’s law gives expression for energy radiated per second per unit area by any hot black body say a star. According to this law, the energy radiated per second per unit area (E) by a perfectly black body is directly proportional to fourth power of absolute temperature (T) i.e.

E ∝T 4 ⇒ E =σT 4

Therefore; σ = ET 4 =

Energy per sec per unit area( )Temp( )4 =

ML2T −2

T L2⎡

⎣⎢

⎤

⎦⎥

K 4⎡⎣ ⎤⎦= M T −3K −4⎡⎣ ⎤⎦

♦Units of Stefen’s constant:

σ = ET 4 =

Energy per sec per unit area( )Temp( )4 = Joule

sec m2( ) K 4( ) = Js−1m−2K −4 =Watt −m−2K −4

♦Solar Constant (S): It is defined energy radiated per second per unit area by sun. Therefore it is nothing but E in Stefen’s law. Thus;

S = Energy per sec per unit area( ) = ML2T −2

T L2

⎡

⎣⎢

⎤

⎦⎥ = M T −3⎡⎣ ⎤⎦

♦Wein’s constant (b): It is defined by the formula . Thus;

λ = bT

⇒ b = λT = L[ ] K[ ] = LK[ ] Different types of Variables and Constants: ♦From the study of dimensional formulae of physical quantities, we can divide them into four categories:

Ø Dimensional Variables: The quantities that have dimensions and do not have a constant value e.g. area, volume, speed, force etc.

Ø Dimensionless Variables: The quantities that are dimensionless and do not have constant value e.g. angle, specific gravity, refractive index etc.

Ø Dimensional Constants: The quantities that have dimensions and do have constant value e.g. Gravitational constant, Planck’s constant etc.

Ø Dimensionless Constant: The quantities that are dimensionless and do have constant value e.g. π.

♦A physical quantity may be dimensionless but may still have units e.g. angle is dimensionless but have units as radians. ♦A physical quantity that is unitless is always dimensionless. Principle of Homogeneity of Dimensions ♦According to this principle “In a physical relationship involving quantities, the dimensions of physical quantities on the RHS should be equal to the dimensions of quantities on the LHS.” ♦In other words, we can add, subtract or equate only those quantities that have same dimensions. But, we can multiply or divide any two quantities. Four Important Dimensional calculations These are:

Ø [L] + [L] = [L] Ø [L] − [L] = [L] Ø [L][L] = [L2] Ø [L] ÷ [L] = unitless


Assignment # 2 Q1. State True or False:

(a) Two or more physical quantities can be added or subtracted only when their dimensions are same.

(b) If A + B = C − D, then dimensions of A = dimensions of B = dimensions of C = dimensions of D.

(c) Supplementary units have no dimensions. (d) A physical quantity having unit must have dimensions. (e) A unitless quantity is always dimensionless. (f) Velocity and speed have same dimensions.

Q2. Fill in the blanks. (a) The dimensions of power are _____________ (b) The dimensions of angular acceleration are _____________ (c) The dimensions of electric charge are _____________ (d) The dimensions of specific heat are _____________ (e) The dimensions of magnetic force are _____________ (f) The dimensions of latent heat are _____________

Q3. Give an example of (a) A physical quantity which has a unit but no dimensions (b) A physical quantity which has neither unit nor dimensions (c) A constant which has a unit. (d) A constant which has no unit

Q4. Write down the dimensions of (a) Volume of a sphere of radius ‘r’ (b) Volume of a cube of edge length ‘a’ (c) Ratio of volume of a sphere of radius ‘r’ to volume of a cube of edge length ‘a’

Q5. The unit of impulse is the same as that of (a) Moment of force (b) Linear momentum (c) Rate of change of linear momentum (d) Force

Q6. Which pair of quantities has dimensions different from the other three pairs? (a) Impulse and linear momentum (b) Planck’s constant and angular momentum (c) Moment of inertia and moment of force (d) Young’s modulus and pressure

Q7. The dimensions of the coefficient of thermal conductivity are: (a) [MLT-3K-1] (b) [MLT-2K-1] (c) [MLT-1K-1] (d) [MLT-2K-2]

Q8. The SI unit of Stefan’s constant is (a) W s-1 m-2 K–4 (b) J s m-2 K–4 (c) J s-1 m-2 K-1 (d) Wm-2 K–4

Q9. What is the physical quantity whose dimensions are [ML2T-2]? (a) Kinetic energy (b) Pressure (c) Momentum (d) Power

Q10 The dimensions of angular momentum are (a) [MLT-1] (b) [ML2T-1] (c) [ML-1T] (d) [ML0T-2]

Q11. The equation of state for n moles of an ideal gas is PV = nRT; where R is the universal gas constant and P, V and T have the usual meanings. What are the dimensions of R?


(a) [M0LT-2K-1 mol-1] (b) [ML2T-2K-1 mol-1] (c) [M0L2T-2 K-1 mol-1] (d) [ML-2 T-2 K-1 mol-1]

Q12. The gravitational force F between two masses m1 and m2 separated by a distance

r is given by F= 1 22

Gmmr

where G is the universal gravitational constant. What are the

dimensions of G? (a) [M-1 L3T-2] (b) [ML3T-2] (c) [ML2T-3] (d) [M-1L2T-3]

Q13. The SI unit of the universal gas constant R is (a) erg K-1 mol-1 (b) watt K-1 mol-1

(c) Newton K-1 mol-1 (d) joule K-1 mol-1 Q14. According to the quantum theory, the energy E of a photon of frequency υ is given by E = hυ; where h is Plank’s constant. What is the dimensional formula for h?

(a) [M L2 T-2] (b) [M L2 T-1] (c) [M L2 T] (d) [M L2 T2]

Q15. What is the SI unit of Planck’s constant? (a) Watt – second (b) Watt per second (c) Joule – second (d) Joule per second

Q16. The dimensions of Plank’s constant are the same as those of (a) Energy (b) Power (c) Angular frequency (d) Angular momentum

Q17. The dimensional formula of which of the following pair is not same? (a) Impulse and momentum (b) Torque and work (d) (c) Stress and pressure (d) Momentum and angular momentum

Q18. Which of the following does not have the same dimensions? (a) Electric flux, electric field, electric dipole moment. (b) Pressure, stress, Young’s modulus (c) Heat, potential energy and work done. (d) EMF, p.d., electric voltage. (a)

Q19. The ratio of Planck’s constant and moment of inertia has the dimensions of (a) Time (b) Frequency (c) Angular momentum (d) Velocity (b) Q20. E, m, L, G denote energy, mass, angular momentum and gravitation constant

respectively. The dimensions of 2

5 2

ELm G

will be that of

(a) Angle (b) Length (c) Mass (d) Time (a) Q21. Which of the following combination of three dimensionally physical quantities P, Q, R can never be a meaningful quantity?

(a) PQ – R (b) PQ/R (c) (P – Q)/R (d) (PR – Q2)/QR (c)

Q22. If P and Q are having different non-zero dimensions, which of the following operations are possible?

(a) P + Q (b) PQ (c) P – Q (d) 1 PQ

− (b)

Q23. The quantities A and B are related by the relation A/B = m, where m is linear density and A is force, the dimensions of B will be: (a) Same as that of pressure (b) Same as that of work (c) Same as that of momentum (d) Same as that of latent heat


Q24. A dimensionless quantity (a) Never has a unit, (b) Always has a unit

(c) May have a unit (d) Does not exist (c) Q25. A unit-less quantity

(a) Never has a non-zero dimensions (b) Always has a non-zero dimensions (c) May have a non-zero dimensions (d) Does not exist. (a)

Q26. The dimensions [M L – 1 T – 2] may correspond to (a) Work done by a force (b) Linear momentum (c) Pressure (d) Energy per unit volume. (c) and (d)

Q27. Choose the correct statement(s): (a) A base quantity cannot be represented dimensionally in terms of the rest of the

base quantities (b) The dimension of a base quantity in other base quantities is always zero. (c) The dimension of a derived quantity is never zero in any base quantity. (d) All quantities may be represented dimensionally in terms of the base quantities.

[a,b,d] Q28. Which of the following dimensions are correctly matched?

(a) Angular momentum = [M L2 T – 1] (b) Torque = [M L2 T – 2] (c) Stefan’s constant = [M T– 3θ–4] (d) Planck’s constant =[M L2 T – 2] [(a), (b) and (c)]

Q29. Turpentine oil is flowing through a tube of length l and radius r. The pressure difference between the two ends of the tube is p. The viscosity of the oil is given by

( )2 2

4p r x

vlη

−= , where v is the velocity of oil at a distance x from the axis of the tube. The

dimensions of η are (a) [M0L0T0] (b) [MLT1] (c) [ML2T-2] (d) [ML-1T-1] Q30. If σ is Stefen’s constant and b is Wen’s constant then what is dimensions of σb4? Q31. Dimensions of Boltzman constant are same as that of

(a) Pressure density (b) Stefan’s constant (c) Planck’s constant (d) Entropy

Q32. Justify the equation [L] + [L] = [L]. Ans.: In this equation [L] is not a numerical value but it represents a physical quantity called length. Due to this reason, length when added with length will always produces a new quantity called length. Therefore, [L] + [L] = [L]. Q33. Justify the equation [L] − [L] = [L]. Ans.: In this equation [L] is not a numerical value but it represents a physical quantity called length. Due to this reason, length when subtracted from another length will always produces a new quantity called length. Therefore, [L] − [L] = [L].


Applications # 1 of Dimensional Analysis To find dimensions of unknown variables:

Q1. If x = a + bt + ct3, where x is in meter and t is in second, find dimensions of a, b and c. Q2. A position of a particle moving along x-axis depends upon time according to equation x = at2 +bt3, where x is in meter and t is in second. What are the units and dimensions of a and b? Q3*. The velocity of a particle depends upon time t according to the equation

cv a btd t

= + ++ .

Find the dimensions of a, b, c and d.

Q4*. In the Vander Wall’s equation 2 ( )aP V bV

⎛ ⎞+ −⎜ ⎟⎝ ⎠

= RT, what are the dimensions and

MKS units of a and b? Here, P is pressure, V is volume, T is temperature and R is gas constant.

Q5*. Write down the dimensions of a and b in the relation 2b xP

at−

= , if P is power and t

is time.

Q6*. Write down the dimensions of ‘ab’ in the equation 2b xE

at−

= Where E is energy, x is

displacement and t is time.

Q7*. Write down the dimension of a/b in the relation; 2a tP

bx−

= Where P is pressure, t is

time and x is distance. Q8*. Write down the dimensions of a/b in the relation F = a√x + bt2 where F is force, x is distance and t is time. Q9**. Find x in the equation: (Velocity)x = (pressure diff.)3/2(density)-3/2.

Q10** Identify the physical quantity X defined by the relation 2

3

IFvXWL

=.Where, F is

force, I is moment of inertia, v is speed, W is work and L is length.

Q11*. The number of particles given by; 2 1

2 1

n nn Dx x

⎛ ⎞−= − ⎜ ⎟−⎝ ⎠

Are crossing a unit area

perpendicular to the x-axis in unit time. If n1, n2 are number of particles per unit volume for the values of x meant to be x1 and x2, what is the dimension of diffusion constant D?

Q12**. In the equation: ( )( )

sin lnvF A Bt

C Dx= + ; if F is force, v is velocity and x is

distance; find the dimensions of A, B, C and D.

Q13**. In the equation: ( )B lnvX AtCt

= + ; if v is velocity, t is time and x is distance; find

the dimensions of A, B, C and D.

Q14*. The dimensions of a/b in the equation; 2a tP

bx−

= where P is pressure is:

(a) [M2 L T– 3] (b) [M T– 2] (c) [L T– 3] (d) [M L3 T– 1] (b)


Q15**. Pressure depends upon distance as exp zPk

α αβ θ

−⎛ ⎞= ⎜ ⎟⎝ ⎠

, where α and β are

constants, z is distance, k is Boltzmann’s constant and θ is temperature. The dimensions of β are

(a) [M0 L0 T0] (b) [M– 1 L– 1 T– 1] (c) [M0 L2 T0] (d) [M– 1 L– 1 T2] (c) Q16**. In the relation F = αt –1 + βt2, t denotes time and α and β are constants. Dimensions of α and β are (a) [M L T – 3] and [M L T – 4] (b) [M L T – 1] and [M L T – 3] (c) [M L T – 1] and [M L T – 4] (d) [M L T – 2] and [M L T – 4]

Q17*. The displacement x in time t is given by the relation; 2t a bx tb c+

= + , where a, b

and c are constants. The dimensions of a, b and c are: (a) T, L T – 1 and L– 2 T3 (b) T, L– 1 T and L2 T– 3 (c) T, L– 1 T and L– 2 T3 (d) T, L– 1 T – 1 and L – 2 T3 Q18**. The number of particles crossing the unit area perpendicular to the Z-axis per

unit time is given by the relation: 2 1

2 1

N NN DZ Z−

= −−

, where N2 and N1 are number of

particles per unit volume at Z2 and Z1 respectively. What is the dimensional formula for D?

(a) L2 T 2 (b) L – 2 T – 1 (c) L2 T – 1 (d) L – 2 T1

Q19**. A hypothetical experiment conducted to find Young’s modulus 3

cosxTYLτ θ

=

where τ is torque and L is length then find x is (a) 1 (b) – 1 (c) Zero (d) 2 (c) Q20**. In relation P = P0exp(– a t2) where t is time, the dimensions of a is (a) T (b) T – 2 (c) T2 (d) Depends upon P

Q21**. In the relation 1

2sin 1

2ndx xa

aax x− ⎡ ⎤= −⎢ ⎥⎣ ⎦−

∫ ; value of n is

(a) 0 (b) – 1 (c) 1 (d) None of these (a)

Q22**. In the relation2

1

2 4sin 1

2ndx xa

aax x− ⎡ ⎤

= −⎢ ⎥− ⎣ ⎦

∫ ; value of n is

(a) 0 (b) – 1/2 (c) ½ (d) None of these (b)

Q23**. In the relation 1

2sin 1

2xdt ta

aat t− ⎡ ⎤= −⎢ ⎥⎣ ⎦−

∫ ; value of x is

(a) 0 (b) – 1 (c) 1 (d) None of these (a)

Q24**. If !"!!!!"!

= 𝑎! sin!! !!

!; x = distance, then n equals

(a) 1 (b) – 1 (c) – ½ (d) ½ (c) Q25**. Suppose a physical quantity can be dimensionally represented in terms of M, L and T, that is [x] = MaLbTc. The quantity mass;

(a) Can always be dimensionally represented in terms of L, T and x.


(b) Can never be dimensionally represented in terms of L, T and x. (c) May be dimensionally represented in terms of L, T and x if a = 0. (d) May be dimensionally represented in terms of L, T and x if a ≠ 0. (d)

Q26**. In the formula aRTVnRTP e

V b−

=−

; find the dimensions of a and b where P is

pressure, n is number of moles, T is temperature, R is universal gas constant and V is volume. [M L5 T – 2mol – 1] and [L3]

Q27**. If MghynxTnx TP e

V−

= , where n is number of moles, P is pressure, T is temperature, V

is volume, M is mass, g is acceleration due to gravity and h is height. Find dimensions of x and y. [M L2 T – 2mol – 1] and y = 1.

Q28**. If force 2

brKeFr

−

= varies with distance r. Then write the dimensions of K and b.

Q29**. The formula W = (F + 2Ma)vn where W is work done, M = mass, a = acceleration and velocity can be made dimensionally correct for

(a) n = 0 (b) n = 1 (c) n = – 1 (d) For no value of n. (d) Q30**. The mass of a liquid flowing per second per unit area of cross-section of the tube is proportional to (pressure difference across the ends)n and (average velocity of the liquid)m. Which of the following relation between m and n is correct? (a) m = n (b) m = – n (c) m2 = n (d) m = – n2

Q31**. The van der Waals’ equation for a gas is 2

a + V

P⎛ ⎞⎜ ⎟⎝ ⎠

(V – b) = nRT

Where P, V, R, T and n represent the pressure, volume, universal gas constant, absolute temperature and number of moles of a gas respectively, ‘a’ and ‘b’ are constants. The ratio b/a will have the following dimensional formula: (a) [M-1 L-2 T2] (b) [M-1 L-1T-1] (c) [ML2 T2] (d) [MLT-2]

Q32**. The velocity v of a particle at time t is given by v = batt c

++

then the

dimensions of a, b, c are respectively; (a) LT–2, L, T (b) L2, T and LT2 (c) LT2, LT and L (d) L, LT and T2


Applications # 2 of Dimensional Analysis To check Correctness of a Physical Relation

Q1. Check the correctness of following relations;

(a) 212

S ut at= +

(b) 2 2 2v u as− =

(c) hmv

λ = ; where h is Planck’s constant, m is mass, v is velocity and λ is wavelength.

(d) 2 cosThr

θρ

= , where h is height of capillary rise, T is surface tension, θ is angle of

contact, r is radius of capillary tube and ρ is density of liquid.

(e)** (2 1)2nthaS u n= + −

(f)* 12

TL m

γ = ; where γ is frequency, L is length, T is tension in the string and m is

mass per unit length of string.

(g)* 1 Kv

dλ= ; where v is velocity of longitudinal waves, λ is wavelength of wave, K is

coefficient of volume elasticity and d is density of the medium.

(h)* 2

2GMvR

= ; where v is escape velocity, G is Gravitational constant, M is mass and R

is radius of planet. Q2. The volume V of water which passes any point of a canal during t seconds is connected with the cross-section A of the canal and velocity u of water by the relation V = KAtu, where K is dimensionless constant. Verify the correctness of the relation.

Q3*. The escape velocity from the surface of earth is given byGMvR

= , where M is

mass and R is radius of earth. Check the correctness of the formula. Q4. The viscous force F acting on a body of radius r moving with a velocity v in a medium of coefficient of viscosity is given by F = 6πηrv. Check the correctness of the relation. Q5*. The rate of flow (V) of a liquid through a pipe of radius r under a pressure gradient

(P/L) is given by 4Pr

8V

Lπη

= where η is coefficient of viscosity of the liquid. Check

whether the formula is correct or not.

Q6*. Test if the equation12

mgLI

γπ

= is dimensionally correct. Here, γ is frequency and I

is moment of inertia.


Applications # 3 of Dimensional Analysis To Derive a Physical Relation

Important Note: Dimensional analysis can be used to derive a relation for a physical quantity, if it depends upon three or less than three physical quantities. However; it cannot be used to derive a physical relation, if the given physical quantity depends upon more than four quantities and if physical relation involves sum or difference of physical quantities. Q1. The time period T of a simple pendulum may depends upon (a) mass of bob ‘m’ (b) Length of thread ‘L’ and (c) acceleration due to gravity ‘g’. Deduce, dimensionally, the formula for T. Q2*. The frequency of vibration (γ) of a stretched string depends upon length of string (L), tension (T) in the string and mass per unit length (m) of the string. Using the method of dimensions, derive the formula for γ. Q3. The centripetal force F acting on a particle moving uniformly in a circle may depend upon mass (m), velocity (v) and radius (r) of the circle. Derive the formula for F using the method of dimensions. Q4*. If force, velocity and time are taken as fundamental quantities, what would be the dimensions of work? [Fvt] Q5*. If the units of force, length and time are taken as fundamental units; what would be the dimensions of mass? [FL–1T2] Q6*. A jet of water of cross-sectional area A and velocity v impinges normally on a stationary flat plate. The mass per unit volume is ρ. By dimensional analysis, determine the expression for the force F exerted by the jet against the plate. [KρAv2] Q7. Experiments shows that, the frequency (n) of a tuning fork depends upon length (L) of the prong, density (d) and the Young’s modulus of elasticity (Y) of material. On the basis of dimensional analysis, derive an expression for the frequency of tuning fork.

[K YnL d

= ]

Q8*. Calculate the dimensions of linear momentum and surface tension in terms of velocity, density and frequency. {[ρv4γ–3] & [ρv3γ–1]} Q9*. The wavelength (λ) of a matter wave depends upon Plank’s constant (h), mass (m)

and velocity (v) of particle. Use method of dimensions to derive the formula. [Khmv

λ = ]

Q10. The velocity (v) of a transverse wave on a string may depends upon length (l) of the string, tension (T) in the string and mass per unit length (m) of the string. Using

dimensional analysis, obtain the expression fro velocity. [Tv km

= ]

Q10. The centripetal force acting on a body may depend upon mass (m) of body, radius of body (r) and angular velocity (ω). Derive formula for F dimensionally. [F = mω2r] Q11*. Assuming that the mass of the largest stone that can be moved by a flowing river may depend upon velocity v, the density ρ and acceleration due to gravity g, show that m varies as the sixth power of velocity of flow. Q12*. The frequency (n) of an oscillating drop may depend upon radius (r) of the drop, density (ρ) of liquid and surface tension (S) of the liquid. Deduce the formula

dimensionally. [ 3

Sv krρ

= ]


Q13*. Using the method of dimensions, derive an expression for rate of flow (V) of a liquid through a pipe of radius (r) under a pressure gradient (P/l). Given that V also

depends upon the coefficient of viscosity (η). [4PrV klη

= ]

Q14. The depth x to which a bullet can penetrate in a human body depends upon coefficient of elasticity (η) and kinetic energy (E). Establish the relation between these

quantities using the method of dimensions. [ ( )1/3/x E η∝ ]

Q15*. Assuming that the critical velocity (vc) of a viscous liquid flowing through a capillary tube depend upon the radius (r) of the tube, density (ρ) of the liquid and coefficient of viscosity (η) of the liquid. Use methods of dimensions to obtain the

relation between these quantities. [ cv rηρ

∝ ]

Q16*. Assuming that the in case of motion of blunt bodies in air, aerodynamical drag D depends upon effective area of the body (A), the speed v of the body relative to air and density of air (σ); show by methods of dimensions that D ∝ σAv2. Q17. A large fluid star oscillates in shape under the influence of its own gravitational field. Using dimensional analysis, find the expression for time period of oscillations (T) in terms of radius of star (R), mean density of fluid (ρ) and universal gravitational constant (G). [T = K(Gρ)– 1/2] Q18**. If Planck’s constant (h), universal gravitational constant (G) and speed of light are taken as standard; then obtain the dimension of mass, length and time respectively in terms of h, G and c. Q19**. Assuming that the time period of oscillations (T) of bubble produced under water due to explosion under water may depend upon (a) radiation pressure (P); total energy of explosion (E) and density of liquid (ρ); deduce the expression for T. Q20**. The viscosity η of a gas depends upon mass m, effective diameter D and mean speed v of molecule present in gas; then

(a) 𝜂 = 𝑘𝑚𝑣!/𝐷! (b) 𝜂 = 𝑘𝑚𝑣/𝐷! (c) 𝜂 = 𝑘𝑚𝑣/𝐷! (d) 𝜂 = 𝑘𝑚𝑣!/𝐷! (b)

Q21**. The form of dependence of the lift force per unit wingspan F on an aircraft wing of width L, velocity v and air of density 𝜌 is

(a) 𝐹 = 𝑘𝐿𝑣!𝜌 (b) 𝐹 = 𝑘𝑣!𝜌 (c) 𝐹 = 𝑘𝐿!𝑣!𝜌! (d) 𝐹 = 𝑘𝐿𝑣𝜌! (a)

Q22**. If velocity v, acceleration a and density d are taken as fundamental quantities, then dimensional formula of kinetic energy is

(a) [v3a –3d1] (b) [v8a –4d1] (c) [v8a –3d3] (d) [v8a –3d1] Q23**. If velocity v, acceleration a and force F are taken as fundamental quantities, then the dimensions of Young’s modulus of elasticity Y will be: (a) [F a2 v-2] (b) [F a2 v-3] (c) [F a2 v-4] (d) [F a2 v-5] Q24**. Frequency n of a tuning fork depends upon length L of its prongs, density ρ, and modulus of elasticity E of the material, then (a) n ∝ E (b) n ∝ E1/2 (c) n ∝ E-1/2 (d) n ∝ E-1


Q25**. If P represents radiation pressure, C represents speed of light and Q represents radiation energy striking a unit area per second, then the non-zero integers, x, y and z, such the PxQyCz is dimensionless are (a) x = 1, y = 1, z = 1 (b) x = 1, y = -1, z = 1 (c) x = -1, y = 1, z = 1 (d) x = 1, y = 1, z = -1 Q26**. If the energy, E = Gp hq cr, where G is the universal gravitational constant, h is the Planck’s constant and c is the velocity of light, then the values of p, q and r are, respectively (a) -1/2, 1/2 and 5/2 (b) 1/2, -1/2 and –5/2 (c) -1/2, 1/2 and 3/2 (d) 1/2, -1/2 and –3/2 Q27**. The time period of oscillations of body under SHM is represented by T = PaDbSc, where P is pressure, D is density and S is surface tension, then values of a, b and c are: (a) – 3/2, ½, 1 (b) –1, –2, 3 (c) ½, -3/2, -1/2 (d) 1, 2, 1/3 Q28**. A gas bubble from an explosion under water oscillates with a time period T proportional to PadbEc, where P is static pressure, d is density and E is total energy of explosion. The values of a, b and c are: (a) 0, 1, 2 (b) 5/6, -1/2, 1/3 (c) –5/6, ½, 1/3 (d) –5/6, -1/2, 1/3

Applications # 4 of Dimensional Analysis To convert one system of units to another

♦This is based on the fact that magnitude of a physical quantity remains the same, whatever be the system of measurement i.e. n1u1 = n2u2 Where u1 and u2 are two units of measurement and n1 and n2 are their respective numerical values. ♦If M1, L1 and T1 be the fundamental units of mass, length and time in u1 system of units and the corresponding fundamental units in system u2 be M2, L2 and T2, then u1 = [M1aL1bT1c] And u2 = [M2aL2bT2c] Then; by definition;

1 1 12 1

2 2 2

a b cM L Tn nM L T⎡ ⎤ ⎡ ⎤ ⎡ ⎤

= ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

Q1*. Convert 1 newton into dyne, where newton is SI unit of force and dyne is CGS unit of force. Q2*. Convert 1 joule into ergs, where joule is SI unit of energy and ergs is its CGS unit. Q3*. Find the value of force of 100N on a system based upon the meter, the kilogram and minute as fundamental units. Q4. The velocity of sound in air is 332 m/s. If the units of length are in km and unit of time is in hour, what would be the value of velocity? Q5. Young’s modulus of steel is 19 x 1010 N/m2. Express it in CGS unit. Q6. Convert a velocity of 72 km/h into m/s. Q7. A body has an acceleration of 10km/h. Find its value in CGS system. Q8*. The value of Stefan’s constant is σ = 5.67 x 10–8 Js–1m–2 K–4. Find its value in CGS system.


Q9. Convert a power of one mega watt on a system whose fundamental units are 10kg, 1dm and 1 minute. Q10. Convert one atmospheric pressure (1 N/m2) into dyne/cm2. Q11**. When 1 meter, one kg and one minute are taken as fundamental units, the magnitude of force is 36units. What is the value of this force in CGS system? Q12*. If the units of force, energy and velocity are 10N, 100J and 5m/s, find the unit of length, mass and time. Q13**. In a system called 1 starsystem, we have 1 starkillogram = 1020 kg, 1 starmeter = 108m and 1 starsecond = 103 sec, then in this system 1J equals (a) 10 – 22 starjoule (b) 10 – 30 starjoule (c) 10 22 starjoule (d) 1030 starjoule (b) Q14*. If unit of mass, length and time are all doubled, how will the unit of energy be affected? Limitations of Dimensional Analysis The dimensional analysis has the following limitations ♦This method gives us no information about the dimensionless constants in the formula e.g. 1, 2. 3…π, e etc. ♦If a quantity depends upon more than three factors, having dimensions, the formula cannot be derived. ♦We cannot derive the formula containing trigonometrical, logarithmic and exponential functions, as they are dimensionless. ♦The method of dimensions cannot be used to derive physical relations involving sum and difference of two quantities i.e. it can not be used to derive the relation S = ut +

212at .

♦It only tells us that the given relation is dimensionally correct. It may or may not be experimentally correct. ♦The dimensions of a quantity does not give any information about the type of quantity i.e. it does not tell us if a quantity is scalar or vector. ♦When dimensions are given, the quantity may or may not be unique i.e. different physical quantities can have same dimensions

Additional Questions Q1**. The method of dimensional analysis can be used to derive which of the following relationship: (a) y = a sin(wt + kx) (b) N = N0 e – λt (c) E = ½(mv2) + ½(Iω2) (d) None of these. Q2**. In two systems of units, the relation between velocity, acceleration and force is

given by 2

12vv ετ

= ; 2 1a a ετ= and 12

FFετ

= where ε and τ are constants then find in this

system:

(a) 2

1

mm

2 2

1ε τ⎡ ⎤⎢ ⎥⎣ ⎦

(b) 2

1

LL

3

3

ετ⎡ ⎤⎢ ⎥⎣ ⎦


Q3**. The units (U) of velocity, acceleration and force in two systems are related as

under:

2'v vU Uα

β= ; ( )'

a aU Uαβ= ; ' 1F FU U

αβ⎛ ⎞

= ⎜ ⎟⎝ ⎠

. All the primed symbols (U’) belong to one

system and un-primed ones (U) belong to the other system. Also, α and β are constants.

How momentum units of the two systems are related? '3

1p pU U

β

⎡ ⎤⎛ ⎞=⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

Q4**. In two systems of units; relations among velocity, acceleration and forces are 2

2 1v vαβ

= , a2 = αβ a1 and F2 = F1/αβ respectively. If α and β are constants then relation

among mass, in two systems are

(a) M 2 =αβM1 (b) 2 12 2

1M Mα β

=

(c) M 2 =α 3

β 3 M1 (d ) M 2 =α 2

β 2 M1

End of Chapter # 1

class xi - regular notes unit 1 chapter 1 physical …...fundamental and derived quantities...

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