physics notes

H2 Physics (J1 Only) Prepared by Ang Ray Yan (HCI 11S7B) All Rights Reserved

of 34

Foreword

I feel that amongst all Singapore students, many of us may not have the privilege of receiving quality education in the subject of physics due to differing teaching pedagogies used by various institutions and teachers/mentors.

Despite my limited ability, I hope that these notes will assist you in your learning journey for physics, be it the ‘A’ you are aiming for, or to sustain your genuine interest in the subject.

Unlike many other subjects, physics has

apparently been one where many misconceptions

arise. Furthermore, being an applied subject, it is

one where memorizing gets you the ‘U’. It is the

understanding, deduction and math that count.

I do hope you see where physics is around you in

this world. From your air-conditioners to cars to

infrastructure, physics is everywhere. If you can

learn to appreciate the greatness of mankind’s

inventions, surely you can appreciate the beauty

of physics.

With that understanding, I wish you all the best for H2 Physics for your promotional exams.

Ang Ray Yan

Hwa Chong Institution (11S7B)

Disclaimers / Terms and Conditions

- Physics needs tons of practice. This note gets you the ‘U’ grade if you only read it.

- g on Earth is defined as 9.81ms-2 unless specified otherwise.

- There might be errors. Please use some discretion when reading through. This note is definitely not the best.

- Definitions are given in boxes

- Even at A’ levels, due to the nature of the subject, only concepts appear here.

- Drawing and graphs are equally important in terms of scoring. It is after all, representations and interpretations of our real world.

- The use of the any calculator is not covered in this note. It is assumed that you have prior knowledge on its use.

- I don’t believe in strong O’ level concepts because I learnt little in my high school years. This note tries to include the basics.

- All content in this set of notes may or may not be accurate in the real world since most of it comprises classical mechanics.

- These notes serve main as a concept check. Applying the concepts is another issue.

- Distribute only to students by email or thumbdrive. The usage of these notes by any school or tuition teacher is strictly prohibited.

- This is meant for J1 students only. I strongly recommend all J2 students to practice on problems instead of wasting time here.

- If you bought a copy of this, please ask for a refund. It is free!


of 34

Contents Page

Measurements 3r-6r

- Physical Quantities and Units 3r-4r

- Errors and Uncertainties 4r-5l

- Precision and Accuracy 5r

- Random and Systematic Errors 5r

- Scalars and Vectors 6l-6r

- Rules of Significant Figures 6r

- Homogeneous Equation 6r

- Dimensional Analysis 6r

Kinematics 7l-9r

- Terminologies 7l

- Describing Motion with Diagrams 7l-7r

- Describing Motion with Graphs 7r

- Equations of Motion 8l

- Free-fall Bodies 8l

- Drag Force 8r

- Projectile Motion 8r-9r

Dynamics 9r-14l

- Types of Forces 9r-10l

- Newton’s 3 Laws 10l-11l

- Conservation of Linear Momentum 11r-12r

- Collisions 12r-13r

- Coefficient of Restitution 13r

- Static and Kinetic Friction 14l

Forces 14l-15r

- Hooke’s Law 14l

- Upthrust / Buoyant Force 14l-14r

- Translational Equilibrium 15l

- Moments 15l

- Rotational Equilibrium 15l

- Static and Dynamic Equilibrium 15l

- Three-force Systems 15r

- Couples 15r

Work, Energy, Power 14r-16l

- Definitions 14r

- Work done 15l

- Mechanical Energy 15l-15r

- Conservation of Energy 16l

- Power and Efficiency 16l

Circular Motion 16r-18l

- Kinematics of Circular Motion 16r-17l

- Uniform Circular Motion 17l

- Centripetal Acceleration / Force 17l-17r

- Vertical Circular Motion 17r-18l

Gravitation 18l-21r

- Law of Universal Gravitation 18l-19l

- Geostationary Satellites 19l

- Gravitational Field Strength 19r-20l

- Weightlessness 20l

- Gravitational Potential 20l

- Gravitational Potential Energy 20r-21l

- Escape Speed 21r

- Binary Star System 21r

Oscillations 22l-24l

- Introduction 22l

- Simple harmonic Motion (S.H.M) 22l-23l

- Damping 23l-23r

- Resonance 23r-24l

Waves 24r-27l

- Introduction (Terms and Graphs) 24r

- Wave Equation 24r-25l

- Transverse vs. Longitudinal Waves 25l-25r

- Phase Difference 26l

- Electromagnetic Waves 26l-26r

- Intensity of Waves 26r

- Polarization 27l


of 34

Superposition 27r

- Principle of Superposition 27r

- Interference 27r-28l

- Diffraction and

Huygen’s-Fresnel Principle 28l-28r

- Young’s Double Slit Experiment 28r-29l

- Diffraction Grating 29r-30l

- Stationary Waves 30l-30r

- Stretched Strings 30r-31l

- Air Columns 31l-31r

Miscellaneous 33l-34l

- Useful Knowledge / Summary 33l-34l

Credits 34l

Chapter 1: Measurements

1.1 Physical Quantities and Units

Physical quantities are properties that can be measured/calculated and expressed in numbers.

1.1.1 International System of Units (SI)

Established in 1960 by the 11th General

Conference on Weights and Measures, the

following are the 7 SI Base Quantities and Units:

Base Quantity Base Unit Symbol

length metre m

Length of path travelled by light in vacuum during a time interval of 1/(299,792,458) of a second.


Mass kilogram kg

Mass of the international prototype of kilogram (made of platinum-iridium, kept at BIPM)


Time second s

Duration of 9,192,631,770 periods of the radiation corresponding to the transition between 2 hyperfine levels of the ground state of the Casesium 1333 atom.


Electric Current Ampere A

The constant current which, if maintained in 2 straight, parallel, 1m apart conductors of infinite length and negligible circular cross section, would produce between the conductors a force equal to 2 x 10-7 Nm-1.


Thermodynamic Temperature

Kelvin K

It is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.


Amount of Substance

mole mol

The amount of substance in a system containing as many elementary entities as there are atoms in 0.012kg of carbon-12.


of 34


Luminous Intensity

Candela cd

Candela is the luminous intensity in a given direction of a source emitting monochromatic radiation of frequency 540 x 1012 Hz having a radiant intensity in that direction of 1/683 Wsr-1

From the base quantities and units, we can obtain

derived quantities and units:

Quantity Formula Units Usual Units

Volume - -

Density

-

Velocity

-

Acceleration

-

Force

Momentum Pressure

Energy - Moment -

Power

Electric Charge

Voltage

V

Resistance

Frequency

1.1.2 Prefixes (Common Ones only)

Factor Prefix Symbol

pico p

nano n

micro

milli M

centi C

deci D

- -

kilo K

mega M

giga G

tera T

1.1.3 Rapid Estimation

Known as ‘Fermi’ problems, estimation uses

simple numbers (e.g. 2, 5) with the correct order

of magnitude (e.g. 10-3 or 104):

This goes by a 3-step process:

First, identify the unknown:

Next, identify the known:

Lastly, find relation between known and unknown:

Note: Actual area of Singapore is 682.7km2.

1.2 Errors and Uncertainties

The experimental error in measuring a physical quantity can be interpreted as the difference in the measured and true value of it.

Do note that we usually do not know what the

true value is. (Hence the need to measure)


of 34

1.2.1 Absolute Uncertainty

Hence, we express values as such:

Where is the measured value and is the

estimated certainty. In this case, is the

absolute uncertainty of .

Do note that each reading taken has its own

estimated uncertainty.

Do note that all absolute uncertainties should

have 1 significant figure and should have the

same decimal places as .

1.2.2 Fractional / Percentage Uncertainty

1.2.3 Combining Uncertainties

| |

For any other functions, we calculate as follows:

Examples of Z include sine, cosine, and any

function with a maximum and minimum.

1.3 Precision and Accuracy

Precision refers to the closeness of a set of measurements.

Accuracy refers to the agreement between the measured and true value of a quantity.

The following illustrates the idea:

Target Accurate Precise

no yes

yes no

yes yes

no no

1.4 Random and Systematic Errors

Random errors occur as a scattering of readings about the average value of measurements. They have varying signs and magnitudes. It can only be reduced by combining measurements (e.g. thickness of 100 A4 paper, not 1) or by repeating measurements and taking averages.

Systematic errors occur as a shift of value from the true value of measurements. They have similar signs and magnitudes. It can eliminated by accounting for it (e.g. zero errors in instruments, calibration etc.)


of 34

1.5 Scalars and Vectors

Scalars only have magnitude.

Vectors have magnitude and direction.

The parallelogram law of addition (left)

demonstrates how to sum 2 vectors, and the

polygon law of addition shows how to sum all

vectors, giving the resultant vector. Also,

| |

1.5.1 Finding Resultant Vectors

To find the resultant vector (green):

(

)

1.5.2 Resolution of Vectors

As shown above, a vector can be resolved into the

vertical and horizontal components. This is to

ensure that they the components are

perpendicular and independent of each other.

Usually, rightwards and upwards is positive and

leftwards and downwards is negative. However,

the question definition takes priority.

1.6 Rules of Significant Figures

This is generally summarized into 3 rules:

For multiplication/division, use least s.f. for result.

For addition/subtraction, use least d.p. for result. For logarithms, the number of s.f. we take logarithms is the number of d.p. for the solution:

1.7 Homogeneous Equation*

Homogeneous equations are equations where units on LHS=RHS.

There are 2 reasons why a homogeneous

equation may not be physically correct:

Coefficient

Missing terms

1.8 Dimensional Analysis*

Dimensions correspond directly with base units.

For instance:


of 34

Chapter 2: Kinematics

Kinematics (Greek: ) is the branch of

mechanics that describe motion.

2.1 Terminologies

Distance [x] (red) is the total length travelled by a moving object irrespective of direction of motion.

The displacement [s] (black) from a reference point, O, is the linear distance and direction of the object from O.

The speed of an object is the rate of change of distance travelled by an object with respect to time.

The velocity of an object is the rate of change of displacement with respect to time.

The acceleration of an object is the rate of change of velocity with respect to time.

∫ ∫

2.2 Describing Motion with Diagrams

2.2.1 Ticker Tape Diagrams

A ticker places a tick on tape dragged by a moving

object. The distance between dots represents the

object’s position change during a defined time

interval (e.g. 0.1s). Hence, we see the top object

travels at a constant speed, whilst the bottom is

accelerating.

2.2.2 Vector Diagrams

Vector arrows are used to depict direction and

relative magnitudes of an object’s velocity. Thus,

we see the top object travels at a constant speed,

whilst the bottom is accelerating.

2.2.3 Stroboscopic Photographs

Stroboscopic photographs are photos taken by

cameras with an open shutter. With a flashing

light at fixed frequencies, a fixed duration

between illuminations produces the different

positions of the object.

2.3 Describing Motion with Graphs

2.3.1 Displacement Time Graphs (s-t)

The gradient of the graph

gives the instantaneous

velocity (reddish-brown

at ). The slope of

connecting line gives the

average velocity. (green

between and ).

2.3.2 Velocity Time Graphs (v-t)

The gradient of the graph

gives the instantaneous

acceleration (reddish-

brown at ). The slope of

connecting line gives the

average acceleration.

(green between and ). The area under the

graph gives the displacement.


of 34

2.4 Equations of Motion

All equations here assume a constant

acceleration.

Hence, we can deduce the equation of velocity:

∫

Finally, we can integrate to get the displacement:

∫

To express displacement in terms of velocity,

2.5 Free-fall Bodies

Freely falling objects is any object moving only under the influence of gravity (i.e. ignore air resistance etc.). They accelerate downwards at 9.81ms-2

2.6 Drag Force

When a body moves through liquid or gas, a drag

force is experienced. It depends on the velocity of

the body. Other factors include shape and

dimension of the body and the viscosity of liquid.

Source:http://www.equipmentexplained.com/images/physics_image

s/fluid_images/flow_images/basics/laminar_turbulent_flow.gif

For Laminar flow (low velocity), the drag force (FD)

is given by:

For turbulent flow (high velocity), the drag force

(FD) is given by:

Hence, applying it to air resistance, we have the

following:

For a body in free fall with air resistance,

the drag force will increase until it is

equal to the weight. Since the net force

will be zero, the object reaches terminal

velocity:

2.7 Projectile Motion

Using the resolution of vectors, we know that the

horiztonal and vertical motions are independent

of each other.

Using the equations of motion,

Horizontal Vertical


of 34

Hence, the impact velocity is given by:

√

2.7.1 Maximum Height (H)

At maximum height,

2.7.2 Duration of Flight (tflight)

Assuming projectile lands on a level ground as it is

initially fired at:

2.7.3 Horizontal Range (R)

For horizontal motion, we know that:

Hence, to get maximum range,

2.7.4 Trajectory Equation

Trajectories are parabolic, as proven below

(

)

(

)

(

)

2.7.5 Projectile Motion (with Air Resistance)

On the flight upwards, air resistance acts in the

same direction as weight, hence the maximum

height is lowered (total downward force larger).

On the flight downwards, air resistance acts in the

opposite direction as weight. Hence, time to

travel up is greater than time to travel down.

Also, note that the path is asymmetrical and the

horizontal range is lower.

Chapter 3: Dynamics

This topic studies the cause of motion and

changes in motion due to forces.

3.1 Types of Forces

There are contact forces and non-contact forces.

Contact forces are in physical contact.

Contact forces are in not in physical contact, and act at a distance.


of 34

3.1.1 Centre of Gravity

The centre of gravity (cg) is the point at which the weight of an object appears to act on.

Suppose we have particles

denoted by and their

mass is , then it follows

that:

∑

∑

3.1.2 Contact Force and Friction

The normal contact force is due to electrostatic

repulsion between molecules of the surface and

the object. It balances the weight, directed

perpendicular to the surface.

Friction always acts in the opposite direction of

relative motion. It will be discussed further in 3.6.

3.2 Newton’s 3 Laws

3.2.1 Newton’s First Law

Newton’s First law states that a body stays at rest or continues to move with a constant speed in a straight line unless a net external force acts on it.

An object’s resistance to change in its state of

motion is known as inertia. Note that the larger

the mass, the higher the inertia.

3.2.2 Newton’s Second Law

First we must understand linear momentum.

The linear momentum of a body is the product of its mass and velocity.

Hence, we can now define the following:

Newton’s second law of motion states that the rate of change of linear momentum is in the same direction and directly proportional to the resultant force acting on it.

Note that if we differentiate it,

Lastly, we define a new term:

The impulse a force is the product of the force and the time interval over which it is applied.

Impulse is the area under a force-time graph. The

average force is represented by a rectangle (e.g.

green, left) Do note that large force applied over a

short time (yellow) hence the same impulse as

small force applied over a long time (green, right).


of 34

We also have the impulse-momentum theorem:

∫ ∫

∫ ∫

∫

To effectively solve problems, the usage of free

body diagrams (below) is crucial. Thus, we label

all forces acting on an object (the block).

Resolving vectors might be necessary.

3.2.3 Newton’s Third Law

Newton’s third law states that if body A exerts a force on body B, body B will exert an equal and opposite force of the same nature on body A. Note: both forces must act on different bodies

Also, note that for all connected components (be

it by string, contact, etc.), they have the same

acceleration:

(

)

3.3 Conservation of Linear Momentum

During collisions, we observe that forces act on

opposite bodies without external forces. (e.g. for

2 billard balls as shown above).

∫ ∫

∑

∑ ∑ ∑

The principle of conservation of linear momentum (PCOM) states that the total linear momentum of a system is conserved if no net external force acts on the system.

∑ ∑

3.4 Collisions

3.4.1 Head-on Collision vs Glancing Collision

A collision is an isolated event where 2 or more colliding bodies exert relatively strong forces on each other for a relatively short time.

For head-on collisions, the direction of motion of

both bodies before and after collision is in the

same line of motion.


of 34

3.4.2 Elastic and Inelastic Collision

An elastic collision is when 100% of kinetic energy

is being conserved.

A completely inelastic collision is when 2 objects

coalesce (stick together) and move with common

velocity after collision. It represents the

maximum possible loss of KE (not loss of all KE).

3.4.3 Relative speed of Approach / Separation

For an elastic 2-body head on collision,

Since KE is 100% conserved for elastic collisions,

That is, for elastic collision, the relative speed of

approach [RSOA] (LHS) equals the relative speed

of separation [RSOS] (RHS).

If 1 of the bodies is initially at rest, then:

3.4.4 Solving Collision Problems

To solve problems, we use PCOM and RSOA/RSOS.

Let A and B be 1.0kg and 3.0kg respectively.

For elastic collision,

For completely inelastic collision,

3.5 Coefficient of Restitution*

The elasticity of a collision is quantified by the

coefficient of restitution,

| |

| |


of 34

3.6 Static and Kinetic Friction*

Static friction is the force opposing motion

between 2 bodies at rest relative to each other.

Kinetic friction is the force opposing motion

between 2 bodies moving relative to each other.

Hence, there is no static friction when there is

kinetic friction, and vice versa.

Chapter 4: Forces

4.1 Hooke’s Law

Hooke’s law states that the magnitude of the force F exerted by a spring on a body attached to the spring is proportional to the extension x of the spring from equilibrium provided the proportionality limit of the spring is not exceeded.

∫

4.2 Upthrust / Buoyant Force

To understand upthrust, we must first know the

pressure exerted by a fluid.

Using the fluid force acting

on the surface bottom that

offsets the weight of the

water column (dark blue),

Source:http://images.tutorvista.co

m/content/fluids-pressure/liquid-

pressure.gif

Note that the pressure

of fluid acts in all

directions.

Source:http://img.sparknotes.com/fig

ures/0/0a1c01f07d0a0e51105b2065c1

36cda0/ideal_p1_3.gif

The left diagram shows

typical mercury

manometers, measuring

the difference in pressure.

For atmospheric pressure, it

is usually at 760 mmHg. The deeper down the

tube, the higher the pressure (due to extra weight

of column of mercury). in this case gives us the

difference in pressure for 2 gases.

Now, we move on to understand upthrust:

Upthrust is the net upward force exerted by a fluid on a body fully or partially submerged in the fluid.

Hence, to find the upthrust acting on a cube (dark

blue here):

This is Archimedes’ Principle, stating that a body

submerged in liquid has an upthrust equal to the

weight of fluid displaced.


of 34

4.3 Translational Equilibrium

When a body is either stationary or moving at

constant velocity, the body is in translational

equilibrium. The condition is that:

∑

4.4 Moments

The moment of a force about a point is the product of the magnitude of the force and the perpendicular distance of the line of action of the force to the point.

With the example of trying to open a door, we see

that the moment about the hinge is given by:

4.5 Rotational Equilibrium

For a body to be at rotational equilibrium, the net

moment of the body about any point is zero, i.e.:

∑

The principle of moments states that for a body to be in rotational equilibrium,

∑ ∑

must be true for any point on the body.

4.6 Static and Dynamic Equilibrium

Static Dynamic

∑ ∑

e.g. Hanging Picture

e.g. Sliding Ice Block

4.7 Three-force Systems

For stationary bodies experiencing only 3 co-

planar forces, then the lines of action of all 3

forces must intersect at 1 point. (net about that

point must be zero).

Hence, we see that for the bridge to be stable

(suspended by the rope), the direction of force

acting on the bridge by the hinge must meet the

intersection of the other 2 forces.

4.8 Couples

A couple is a pair forces equal in magnitude but opposite in direction whose lines of action are parallel but separate.

Couples only produce rotation and no translation.

The resultant torque is given by:

(

) (

)

Chapter 5: Work, Energy, Power

5.1 Definitions

Work is the transfer and transformation of energy between one body and another.

The energy of a system is a measure of its capacity to do work.

Similar to using momentum and impulse, we can

find the change in energy using work done

without knowing the time interval when the

force is applied.


of 34

5.2 Work Done

The work done on a body is the product of the force and its displacement in the direction of the force.

Hence, given the following diagram,

We conclude that:

Hence, negative work done is doing work in the

opposite direction of displacement. (In the above

case, it could be work done by friction).

Note that the total work done is the area under

the force displacement graph:

∫

For an expanding gas, do note that there is

another formula for the work done:

Provided pressure is constant during expansion,

the force exerted on the piston is constant:

5.3 Mechanical Energy

The total mechanic energy of a system is the sum

of kinetic and potential energy in the system:

5.3.1 Kinetic Energy

Kinetic energy of a body is a measure of energy possessed by the body by virtue of its motion

Using the Newton’s 2nd Law and kinematics

equation for uniform acceleration:

(

) (

)

Hence, a decrease in KE is negative work done,

and the increase in KE is positive work done.

5.3.2 Potential Energy

Potential energy of a body can be defined as the amount of work done on it to give it the current position it occupies.

For an object to exist at its current

position, it needs to overcome the

earth’s attraction:

This is also the gravitational potential energy

(G.P.E) since the object is in a gravitational field.

For an object falling through a distance of :

Hence in general for non-uniform fields


of 34

5.4 Conservation of Energy

The principle of conservation of energy states that energy is a quantity that can be converted from one form to another but cannot be created or destroyed. The total energy of an isolated system is constant.

It is an effective method for dealing with various

problems in mechanics, for instance:

To find the maximum compression of spring,

√

√

5.5 Power and Efficiency

Power is defined as the rate of work done.

Also note the following relation with velocity:

(

)

Efficiency is the ratio of useful output power to total input power, i.e.:

The efficiency is usually less than 1 since the input

energy is converted to other non-useful forms of

energy (e.g. heat energy in light bulbs).

Chapter 6: Circular Motion

6.1 Kinematics of Circular Motion

6.1.1 Circular Measure

Given this diagram, we know that

the arc length (red) is given by:

Hence, the radian is defined as the ratio between

arc length and the radius of the circle.

(

)

6.1.2 Angular Displacement and Velocity

If an object moves from to ,

then the angular displacement is

the change in angle ( ).

So similarly, to find angular velocity,

6.1.3 Tangential Speed

Knowing that , we

differentiate w.r.t time:

(

)


of 34

6.1.4 Period and Frequency

Period is the time for 1 complete cycle (or revolution)

Frequency is the number of revolutions per unit time.

6.2 Uniform Circular Motion

For a uniform circular motion,

the tangential speed remains

constant, but the direction of

velocity is always changing.

6.3 Centripetal Acceleration / Force

With changing direction and same speed, there

must be acceleration perpendicular to the

velocity vector, known as the centripetal

acceleration (ac). To derive it,

Since the position and velocity vectors move in

tandem, they go around the circle in the same

time, equal to the distance travelled divided by

the velocity:

| |

| |

| |

| |

By equating both equations, we get:

| |

| |

| |

| |

Hence, using Newton’s 2nd Law, the centripetal

force is given by:

Source:http://www.borzov.net/Pilot/FSWeb/Lessons/Student/image

s/Lesson2Figure01.gif

The above shows the banking of a plane, where

tilting the plane gives the horizontal component

of lift responsible for turning (centripetal force).

(

)

6.4 Vertical Circular Motion

When dealing with vertical circular motion, the

conservation of energy becomes very useful:

For instance, if the roller coaster (blue) and its

passengers are 170kg, is travelling at 33ms-1 and

the loop is of radius 19m, we can determine the

normal contact force at the top and bottom and

minimum speed for the roller coaster to pass the

loop safely at the top.


of 34

(

)

(

)

√ √

Chapter 7: Gravitation

7.1 Law of Universal Gravitation

Newton’s Law of universal gravitation states that every particle attracts every other particle with a force directly proportional to their masses, and inversely proportional to the square of the distance between them, i.e.:

Note: Particles are point masses and of negligible dimensions. Objects with radial symmetry can also be treated as a point mass. (Shell theorem). G is the gravitational constant, experimentally determined to be 6.67 x 10-11 N m2 kg-2

7.1.1 Weighing the Earth

Since the moon orbits the moon, we can weigh

the earth using this law and circular motion.

(

)

The same technique applies for the Sun, satellites,

moons and various objects in space. Note that

this is only an estimation.

7.1.2 Acceleration of the Earth

Using the law, we know that the earth would

accelerate towards the apple. Using Newton’s d

2nd 3rd Law,

Hence, we see that the Earth has negligible

acceleration due to its large mass.


of 34

7.1.3 Inverse Square Relationship

Since from the previous example we know that:

(

)

Kepler’s 3rd Law helps to explain how the inverse

square relationship is derived:

(

)

(

)

7.2 Geostationary Satellites

Geostationary satellites are satellites with orbits such that they are always positioned over the same geographical spot on Earth.

Note that it must be in the

same plane as the equator

such that the orbit’s centre

and centre of the Earth is

concentric.

Assuming a circular orbit and that the radius of

the earth is 6.58 x 106m and the mass of the earth

to be 5.98 x 1024 kg,

(

)

√

With such high altitudes, the whole Earth disk is

viewable, but the spatial resolution (amount of

details) is poor. Places further away from the

equator have poorer resolutions.

7.3 Gravitational Field Strength

The gravitational field strength at a point is defined as the gravitational force per unit mass acting on a small mass placed at the point.

If a gravitational field is set up around M and

attracts m which is distance r away from M,

For any spherical body, the acceleration inside it

is zero. For these situations,

At P, the gravitational field

due to solid spherical shell A

is zero. However, the

gravitational field at P due to

spherical mass B (dotted) is

given by:

However, since B is a mass in the shell,

(

) (

)

Hence, if we were to sketch g against r,


of 34

Note that all gravitational field lines are

perpendicular to the gravitational field vector g.

Also, when g is large, the gravitational field lines

are closer.

However, it is important to note that g is not

uniform on earth. First, the earth is an imperfect

sphere. Since we know that:

Also, note that the density of the earth is not

uniform. With the earth rotating, the

gravitational pull has to also provide for the

centripetal acceleration, lowering g.

7.4 Weightlessness

We know that the weighing balance measures the

normal contact force acting on the object. Hence,

there are 2 types of weightlessness.

True weightlessness is when there is no net gravitational force acting on an object.

We realize that by Newton’s 2nd

Law,

However in this case, since , . Thus,

the reading on the weight machine is zero.

Apparent weightlessness is observed when an object exerts no contact force on its support.

7.5 Gravitational Potential

The gravitational potential at a point in a gravitational field is the work done per unit mass by an external force, in bringing the mass from infinity to that point.

Note: Points of equal distance away from the centre of the Earth are equipotential.

7.6 Gravitational Potential Energy

Assume that point A is infinity, then to move mass

from point A to point B:

At infinity, the gravitational potential energy is 0,

The gravitational potential energy (G.P.E.) of a mass at a point in a gravitational field is the work done by an external force in bringing the mass from infinity to that point.

Since increasing separation distance results in a

gain in G.P.E (gravitational force is attractive in

nature), and infinity is the reference point (U=0),

hence G.P.E is always negative.

7.6.1 G.P.E of a system

To find the number of G.P.E. of

a system with n masses, we

have the following:

For 3 mass, we see that:

This represents the G.P.E. between every 2 point

masses. Hence, for n masses,

∑( ∑

)


of 34

7.6.2 G.P.E. near Earth’s Surface

At the Earth’s surface, the change in G.P.E is:

(

)

7.6.3 Relationship between G.P.E. and Fg

To move a point further from mass M,

∫ ∫

7.6.4 Total Energy

For any mass m (e.g. satellites) moving in circular

orbit around spherical M, the total energy is:

(

)

7.7 Escape Speed

The escape speed is the minimum speed to project a mass to escape a gravitational field.

√

√ (

) √

7.8 Binary Star System

A binary star system

contains 2 stars. We

know that the force

acting on each other

is:

By using circular

motion, we can

equate them:

(

)

(

)

An object at P experiences true weightlessness.

(g=0 as shown from the . Hence, to

reach from , we only need K.E. sufficient to

reach P:


of 34

Chapter 8: Oscillations

8.1 Introduction

Oscillation is the repetitive variation of some measure about a point of equilibrium or 2 or more different states.

Free oscillations are systems oscillating at the natural frequency of the system, the frequency characteristic of the system.

Here are some

examples of free

oscillations. In the real

world, they are

subjected to dissipative

forces, known as the

damping effect.

8.2 Simple harmonic Motion (S.H.M)

Assuming we have a particle vibrating along the

lines of XY and the displacement is recorded to

the right. (In a displacement-time graph)

Observing the above, we make some observations

using trigonometry:

Hence, given this generic displacement equation,

we can begin to work out the rest.

8.2.1 Equations in S.H.M

The above graph plots against

Hence, we can now define simple harmonic

motion (S.H.M):

Simple harmonic motion is a periodic motion where an oscillator is subjected to a restoring force directed towards the equilibrium point.

Also, note that to express :

(

)

(

)

√

Hence, we have the following graph (v against x):

√

Note: the red graph ‘moves’ in the

clockwise direction (think about the motion) [For

instance, when , the next moment must

have ]


of 34

8.2.2 Energy in S.H.M

Given the previous mentioned equations, we can

derive the energy in the oscillator:

Plotting energy against time gives the following:

In essence, it is a sine squared graph (for P.E.)

and cosine squared graph (for K.E.).

8.3 Damping

The progressive decrease in amplitude of any oscillatory motion caused by dissipative forces is also known as damping.

Examples include attaching cardboard (for more

air resistance), immersing oscillators in fluids

(more viscous) and eddy currents.

8.3.1 Light Damping

Oscillating under resistive forces, the amplitude

decreases by the same proportion after each

cycle. Note that the period is slightly longer than

that of the ‘undamped’ value.

8.3.2 Critical Damping

Larger resistive force results in critical damping,

where the oscillator returns to the equilibrium

point in the shortest time without overshooting.

This is used is balances, ammeters/voltmeters to

indicate readings in the shortest time.

8.3.3 Heavy Damping

An even stronger damping force will cause the

oscillator to take a longer time to reach

equilibrium. For instance, over-damped car fuel

gauge indicators are used to give reasonable

indications despite car movement.

8.4 Resonance

Firstly, we must know what forced oscillations are:

Forced oscillations are oscillations under the influence of an external periodic force with a driving frequency.

Next, we move on to investigate resonance.


of 34

Resonance is the phenomenon in which an oscillatory system responds with maximum amplitude to an external periodic force when the driving frequency equals natural frequency of the driven system.

The graph shows various degrees of damping

(light, heavier, and even heavier). Hence, we see

that amplitude of lightly damped systems is very

large at resonance. Damping lowers resonant

frequency to below natural frequency.

With increasing damping, we realize that:

1) The amplitude of oscillation decreases

2) The Resonance peak becomes broader

3) The resonance peak shifts leftwards

4) The graph does not cut at 0. This is because

driving frequency of 0 means there is 1 swing.

Examples of useful resonance include:

1) Microwave cooking (microwave frequency is

natural frequency of water), cooking food

without heating plastic containers too much.

2) Magnetic Resonance Imaging (MRI) allows

analysis of energy absorption using strong EM

fields to produce images (similar to X-rays).

Examples of destructive resonance include:

1) When an opera singer projects a high-pitched

note matching the natural frequency of glass,

glass vibrates at large amplitudes, breaking it.

2) Collapse of bridges (e.g. Tacoma Narrows

suspension bridge). High winds results in

resonance. Hence, the bridge vibrates at

exceptionally large amplitudes and collapses.

Chapter 9: Waves

9.1 Introduction (Terms and Graphs)

Wave is a disturbance of some physical quantity. As the disturbance propagates through space or medium, energy and momentum can be transferred from 1 region to another.

Source:http://rpmedia.ask.com/ts?u=/wikipedia/commons/thumb/7

/77/Waveforms.svg/350px-Waveforms.svg.png

Waves can come in many waveforms (above). We

will use only sinusoidal waves for simplicity.

9.2 Wave Equation

source: http://www.a-

levelphysicstutor.com/images/wa

ves/sinus-graph01.jpg

The left shows the

displacement-time

graph (1 particle, above)

and displacement-

distance graph (whole

wave, below)

Period is time taken for a point on the wave to complete one oscillation

Frequency is the no. of oscillations per unit time made by a point on the wave.

Wavelength is the distance between 2 adjacent points that are in phase.

Displacement of a particular point is the distance and direction of that point from its equilibrium position.

Amplitude is the maximum displacement of a point on the wave.

Crests are points with maximum, positive displacement.

Troughs are points with maximum, negative displacement.


of 34

For a periodic wave, it travels one wavelength

during 1 period.

Hence, we can determine its speed:

Waves can be categorized into the following:

Mode of Vibration Longitudinal

Transverse

Motion Progressive

Stationary / Standing

Medium Mechanical

Electromagnetic

Matter

It is important to note that waves usually transfer

energy and not matter. Some waves require a

medium (e.g. sound and air/water) whereas

others can occur in vacuum (e.g. light rays from

Sun).

9.3 Transverse vs. Longitudinal Waves

9.3.1 Transverse Waves

Source:http://sciencecity.oupchina.com.hk/npaw/student/suppleme

ntary/images/graph-1b_8.jpg

Transverse waves are waves where displacement of particles is perpendicular to the direction of wave propagation.

Transverse waves are similar to their wave

profiles and they can (obviously) exist in many

planes. Examples include all electromagnetic

waves.

9.3.2 Longitudinal Waves

http://sciencecity.oupchina.com.hk/npaw/student/supplementary/i

mages/graph-1b_7.jpg

Longitudinal waves are waves where the displacement of particles is parallel to the direction of wave propagation.

Sound is a good example of longitudinal waves:

Source:http://hyperphysics.phy-

astr.gsu.edu/hbase/sound/imgsou/lwav2.gif

9.3.3 Progressive Waves

The displacement-distance graph shows the same

wave travelling left to right when t=0, t=1 and t=2.

Hence, by analyzing the particle at t= , we can

plot the displacement-time graph for the single

particle:

The same technique can be applied for

longitudinal waves by analyzing its wave profile.


of 34

9.4 Phase Difference

The phase of an osicillation is the stage of oscillation that is represented by the phase angle, where 2 radians or 360 represents one complete cycle.

Note: points chosen must be in phase (black):

(

) (

)

9.5 Electromagnetic Waves

Source:http://micro.magnet.fsu.edu/primer/java/wavebasics/basicw

avesjavafigure1.jpg

Electromagnetic (EM) waves consist of the electric (E) and magnetic (B) field oscillating perpendicular to the direction of wave propagation. No medium is required. They travel at the speed of light (which is an EM

wave), where .

The EM spectrum classifies various EM waves:

Source: http://amazing-

space.stsci.edu/resources/qa/graphics/qa_emchart.gif

This table contains some uses of EM waves:

Name Detection Uses

Radio Radio Aerials Communications

Micro Tuned Cavities Communications and cooking

Infra-Red (IR)

Photography / Heating Effect

Satellite, TV controls

Visible Light

Eye / Photography

Sight, communication

Ultra Violet (UV)

Fluorescence, solid state detectors

Food sterilization

X-rays Fluorescence Diagnosis

Gamma ( rays

Scintillation counter

Radiotherapy

9.6 Intensity of Waves

Source:http://toonz.ca/bose/wiki/images/1/1e/IntensitySurfaceSphe

re.gif

The intensity is defined as the power per unit area that passes perpendicularly through a surface area, i.e.:

Since intensity is the energy per unit time per unit

area, we can thus conclude that:

Also, the diagram shows that we can apply the

inverse square law for intensity:


of 34

9.7 Polarization

Source:http://www.exo.net/~pauld/summer_institute/summer_day

8polarization/polarizerfencemodel600.jpeg

Polarization is the phenomenon where a transverse wave is made to oscillate in a single plane, the plane of polarization.

The first polarizer is known as the ‘polarizer’ and

the second is known as the ‘analyzer ’.

All polarization filters

only allow planes in the

plane of polarization to

pass. Hence, we can

resolve the electric field

to give a vertical and

horizontal component.

(

)

(

)

Hence, we observe the following:

If the polarizer and analyzer have planes of

polarization perpendicular to each other, then no

light passes through.

Chapter 10: Superposition

10.1 Principle of Superposition

The principle of superposition states that when 2 or more waves of the same kind overlap, the resultant displacement at any point any instant is given by the vector sum of individual displacements that each individual wave would cause at that instant, i.e.:

10.2 Interference

Inteference is the combination of waves in the same region of space at the same time to produce a resultant wave.

10.2.1 Constructive, Destructive Interference

There are 2 types of interference, constructive

and destructive interference:

10.2.2 Path Difference and Phase Difference

Source:http://roncalliphysics.wikispaces.com/file/view/nodal_lines.g

if/233899502/nodal_lines.gif

Plotting lines that join constructive interference

(red, anti-nodal lines) and destructive

interference (blue, nodal lines), we obtain the

above diagram.


of 34

Path difference is defined as follows:

| | | |

For instance, the above diagram shows that:

| |

If they are in phase, constructive interference

occurs (as above). However, if they are anti-phase,

then destructive interference occurs.

Generalizing, we have the “final” phase

difference, given by:

Hence, for constructive interference,

Hence, for constructive interference,

(

)

10.3 Diffraction and Huygen’s-Fresnel

Principle

Diffraction is the apparent bending of waves around small obstacles and the spreading out of waves past small openings.

And we use the Huygen’s-Fresnel principle to

explain that phenomenon:

Huygen’s-Fresnel principle states that every point of a wave may be considered a secondary source of wavelets spreading out in all directions with a speed equal to the speed of propagation of the wave.

The following shows how it can apply to waves,

for both refraction and diffraction.

The new wave front is thus the envelope of

wavelets (green). Hence, for smaller apertures,

the envelope of wavelets is more spherical.

10.4 Young’s Double Slit Experiment

Thomas Young used the double-slit experiment in

1803 to show that light was a wave by

demonstrating intereference patterns predictable

by wave theory after his paper was rejected in

1799 by the royal society.

The single slit ensures the coherency of the wave.

2 waves are coherent if they have a constant (not necessarily 0) phase difference between them.

Coherent waves have the same wavelength and

frequency, and hence the same speed.

In order to determine maxima and minima, we

must first observe that for this experiment:

P


of 34

Source:http://www.u.arizona.edu/~mas13/draft4.310_files/image03

4.jpg (left)

First, we make the following observation:

For bright fringes (maxima),

For dark fringes (minima),

(

)

To find the exact positions of dark and bright

fringes (as shown in the initial experimental setup

diagram):

(

)

(

) (

)

Hence, the fringe separation, distance between 2

adjacent bright or dark fringes, is given by:

( )(

) (

)

10.5 Diffraction Grating

After seeing the double slit experiment, we now

use diffraction grating, adding many more

parallel, closely spaced and equidistance slits.

Diffraction grating usually involves hundreds or

thousands of slits.

Source:http://nothingnerdy.wikispaces.com/file/view/diffraction_gra

ting_geometry.jpg/213547792/diffraction_grating_geometry.jpg

Source: http://www.a-levelphysicstutor.com/wav-light-diffr.php

The above diagram represents the various

interference patterns with varying slits.

We observe that:

1) Maxima increases (more slits)

2) Better contrast in fringe pattern

3) Position of maxima is the same

Hence, using property 3, we can adapt the

equation for Young’s experiment to find the nth

order maxima for a diffraction grating, i.e.:


of 34

Source:http://hyperphysics.phy-

astr.gsu.edu/hbase/phyopt/imgpho/diffgrat.gif

It is important to observe that different

wavelengths of light have different maximas (e.g.

that of red and blue.

This is demonstrates that small angle

approximation does not hold (due to increasing

angle). Since fringes are irregularly spaced,

(

)

Commercially, gratings are labelled by no. of lines

per unit length, N.

10.6 Stationary Waves

Source:http://tap.iop.org/vibration/superpostion/324/img_full_4680

0.gif

Firstly, we note a phenomenon that when a wave

hits a fixed / denser surface (e.g. mirrors), they

undergo a phase change of radians.

This is because the wave exerts an upward force

(above diagram) on the fixed surface. Hence, by

Newton’s 3rd law, the wall exerts an equal and

opposite (downward) force on the medium (e.g.

string), resulting in a negative displacement.

Hence, when we have 2 identical waves moving in

opposite direction, we have a stationary wave

(red) that is being formed. Stationary waves

obviously have no translation of energy. It has the

following properties compared to a normal wave:

Nodes are points that never move and antinodes

are points having the greatest amplitude of

vibration.

10.7 Stretched Strings

Source:http://learn.uci.edu/media/OC08/11004/OC0811004_Standi

ngWave04.jpg

A string that is fixed on 2 ends can vibrate (above).

Like in simple harmonic motion, when the string

vibrates at its natural frequency, it obtains the

resonant modes of vibration (below):

Source:http://www.miqel.com/images_1/jazz_music_heart/harmoni

cs.jpg

http://tap.iop.org/vibration/superpostion/324/img_full_46800.gif

http://tap.iop.org/vibration/superpostion/324/img_full_46800.gif


of 34

Harmonics are all resonant frequencies of vibrations that can be generated.

( )

√

Note: the 1st harmonic is known as the

fundamental frequency.

Overtones are frequencies that can be produced by an instrument accompanying the 1st harmonic that is played.

10.8 Air Columns

There are 2 types of air columns:

10.8.1 Open pipes

Source:http://labspace.open.ac.uk/file.php/7027/ta212_2_015i.smal

l.jpg

Generalizing, we can deduce that:

10.8.2 Closed Pipes

Generalizing, we can deduce that:

10.8.3 End Corrections

End corrections occur because in practice, the

open end of a pipe is set into vibration and the

displacement antinode occurs at a distance c

(above).

From the above,

It has been found that end correction is

approximately 50-60% of the radius of the cross

sectional area of pipe. It might be better to take

them into consideration for large pipes.


of 34

Chapter 11: Miscellaneous

11.1 Useful Knowledge / Summary *

11.1.1 List of Useful Formulas by Topic

This list is non-exhaustive:

Physical Quantities and Measurements

| |

√

Kinematics

Dynamics

∑

∑

Forces

∑ ∑

Work, Energy, Power

Circular Motion

Gravitation

(

)


of 34

Gravitation (Continued)

∑( ∑

)

√ √

Oscillations / Simple Harmonic Motion

√

Waves

(

) (

)

Superposition

∑

| |

√

11.1.2 List of Useful Constants

These are fundamental constants to be used:

Gravitation constant

Speed of EM Waves

Electron Charge Planck’s Constant Stefan-Boltzmann Constant

Gas Constant Avogadro’s Constant

Boltzmann’s constant


of 34

11.1.3 List of any other useful data (for now)

Credits

This set of physics notes is done by Ang Ray Yan,

Hwa Chong Institution 11S7B.

The following people deserve their due

recognition in making this set of notes:

- Mr Thomas, my physics tutor who rekindled

my interest for physics, showing me that

physics was useful, interesting, applicable and

unlike anything in my high school years.

- Lim Yao Chong for being a reliable helpline in

my weakest topics, particularly dynamics.

- Phang Zheng Xun for giving more accurate

definitions and various explanations.

- Yuan Yu Chuan for correcting my English,

which is of “powder-ful” standard.

physics notes

Documents

ang ray yan

inverse square

high school

completely

simple harmonic

force opposing

external periodic

air resistance