CIE A Level Notes

1 Asif Ahsan| Contact: 01911784608

CIE A‐LEVEL PHYSICS NOTES COURSE INSTRUCTOR – ASIF AHSAN CONT: 01911784608

Chapter 01

Physical Quantities & Units

Measurable quantities are called physical quantities. There are two types of physical quantities:‐

1. Base Quantity

2. Derived Quantity

1. Base Quantity: A quantity which is defined without the help of other quantities is called a base

quantity. There are only 7 base quantities:

i. Mass

ii. Length

iii. Time interval

iv. Temperature difference

v. Electric current

vi. Amount of substance

vii. Luminous intensity

2. Derived Quantity: A quantity which is defined with the help of two or more base quantities is

known as a derived quantity. Example:

i. Force ii. Charge iii. Velocity iv. Density

Units: A unit is a standard measured value compared with which other measurements are expressed. For e.g. 1m is the length defined by someone in science museum and There are 2 types of units:‐

1. Base Unit 2. Derived Unit

1. Base Unit: The unit of a base quantity is called a base unit. Since there are only 7 base quantities,

base units are also 7 in number. They are:‐

i. Kilogram (kg) mass

ii. Meter (m) length

iii. Second (s) time

iv. Kelvin (K) temperature E.g. 1200C = (120+273)K

v. Ampere (A) electric current

vi. Mole (mol) amount of substance

vii. Lux (L) luminous intensity

2. Derived Unit: The unit of a derived quantity is called a derived unit.



Derived quantity Derived Unit Base Unit form

Force N (Newton) F=ma

Charge C (Coulomb) Q=I*t

Speed ms‐1 ms‐1

Density Kgm‐3 kgm‐3

Pressure Pa (Pascal) P = = 2

2= kgm 1s 2

Power = ∗ P =

∗ = kgm2 s‐3

Prefixes:

Units kilo (k) Mega

(M)

Giga

(G)

Milli

(m)

Micro

(µ)

Nano (n)

x103 x106 x109 x10‐3 x10‐6 x10‐9

Homogeneous Equation:

An equation containing the same type of physical quantities is called a homogeneous equation. Example:

v = u +at

ms‐1 = ms‐1 + ms‐2 * s

ms‐1 = ms‐1 + ms‐1

ms‐1 = ms‐1 All correct equations are homogeneous, but all homogeneous equations are not correct!

i) x = ut + 1

2 at2

m = m +m m = m

ii) v2 = u2 + 2ax

m2s‐2 = m2s‐2 + m2s‐2

m2s‐2 = m2s‐2 These equations are homogeneous, but with respect to their units only. Not all the equations are correct in terms of their coefficients.



Vectors

Adding Vectors:

When two vectors act at some angle another than 0° and 180°, they can be added using two rules:

(i) Triangle Rule (ii) Parallelogram Rule

Triangle Rule:

If two vectors are drawn head to tail according to their magnitude and direction, then the third line drawn from the standing point to complete a triangle represents the resultant of the two vectors both in magnitude and direction.



Scale Drawing:

A sprinter runs 50m towards north, then 30m toward north‐east. What is his resultant displacement?

1 10

Resultant displacement is 75 , ° East of North.

R = 105N at yo with 40N force

Polygon of Vectors

Resultant

v2

v1

v3

v4

v1

v2

v3 v4

Resultant



When more than two vectors are drawn from head to tail, according to their magnitude and direction,

the resultant will be represented by the line from the starting point to the ending point in order to

complete a closed diagram or to complete a polygon.



Parallelogram Rule:

If two vectors acting at a point are represented by the adjacent sides of a parallelogram, then the diagonal of the parallelogram drawn from that point will represent the resultant, both in magnitude and direction.

The above three forces are on the same plane and therefore T1, T2 and W and three co‐planar forces. Description: Point A is in equilibrium under the action of three co‐planar forces. So the resultant force acting at A must be zero. The following conditions are applied for the situation described above:

(i) Resultant of the two forces is equal and opposite to the remaining force (ii) When the three forces are drawn head to tail, a closed triangle is formed (iii) The line of action of the three forces passed through a single point. (iv) Sum of the forces in any one direction will be zero (v) Sum of the forces in another direction is zero



•measurelengthsusingaruler,vernierscaleandmicrometer•measureweightandhencemassusingspringandleverbalances•measureanangleusingaprotractor•measuretimeintervalsusingclocks,stopwatchesandthecalibratedtime‐baseofacathode‐rayoscilloscope c.r.o. •measuretemperatureusingathermometerasasensor•useammetersandvoltmeterswithappropriatescales•useagalvanometerinnullmethods•useacathode‐rayoscilloscope c.r.o.

usebothanaloguescalesanddigitaldisplaysc usecalibrationcurvesd showanunderstandingofthedistinctionbetweensystematicerrorsincludingzeroerrors andrandomerrorse showanunderstandingofthedistinctionbetweenprecisionandaccuracyf assesstheuncertaintyinaderivedquantitybysimpleadditionofactual,fractionalorpercentageuncertainties arigorousstatisticaltreatmentisnotrequired .



Chapter 02

Rectilinear Motion

Scalar Quantities: Quantities which have magnitude (size) only, but no direction, are known as

scalar quantities. Example:

1. length

2. distance

3. time

Vector Quantities: Quantities which have both magnitude and direction are known as vector

quantities. Example:

1. force

2. displacement

3. velocity

Distance travelled: This is the change in position in any direction.

Displacement: This is the distance travelled or change in position in a particular direction.

Speed: It is the rate of change of position or rate of change of distance, travelled in any

direction.

Word equation: speed =

s =

Unit : ms‐1

Velocity: It is the rate of change of displacement, or rate of change of distance travelled

in a specified direction.

Word equation: velocity =

Unit: ms‐1

A B

Displacement

2.5 m

BA

5.0 m



When velocity of a body continuously changes, we deal with two types of velocities: 1) Instantaneous velocity & 2) Average velocity

Instantaneous Velocity:

Velocity of an accelerating body at a particular instant is called instantaneous velocity. This velocity continuously changes as the body accelerates or decelerates. It is measured as displacement over a short period of time.

Instantaneous velocity =

Average Velocity:

Average velocity of an accelerating body is defined as the total displacement over total time.

Average velocity =

To measure velocity or acceleration we need some specific apparatuses to get exact result.

1. Friction compensated runways 2. Timing Devices

Friction compensated runway:

a. The runway is comprised of a triangular shaped structure and with air outlets and air inlets. As

the blower starts, air comes out through the outlets and makes the trolley float. As the contact

between trolley/Glider and runway becomes negligible so friction almost disappears and

runway becomes friction compensated.

m1 m2

L1 L2



b. The second types of runways are ordinary types of runway but slightly tilted in order to compensate for friction.

Timing Devices

1. Light gate 2. Stopwatch 3. Ticker timer 4. Multi flash camera 5. Video camera

Light gate:

A light gate consists of a light source producing a narrow beam of light and a light sensor connected to

an electronic timer. The timer is switched ON/OFF, depending on the light beam reaching the detector

or not.

v =



EXPERIMENT: To measure the instantaneous velocity of an accelerating trolley when it passes a

point on the runway. (Video Available)

The mass hanger is released so that the trolley accelerates. As the trolley passes the light gate, the

timer records the interruption ‘T’. Length of the trolley is measured ‘L’ using a meter rule.

The instantaneous velocity of the trolley as it runs down is given by v = .

EXPERIMENT: To measure the average velocity of an accelerating trolley when it passes a distance.

The trolley is released from the top of the runway so that it accelerates. As the interrupts the first

light gate, the timer is switched ON. As the trolley interrupts the second light gate, the timer is

switched OFF. So, the time taken for the trolley to travel the distance AB is recorded from the timer.

Length of AB is measured ‘x’ using a measuring tape/meter rule.

Average velocity over AB is given by v = .

Acceleration:

The rate of change of velocity of a body is called its acceleration.

Word equation: acceleration =

It is a vector quantity; its SI unit is ms‐2.



An experiment to measure acceleration using a light gate and double interrupters:

Lengths of the interrupters are measured ‘L1’ and ‘L2’. The trolley is released from the top of the

sloped runway, so that it accelerates. Time ‘T1’ for the first interruption, ‘T2’ for the second

interruption and ‘T3’ between the interruptions are recorded from the intelligent timer.

Calculations:

Average velocity over the first interruption is u = 1

1 .

Average velocity over the second interruption is v = 2

2 .

∴ Acceleration =

EXPERIMENT: To measure acceleration using a Video Camera

The trolley is allowed to move down the calibrated runway. Motion of the trolley is recorded by

the video camera. If the video camera takes 25 pictures (frames per second), the time between

two consecutive frames is 1

25 or 0.04 seconds.

The video is played back 1 frame at a time. Distance ‘x’ moved by the trolley between successive

frames is measured from the scale.



x3> x2> x1

Average velocity, v, of the trolley between the frames is calculated 0.04

,

where x = x1, x2, x3, etc

If a graph is plotted v against t, it will be a straight line. Gradient (slope) of this line is the

acceleration.

Motion graph using Ticker‐timer & tape:

* The ticker‐timer makes 50 dots/sec.

∴ 1 dot is made in 1

50 sec = 0.02 sec.

∴ 5 dots are made in 0.02 * 5 = 0.1 sec.

Initial velocity: u = 0.1 ms‐1

v (ms‐1)

t (sec)



Final velocity: v = 0.1 ms‐1

Acceleration: a = ms‐2

a = .

.

. ms‐2

Using Motion‐Sensor and Data‐Logger:

t = time differences between pulses

velocity1 = ms‐1

velocity2 = ms‐1

The gradient of the graph of v against t gives acceleration.

Motion graphs for different types of motion

Things to remember:

Gradient of an “x‐t” graph is velocity.

Gradient of a “v‐t” graph is acceleration.

Area under “v‐t” graph is distance.

Area under “a‐t” graph is velocity.

s = 330 ms‐1

t = 0.2 sec

D1 = 330∗0.2

2 m

D2 = 330∗0.4

2 m



Motion Graphs



Motion graph for a bouncing ball



Equations of Motion

i) x= (2

)t

ii) v = u + at

iii) x = ut + 1

2at2

iv) v2 = u2 + 2ax

When a body accelerates uniformly, four equations regarding its motion are used. They are:

i. x =

From the definition of average velocity, v =

If acceleration a is constant, average velocity is the arithmetic mean of the initial and the final

velocities.

v = → (a)

v = → (b)

From (a) and (b),

=

∴ x =

ii. Acceleration is defined as the rate of change of velocity.

a =

or, v – u = at

∴ v = u + at

iii. Equation (i.)

x =

Equation (ii.) x = u = at

Or, x =

Or, x =

∴ x = ut + at2



iv. From equation (ii.), v = u + at

∴ t =

Substituting in equation (i.),

x =

or, x =

or, 2ax = v2 – u2

∴ v2 = u2 + 2ax

EXPERIMENT: To measure the value of g

i) Direct Method:

The time taken by the ball to travel the height, h, is recorded, t from the timer.The height, h, is

measured by using a meter rule.

The experiment is repeated several times by altering the height, h.

h = gt2

The equation of a straight line : y = mx

Here, h y, t2 x, g m

A graph is plotted h against t2.

A straight line passing through the origin is obtained.

gradient = 1

2 g

∴ g = 2 * gradient

h (m)

t2 (s2)

gradient = m = g

Procedure:

When the switch is moved from A to

B, the electromagnet loses its

magneism, the ball starts to fall and

simultaneously the timer is

automatically switched ON. As the

ball hits the trap door, the contact is

broken and the timer stops.

h



Advantage of using graphical method

1) If there is any systematic error, the graph will have a y intercept, but its gradient will remain

the same. So, the value of g will not be affected by systematic errors.

2) A best‐fit line averages the good set of values.

ii) Using Light Gate and Double Interrupter:

The length of identical interrupters is measured ‘l’. The double interrupter is released vertically so

that it accelerates downward due to gravity. Time for 1st interruption is recorded t1, in seconds.

Time taken for 2nd interruption is recorded t2, in seconds. Finally, the time taken between the

interruptions is recorded t, from the intelligent timer.

Calculations:

Average velocity over the 1st interruption is u =

Average velocity over the 2nd interruption is v =

Acceleration of free fall is g = .

Resolution of Vector

To split a single vector into two or more components at some appropriate angle is called a resolution

of vector.

FH = FCosӨ = Horizontal component

Fv = FSinӨ = Vertical component

FH

Fv

Ө

F



Inclined Plane:

Ө

Weight

Friction



Chapter 03

Projectile Motion

An object that has been cast, flung, tossed or thrown is called a projectile. The force of primary importance acting on a projectile is gravity. This is not to say that other forces do not exist, just that their effect is minimal in comparison.

There are three types of projectile:

i) Vertical projectile – a body which is thrown vertically upward ii) Horizontal projectile – a body which is thrown horizontally from a height iii) Angular projectile – a body which is thrown at an angle with the horizontal

In vertical projectile motion, the ball does not move in the horizontal direction.

In horizontal projectile motion, the ball moves horizontally with an initial velocity, but it also moves

downward. The horizontal velocity has nothing to do with the vertical motion, i.e., it is independent of

the vertical motion. The vertical motion is only due to gravity.

Vertical Projectile

A ball is thrown vertically upwards with an initial velocity of 1.5ms‐1.

Find: a) maximum height reached, b) time taken to reach the maximum height, c) time taken to reach

the ground from the maximum height, d) velocity it gains before reaching the ground.

Solution:

a) v2 = u2 + 2ax

or, 2(‐9.81)x = 0 ‐1.52

∴ x = 0.12 m

b) v = u +at

or, 0 = 1.5 + (‐9.81)t

∴ t = 0.15 s

c) t = 0.15s

d) v = u + at

u

x

Өv

Vertical Projectile Horizontal Projectile Angular Projectile



or, v = 0 +(9.81)(0.15)

∴ v = 1.47 ms‐1.

Horizontal Projectile:

Vertical velocity increases at a rate of g because horizontal velocity does not have any component in

the vertical direction.

This is why vertical and horizontal motions are considered independently. Considering vertical motion:

h = gt2

∴ t = √

Considering horizontal motion, where u remains constant,

x = u . t

Given: horizontal velocity, u = 2ms‐1, height, h = 15m. find the horizontal distance travelled after

landing on the earth.

Solution:

t = √ = √∗

. = 1.75 s

Horizontal distance travelled, x = u . t

or, x = 2 * 1.75

∴ x = 3.50 m

Angular Motion

v1

u

v2

v3 v4

x

Ө h

x or R

600

450

300



If T is the time to reach the maximum height, the time spent in the air before returning to the ground

is 2T.

Vertical and horizontal motions are independent of each other.

Considering vertical motion:

Height, h is travelled because of the vertical component of u, i.e., uSinӨ.

In case of vertical motion:

u =uSinӨ

v = 0

g = ‐9.81 ms‐2

v = u + at

0 = uSinӨ – 9.81*t

∴ t = Ө

. ________ (i)

In case of horizontal motion:

Time taken to travel x = 2t

Horizontal velocity = uCosӨ

x = uCosӨ * 2t_________ (ii)

or, x = uCosӨ * 2*Ө

or, x = * 2SinӨCosӨ

∴ x = * Sin2Ө

x will be maximum when, Sin2Ө = 1

or, 2Ө = 900

∴ Ө = 450

∴ xmax =



Forces

Types of Forces:

There are two types of forces:

(1) Contact Force: Forces which cannot act without contact are known as contact forces for e.g. Friction,

Solid fluid drag force (Viscous forces), tension Normal Reaction Force

(2)Non‐contact forces: This is the force between objetcs for which contact is not required i.e. this force

acts at a distance without physical contact e.g. (i) gravitational force (ii) eleectrostatict force

(iii)magnetic force etc.

Gravitational Force: Force between two masses

Force between two masses is directly proportional to the product of the masses and inversely

proportional to square of their distance apart.

Newton’s Laws of Motion:

1st Law: This law gives a qualitative definition of force.

Statement 01: “A body at rest remains at rest and the moving body moves at a constant velocity

unless an unbalanced force acts on it.”

(“A body continues in its state of rest or of uniform motion in a straight line unless

compelled by some external force to act otherwise.” – formal statement)

Statement 02: “A body is in equilibrium when the resultant force acting on it is zero.”

F3

F2 F1

For this body to be at rest,

F1 + F2 + F3 = 0



Statement 03: “For acceleration, an unbalanced force is required.”

Newton’s 1st Law also reveals a common tendency of all bodies which called inertia.

Inertia: It is the reluctance of a body to change its state of motion or rest. Inertia

depends on the mass of the body. The more the mass, the more is the inertia.

2nd Law: This law gives a quantitative definition of force.

Statement 01: “Acceleration of a body is directly proportional to the unbalanced force applied on it

and the acceleration takes place in the direction of the unbalanced force.”

a ∝ F …….(1)

If same force is applied on different masses, acceleration is inversely proportional to mass.

a ∝ ……(2)

Combining (1) & (2),

a ∝

F ∝ ma

F ∝ m( )

F ∝

“Force is directly proportional to the rate of change of momentum and the change takes place in the

direction of the unbalanced force.”

F ∝ ma

F = k*ma, where k = proportionality constant

Defintion1N of force is the force which gives a body of mass 1 Kg an acceleration of 1ms‐2.



EXPERIMENT: To investigate that acceleration is directly proportional to force (when mass is

constant)

Procedure:

When the runway is made friction‐compensated properly, on giving a small push to the trolley, it moves

at a constant velocity and the times of interruptions with the card are found to be equal. The trolley is

pulled down the friction‐compensated runway with a constant force maintained by a Newton‐meter.

Experiment is repeated by using different constant forces.

Measurement:

1) Length of identical interrupters = L

2) Constant pulling force = f (found from the Newton‐meter)

3) Time for 1st interruption = t1,

Time for 2nd interruption = t2

Time between interruptions = t3

Calculations:

Average velocity over 1st interruption = u =

Average velocity over 2nd interruption = v =

∴ Acceleration =

A graph is plotted a against F and a straight line passing through the origin is obtained.

a (ms‐2)

F (N)



EXPERIMENT: To prove acceleration is inversely proportional to mass

Procedure:

The mass of a trolley is measured using a top‐pan balance. When the runway is made friction‐

compensated, on giving a small push to the trolley, it moves with a constant velocity. The times of

interruptions with the cards are found to be equal. A Newton‐meter is tied with the trolley so that

by using the Newton‐meter, the trolley is pulled down the friction‐compensated runway with a

constant force.

The experiment is repeated by loading the trolley with extra known masses.

Measurements:

1) Length of the identical interrupters = L

2) Mass loaded on the trolley = m

3) Constant pulling force = F

4) Time for 1st interruption = t1,

Time for 2nd interruption = t2

Time between interruptions = t3

Calculations:

Average velocity over 1st interruption = u =

Average velocity over 2nd interruption = v =

∴ Acceleration =

A graph of a against is plotted and

a straight line passing through the origin is

obtained

a (ms‐2)

(Kg‐1)



3rd Law: This law gives the idea that a single force does not exist. Forces always occur in pairs.

Statement: “When a body A applies a force on another body B, B also exerts an equal force on A in

the opposite direction for the same length of time and the two forces are of the same

type.

If the force on B is called action, then the force on A will be reaction. So, Newton’s 3rd law can also be

stated as:

“For every action, there is an equal and opposite reaction.”

The pair of forces (also known as Newton’s 3rd law Pair) has the following properties:

i) The forces are equal in magnitude

ii) The forces are of same type

iii) The forces have the same line of action

iv) The forces act for the same length of time

v) The forces act in opposite directions

vi) The forces act on different bodies

Earth pulling Ball downward with

gravitational force

Ball pulling Earth upward

with gravitational force

EARTH (A)

BALL (B)

Similarities

Differences



Free body force diagrams

Newton’s 3rd Law pairs:

A and H

C and B

D and G

E and F

Forces on Cat: A = Earth pulls the Cat down B = Table pushes the cat up

Forces on the Table: C = Cat pushes the table down D = Earth pulls table down E = Earth pushes table up

Forces on Earth: F = Table pushes Earth down G = Table pulls Earth up H = Cat pulls Earth up

Situation: A cat is sitting on a table placed on Earth.



Rocket Propulsion:

Forces and weight inside a lift:

How do we feel our weight?

When we stand on ground we exert a contact force on the ground (contact force is equal to

weight but contact force is not weight itself. Remember, weight is a gravitational force).

Ground exerts an equal force to our body. That force is felt by our body as our weight.

Please Turn Over

Rocket exerts a force on the fuel to the downward direction. By Newton’s 3rd

law, fuel exerts an equal force on the rocket to the upward direction.

The force is greater than the rockets weight and hence the rocket accelerates

upward.

As the rocket goes upward the weight of the rocket decreases due to burning

of fuel and also due to the decrease in gravitational force. So resultant force in

the upward direction increases and acceleration increases.

Normal Reaction Force

Weight

Free body force diagram of the man



When we travel by a lift we either go upwards or downwards. In each case there are three

stages of motion:

Upwards Downwards

Lift starts moving

(Acceleration)

Floor accelerates so

person feels a

resultant force from

the floor of the lift

upwards. So overall

upward force on the

person increases. He

feels his weight has

increased.

Lift starts moving

(Acceleration)

Floor of the lift

accelerates so the

resultant force is

downwards. Overall

upward force on the

person decreases

and he feels his

weight has

decreased

Lift moves with a

constant velocity

(No acceleration)

Floor of the lift

doesn’t exert any

additional resultant

force. So person

feels his weight he

would have felt

normally.

Lift moves with a

constant Velocity Floor of the lift

doesn’t exert any

additional resultant

force. So person

feels his weight he

would have felt

normally.

Lift decelerates as it

reaches a desired

floor

(Deceleration and

stop)

Floor of the lift

decelerates so the

resultant force is

downwards. Overall

upward force on the

person decreases

and he feels his

weight has

decreased.

Lift decelerates as it

reaches a desired

floor

(Deceleration and

stop)

Floor decelerates so

person feels a

resultant force from

the floor of the lift

upwards. So overall

upward force on the

person increases. He

feels his weight has

increased.



Viscous Drag:

Viscous drag is the force between different layers of a fluid that opposes their relative motion.

Greater the speed, more is the viscous drag

The further the fluid is from the fixed surface, the greater its speed.

Solid‐fluid Drag Force:

The force that opposes the motion of a solid in a fluid is called solid‐fluid drag force. The origin of this force is viscous drag. The more the speed of the solid, the more is the drag force.

Up‐thrust remains same since W stay same When D = W, resultant force = 0



As the ball released, it accelerates. The layer of liquid in contact with the ball moves and the adjacent layers oppose its motion. So an upward drag force occurs As the speed increases, drag force increases. At some particular speed, drag (D) becomes equal to weight (W). The body no more accelerates and reaches a constant speed called the terminal velocity. Up‐thrust has been neglected



Law about Fluid Motion:

Rate of flow at any cross‐section in the tube is constant.

Rate of flow =

= ∆

∆

Rate of flow is the volume of fluid moving per unit time. V = A l

Rate of flow =

∴ Rate of flow = A v

(v = velocity = )

∴ A v = constant Therefore in the narrow part of the tube water flows faster.

Aerodynamic Lift

Faster air exerts less pressure than slower air.

As the aircraft moves volume of air entering end A per second relative to the motion is equal

to the volume of air leaving end B per second because of the shape of the wing, air above the

wing has to travel a greater distance than the air beneath it. So the air above moves faster.

Faster air exerts less pressure than slower or stationary air.



Downward pressure of air on the top of the aircraft is less than the upward pressure from

beneath it. This is why the upward force is greater than the downward force and there is a

resultant upward force on the aircraft which is called the aerodynamic lift.

Force Diagram:

Aircraft flying horizontally at a constant velocity:



Aircraft gaining height at a constant velocity:

cos

sin



Momentum

Momentum:

Momentum is the product of mass and velocity. p = mv

Conservation of Momentum:

Total momentum of a system of two or more bodies remains constant before and after collision or explosion, provided no other external forces act upon them.

m1 u1 + m2 u2 = m1 v1 + m2 v2

m1 u1 + m2 u2 = (m1 + m2) v If u2 = 0, m1 u1 = (m1 + m2) v



m1 u1 – m2 u2 = ‐ m1 v1 + m2 v2

Explosion:

Total momentum = 0 ‐m1 v1 + m2 v2 = 0 ∴ m2 v2 = m1 v1 The two trolleys will gain equal momentum in opposite direction. So the net change in momentum is zero. But the total kinetic energy of the trolley increases after the collision. The spring does not work on the two trolleys and energy is transferred from the spring to the trolleys and energy is transferred from the spring to the trolleys.

Total momentum of the gun and bullet before the fire is zero.



After the fire, momentum of the bullet is mv1 For the total momentum to remains zero, the gun mast recoil, that is, move backward with the same momentum as the bullet.

An Experiment to investigate Conservation of Momentum, where two bodies

combine after collision:

G‐1 is given a push to the right. It moves at a constant velocity. After the collision with G‐2, both the gliders move together at a new constant velocity. During these motions, the card interrupts the light gates. Measurements: Mass of the gliders, m1 and m2 Length of the card, l Time of first interruption, t1 Time of second interruption, t2 Calculations:

Velocity of glider‐1 before collision is u =

Common velocity of the gliders after collision is v =

Total momentum before collision = m1 u Total momentum after collision = (m1 + m2) v The results should show that, m1 u = (m1 + m2) v So, the law is verified.



Impulse

Impulse: Impulse is the product of force applied and the time for which the force is applied. Impulse = Force time Usually we deal with impulse when a large force is applied for a short time. For example;

(i) When a cricket ball is hit by a bat (ii) When two mobbing bodies collide and etc.

Unit of Impulse = Ns

F =

Or, mv – mu = F t ∴ Change in momentum of a body = impulse on the body

Force‐time Graphs:

i. Constant Force:

Area under force‐time graph = impulse



ii. Linearly Changing Force:

Change in momentum on impulse = F t

= Average Force time

iii. Non‐uniformly Changing Force:

Impulse = Area under the graph = Number of complete square under the curve Area of one square



Impulses and Conservation of Momentum:

According to Newton’s 3rd Law of Motion, F1 = F2 And the forces act for the same length of time. ∴ F1 t = ‐ F2 t ∴ Impulse on m1 = ‐ Impulse on m2 ∴ Change in momentum of m1 = ‐ Change in momentum of m2 ∴ Net change in momentum is zero and the total momentum remains unchanged. Δ Pm1 + Δ Pm2 = 0 Δ Pm1 + Δ Pm2 = Δ P (change in momentum) Δ P = 0 Therefore there is no change in momentum ∴ Momentum before collision = Momentum after collision Change in momentum on m1 = ‐ Change in momentum on m2 Or, m1 v1 – m1 u1 = ‐ (m2 v2 – m2 u2) Or, m1 v1 – m1 u1 = ‐ m2 v2 + m2 u2 ∴ m1 v1 + m2 v2 = m1 u1 + m2 u2 ∴ Total momentum after collision = Total momentum after collision

To Measure the Force of a Kick:



Measurements: Mass of the ball = m Time of kick = t (from the timer) Height = h (from which the ball is kicked) Horizontal displacement = x Calculations: If t is the time the ball elapses in air, considering vertical motion,

h = gt2

∴ t =

Considering horizontal motion, x = v t

v =

F = =

∴ F =



Chapter 04

Work, Energy & Power

Work: It is defined as the product of the force applied and the distance moved in the direction of

the applied force.

Work = Force * Displacement in the direction of the force

W = F * d(x)

Circular Motion:

For an object which is moving in circle, a force always acts on the object which is directed towards the

centre of the circle. Here, the displacement of the object is along the tangent to the circle and the force

is perpendicular to the displacement. No work is done on the object.

Negative work is done when the displacement occurs in the opposite direction of the unbalanced force.

Its consequence is the deceleration of the body. E.g. work done by a braking force or frictional force

acting on a moving body is negative work.

Force‐Displacement Graphs:

i) Constant Force

Work done = Area under the graph

= F * x

ii) Force changing linearly

Ө

F

F

F

x

x

x

W = Fx W = (FCosӨ) * x W = (FCos900) * x

F (Braking Force)

W = ‐F *x

x

F

x F



Work done = Fx

iii) Force changing non‐linearly

Energy of a small square = 1x1=1J

No. of complete squares = x

No. of incomplete squares = y

Total energy under the graph

= (x*1) + (1

2 * y * 1)

= z J

Work Done by a gas that is expanding against a constant external pressure: PΔV

Energy

It is the ability of doing work. There are two types of mechanical energy:

1) Potential Energy

2) Kinetic Energy

Potential Energy

Whenever work is done against a non‐contact force, the energy gained by the body is called potential

energy. There are two types of potential energy:

a) Gravitational Potential Energy (GPE)

b) Elastic Potential Energy (EPE)

c) Electric Potential Energy

F

x



Gravitational Potential Energy

A body gains gravitational potential energy when its position is changed against gravitational force. A

body loses gravitational potential energy when its position is changed along the direction of

gravitational force.

Here, the gain in gravitational potential energy = mgΔ h.

Elastic Potential Energy

When work is done against elastic force, the gain in energy is known as elastic potential energy.

Example: elastic potential energy in a stretched spring

According to Hooke’s Law, force is directly proportional to displacement.

h

F

mg

W = F * h

GPE = mgh



Hooke’s Law:

F ∝ x

F = kx, where, k = spring constant (stiffness of the spring)

When work is done against a contact force, no potential energy is gained by the body. In this case, the

body usually gains thermal or internal energy.

Electric Potential Energy:

Kinetic Energy

Energy possessed by a body because of its motion is called kinetic energy. If a body of mass m moves

at a velocity v, its kinetic energy, K.E. = mv2.

Derivation of ½ mv2:

A constant force, F, is applied on a body of mass m, which is initially at rest. The body undergoes a

constant acceleration a, and gains a velocity v after a displacement of x.

v2 = u2 + 2ax

F (N)

x (m)

Energy = Area under the graph = Fx

E.P.E = Fx ‐‐‐‐‐‐‐ (i)

E.PE. = (kx)x = kx2 ‐‐‐‐‐ (ii)

FF

v

x

Smooth surface



or, v2 = 2ax

or, ax = v2

Kinetic Energy, K.E. = F * x

= ma * x

= m * (ax) = m * ( v2)

∴ K.E. = mv2

Relationship between kinetic energy and momentum:

Momentum of a body = its mass * its velocity

Momentum = m * v

∴ ρ = m * v Unit : kgms‐1

K.E. = mv2 * K.E. = *

=

Internal Energy: Some of potential energy and kinetic energy of the molecules in matter is called

internal energy.

Power

It is defined as the rate of doing work or the rate of energy transferred.

Power =

(Watt) Power =

(J/s)

∴ P = F * v

Efficiency

It is the ratio of useful energy or useful power output to the total energy or total power input. It is

expressed in both ration and percentage.

Efficiency =

* 100%

=

* 100%



Law of Conservation of Energy

“Energy can neither be created nor destroyed. It can only be transferred from one form to another. The

total amount of energy in the Universe is constant.”

EXPERIMENT: To investigate the law of conservation of energy

Procedure:

As the load falls, it loses potential energy. Both the glider and the falling mass (weight) gains kinetic

energy.

Measurements:

Mass of the glider with card = m1

Mass of the falling load = m2

Length of the card = L

Distance travelled by the glider before it interrupts the light gate = x

Time of interruption = t

Calculations:

Velocity of the glider when it interrupts the light gate: v =

Loss of G.P.E. of the load: m2gh or m2gx

Gain in Kinetic energy of glider:

Gain in Kinetic energy of load:

If the results show that: + = m2gx then it is proved

that the total energy of the system is conserved.

Assumption:



No friction in the pulley

The string is inextensible and weightless.

Elastic Collision

When the kinetic energy of a system of two or more colliding bodies remains constant before and after

the collision, it is called elastic collision.

Inelastic Collision

When the total kinetic energy of a system changes after collision, it is called inelastic collision.

When two bodies collide with springs between them, the collision is approximately elastic. Perfectly

elastic collision occurs between atoms and molecules. Usually, in other collisions, total kinetic energy

decreases because some energy is transferred to internal energy.

EXPERIMENT: To study an elastic collision

Procedure

The two gliders are given a small push towards each other. The gliders collide and rebound. During

these motions, each of the cards interrupts on the light gate twice.

Measurements:

1) Length of the cards = L1 and L2

2) Mass of the gliders = m1 and m2

3) Time of interruption of glider 1 before collision = T1 sec

4) Time of interruption of glider 2 before collision = T2 sec

m1 m2

L1 L2



5) Time of interruption of glider 1 after collision = T3 sec

6) Time of interruption of glider 2 after collision = T4 sec

Calculations:

Before collision: velocity of glider 1 = u1 =

velocity of glider 2 = u2 =

Total kinetic energy = m1u12 + m2u22

After collision: velocity of glider 1 = v1 =

velocity of glider 2 = v2 =

Total kinetic energy = m1v12 + m2v22

If,

m1u12 + m2u22 = m1v12 + m2v22, the collision is

elastic.

EXPERIMENT: To measure the speed of an air rifle pellet

Procedure

The rifle is fired horizontally so that the pellet is embedded in the suspended plastic block.

Measurements:

Mass of pellet = m1



Mass of block = m2

Maximum height gained by the pellet and the block = h

Calculations:

GPE gained by the pellet and the block = (m1 + m2) * g * h.

If the common velocity of the block and the pellet just after the collision is v, their kinetic energy is

= (m1 + m2)v2.

According to the law of conservation of energy,

(m1 + m2)gh = (m1 + m2)v2.

∴ v = √2 h

If the speed of the pellet before collision is u, then the law of conservation of momentum ,

m1u1 = (m1 + m2)v

∴ u1 =

* v



Chapter 05

Fluids

Pressure in fluid

Pressure exerted by a fluid is defined as the weight of fluid per unit area.

P = Weightoffluid

Area =

F

A

P = hρg

Atmospheric Pressure

Atmospheric pressure is the pressure due to the huge amount of air above us.

ρ = 1.2 kgm‐3

Atmospheric pressure = 9000 * 1.2 * 9.81 = 100,000 Pa

Pressure and depth, h

Pressure at a given depth is independent of direction.

The pressure is same in all directions. The pressure on a submerged object is perpendicular to the surface at each point on the surface.

where, h heightoffluid

ρ densityoffluid

g gravitationalfieldstrength

P P

Ph



The pressure at a given depth is independent of direction -- it is the same in all directions. This is another statement of the fact that pressure is not a vector and thus has no direction associated with it when it is not in contact with some surface. The pressure on a submerged object is always perpendicular to the surface at each point on the surface.

Upthrust

Upthrust is the force exerted on an object by a fluid in the upward direction when the object is immersed in the fluid. Upthrust is the result of pressure difference between the top surface and the bottom surface of a solid immersed in a fluid. Upthrust of a fluid on a particular solid at a particular temperature remains constant.

Resultant pressure = P2 –P1

P = (h2 – h1)ρg

= h ρg

F = Ahρg

∴ F = vρg (v = volume of the object, ρ = density of fluid, g= gravity)

Archimedes’ Principle

Up thrust on a body by a fluid is equal to the weight of the fluid displaced by the body.

Volume of can = volume of fluid displaced

Volume of fluid displaced (v) * Density of fluid (ρ) = mass of fluid (m)

Up thrust = (vρ)*g

h2

P2

P1 h1

P1 = h1ρg

P2 = h2ρg

Since,

h2 > h1,

∴ P2 > P1,



∴ Upthrust = mg Where mg is the weight of the fluid displaced.

Floatation

An object will float if its weight is equal to the weight of the fluid it displaces.

The ship will float if,

Weight of the fluid displaced = Weight of the ship

In order to stay afloat, an object (e.g. a ship) in a fluid (e.g. water) needs to displace fluid of weight equal to the object’s own weight. If the weight of the object is large, it will need to displace a large volume of the fluid, else it will sink. On the other hand, if the density of the fluid is high, the object will need to displace less volume of water (since density is more, a smaller volume of the fluid will produce the weight required to balance the object’s weight) and so the object will float more.

TITANIC

Upthrust

Weight



Moments and Couples

There are two types of motion:

1) Translation 2) Rotation

Translation:

Translation occurs when all the points of an object are displaced over the same distance at the same speed.

AA1 = BB1 CC1 A, B, C undergoes the same displacement at the same speed. So, the body undergoes only translation.

Rotation:

The points A, B, and C undergo different displacements at different speeds.

In rotation, different points of a body move through different distances at different speeds, and there is at least a single point which does not displace at all.

Sometimes a body undergoes both translation and rotation. Example; the motion of a top, motion of

the Earth and etc.



Moments:

The turning effect of a force is called its moment. Quantitatively moment is the produce of the force and its perpendicular distance from the turning point. Moment = Force 1 distance from the turning point/ pivot/ fulcrum

Two types of Equilibria: (i) Translational (ii) Rotational The block is at rest on the plane The block does not translate or rotate, so it is in both translational and rotational equilibria.

A ballet dancer spinning on her toe at a fixed point is in a translational equilibrium, but not in rotational equilibrium.

A car moving in a straight line is in rotational equilibrium, but not in translational equilibrium.

Condition for Translational Equilibrium:

Resultant force on the body must be zero

Condition for Rotational Equilibrium:

Resultant moment by the forces on the body about a point is zero. This is called principle of moments which states that: “When a body is in equilibrium, the total clockwise moment about a point is equal to the total anti‐clockwise moment about the same point” ∑ fd = 0 When a fan is moving, moment of the forces on the fan is not zero, but the resultant force is zero. So the fan only rotates and does not undergo translation



Resultant Force = 0 Resultant Moment 0, because both are clockwise moments

Couple:

A couple is formed when two equal forces having different lines of action act on a different body in opposite direction

[d1 = d2] Couple always causes rotation not translation. The moment of the couple: = F d2 + F d1 = F (d1 d2) = F d ∴ Total moment of a couple = one of the forces distance between the forces

Torque ( ):

The resultant moment of a numbers of forces is called a Torque ( ). Moment of a couple or a system of two or more forces is called Torque.



Centre of Gravity (G):

Centre of a gravity of a body is a point through which the total weight of the body seems to act To locate the centre of gravity of irregular shaped objects (thin), two methods are used:

(i) Balancing Method (ii) Hanging Method

Conditions for Equilibrium:

When a static system is acted upon with a number of co‐planar forces, then the conditions for equilibrium are:

(i) Sum of the forces in any one direction (horizontal) is zero (ii) Sum of the forces in another direction (vertical) is zero (iii) Sum of the moments about a point is zero

2m



Matter and Deforming Solids





According to general observation, solids are categorized into the following types: 1) Rigid: materials which do not deform easily when a force is applied. E.g. block of metal, wood 2) Plastic: materials which deform permanently and do not return to their original shape when a

force is applied. E.g. wet clay, dough, etc. 3) Elastic: materials which deform easily when a force is applied and returns to its original shape

when th0e force is removed. E.g. spring, rubber band

Different Types of solids:

Malleable

Materials those can be hammered into thin sheets . Gold, as an example, is malleable material and can

be hammered into Gold leaf.

Ductile

Ductile materials can be drawn into wires. Copper wires are used extensively for electrical connections

and are produced by drawing out cylinders to the desired thickness. Most ductile materials are also

malleable but not all malleable materials are not ductile.

Tough

Tough materials are those which are able to withstand large impact forces without breaking. Tough

metals usually undergo considerable plastic deformation in order to absorb the impact energy.

Brittle

A brittle object will break or shatter with little deformation when subjected to shocks or impacts.

Hard

The materials which resist plastic deformation caused by denting or scratching or cutting are said to be

hard materials.

Strong

An object is strong if it can withstand a large force before it breaks. The strength of a material will

depend on its size. For e.g. thick cotton thread requires a bigger breaking force than a thin wire.

The strength of a material is defined in terms of its breaking force

Hooke’s Law

“Extension of an elastic body is directly proportional to the force applied or directly proportional to the restoring force but in the opposite direction until it exceeds its elastic limit.”



F e or Fs ‐e (F = applied force, Fs = Restoring Force, e = extension also represented by Δx) F = ke Or Fs = ‐ke (The negative sign represents that the extension of the spring and the restoring force are proportional but takes place in the opposite direction.) K = F/e = spring constant/ or stiffness of the spring. Qualitative definition of ‘k’ = How much force is required to extend or compress the spring. Car springs are stiffer than ordinary laboratory springs. Quantitative definition: Spring constant is the force needed per unit extension. Force ‐ Extension graph of average extendable materials:

Limit of proportionality: The force up to which extension is directly proportional to force, that is, ‘F’ against ‘e’ graph is straight line, is called the limit of proportionality.

Elastic Limit: The force beyond which an elastic material behave plastically that is, the material does not return to its original shape or length when the force is removed is called elastic limit. Limit of proportionality and the elastic limit are very close and the limit of proportionality is a little less than the elastic limit. The extension or deformation beyond the elastic limit is called plastic deformation.

Yield Point & Breaking Point

Beyond elastic limit, at a certain point, known as the yield point, the extension of a body increases rapidly with even a small increase in force.

Original unextended length



If the object keeps extending, at a certain point, known as the breaking point, the object breaks down.

E = kx2

or, E = Fx = (kx)x = kx2

Unit : Joule

Stress, Strain & Young’s Modulus

Stress: It is the force per unit area of cross‐section.

k3

k2

1

2

3

F (N)

e (m)

Gradient = k1

∴ k3>k2>k1

∴ Spring (3) is more stiff than spring (2)

∴ Spring (2) is more stiff than spring (1)

Area under a force‐extension graph

= Energy stored in spring

Elastic Potential Energy



i.e. stress = Force

Areaofcross section

There are three types of stress:

1) Tensile2) Compressive3) Shear

Tensile Stress: It is defined as the tensile force per unit area of cross‐section.

Tensile stress = Force

Areaofcross section S.I.

Unit: Nm‐2 or Pascal

Strain: The extension per unit length is called the strain

Strain = L or,

L

Extension, e = L1 –L2

Young’s Modulus

For materials that obey Hooke’s Law, the tensile stress is found to be directly proportional to tensile strain i.e Stress α Strain.

A

F

Tensile Stress Compressive Stress

L1L2

Shear Stress



Stress α Strain

Stress = E * Strain

E = young Modulus

Young’s modulus (E) =

Strain Ԑ

Unit: Pa (Yes ! It is same as Pressure’s unit)

Stress – Strain Graph beyond the elastic limit of a ductile Material:

u =

A – Limit of proportionality: It is the point up to which the stress is proportional to strain and the graph is a straight line. After that the stress is not directly proportional but will come back to original shape after the removal of stress.

B – Elastic limit: It is the stress beyond which the body or the material behaves plastically and it fails to return to its original shape after the stress is removed.

y = Yield Point : The point at which material shows large increase in strain for a small increase in stress.

C – Ultimate tensile stress: Ultimate Tensile Strength = Ultimate tensile strength (UTS), often shortened to tensile strength (TS) or ultimate strength, is the maximum stress that a material can withstand while being stretched or pulled before failing or breaking.

Relationship Between Young Modulus and Stiffness.



Young’s modulus:

Strain Ԑ =

/

/ =

∗

∗ = k*

E = k*

∴ k =

If the gradient of the force‐extension graph or stress‐strain graph is large, that means the Young Modulus of the material is large and the material will be stiffer.

Compressive Force

When a force tries to reduce the volume of a material, the force is called a compressive force.

Compressive Stress

It is the compressive force per unit area.

Compressive stress =

The stress beyond which a material breaks is called its compressive strength.

Energy Density

The energy stored per unit volume or the work done per unit volume in an elastic deformation is called the energy density.

i.e

4πr2 F

F

F

F

Stress

Strain

. Energy density =

Stress =

Strain =

∴ Stress * Strain = * =

Stress = E * strain

Area = ½ * stress * strain

= Energy density



Experiment to measure the Young Modulus of a straight wire

Before adding the loads the length of the wire ‘l’ from the end of the wooden blocks and middle of the

pulley is measured using a meter just hanging the mass hanger only.

The diameter of the wire is also measured at several places of the wire using a screw gauge and an

average diameter ‘d’ is calculated.

A known mass ‘ m’ is hung with the help of the mass hanger and the extensions ‘e’ found with the help

of the meter rule.

The experiment repeated with several known masses and the corresponding extensions of the wire are

measured.

Calculations

Radius of the wire: r =

Cross Sectional Area of the Wire: A = πr2

ForceappliedtothewireF mg

Stressappliedtothewire F/A

Strain e/l

AgraphofStressagainstStraininplottedandthegradientofthegraphgivestheyoungmodulusofthematerial.

SafetyPrecaution:Safetygogglesshouldbewornincaseofsnappingwires.Theapparatusshouldbesetupatlowheightsothatfallingloadcan’tdomuchdamage.Alsofloormatsmustbeusedtoavoidbouncingbackofloadstofeet.

Precaution:Thescrewgaugeshouldbehandledcarefullytoavoiddamageofthewire.



Electric Field & Coulomb’s Law:

Two charged bodies attract or repel each other by a force called electrostatic force. Some points to

remember about electrostatic phenomena and electric charges are:

(i) By friction, electrons are transferred from one substance to another. Example: From

Perspex to cloth or from cloth to polythene.

(ii) Positively charged body means there is a lack of electrons and, negatively charged body

means there is excess of electrons.

(iii) Like charges repel and unlike charges attract.

(iv) Electric charges always transfer discretely, i.e. at whole number of electronic charge

1.6 10

Coulomb’s Law of Electronic Force:

The force of attraction or repulsion between two charged bodies is directly proportional to the product

of the amount of their charges, i.e. ∝ 1 2 (i)

The force is also inversely proportional to the square of their distance apart, i.e. ∝ (ii)

Combining (i) and (ii), ∝

∴ , where is a constant which depends on the medium between the charges. For vacuum

or air, 9 10 .

Calculate the electrostatic force between two spheres, 12 apart in air, having positive charges of

95 and 106 .

.6.29 10

If is positive, the force is repulsive and if is negative, the force is attractive.

In the expression, ,

, where is a constant called the permittivity of the medium between the charged

bodies.



i.e.

For vacuum, permittivity is o 8.85 10

Relative Permittivity:

The relative permittivity of a medium is the ratio of its permittivity, , to the permittivity of vacuum,

o.

∴

For air, o 1.005

For water, o 80

Permittivity of water is about 80 times that of air.

When NaCl is in water, electrostatic force of attraction between Na+ and Cl‐ becomes . So, NaCl

dissolves in water.

Electric Field:

The region around an electric charge, where it exerts an electrostatic force on another charge is called

an electric field.

Electric field is represented by lines of forces. It is a vector quantity; the direction of the field at any

point is defined as the direction of the force on a small positive charge placed at that point.

For an isolated point charge, the electric field is radial.



For a hollow charged conductor, there is no electric field inside and, if the conductor is spherical, then

the field outside is radial.

At a greater distance, flux density decreases, so field strength decreases.

Between two parallel charged plates, the field is uniform.

Electric Field Strength (E):

Electric field strength at a point is the force exerted by the field on 1C charge.

i.e.

;

:

The field strength at is,

Or,

∴ ∝



Experimental Investigations to find the pattern of Electric Field Lines:

(i) Between two point electrodes

(ii) Between two parallel plate electrodes



(iii) Field between two parallel plates

In region , the field lines are parallel and equal spaced. Such a field is called a uniform field.

In a uniform field, field strength, , remains constant everywhere.

Work Done in an Electric Field:

∴

Also,

Since ,

∴ therefore W/q = Ex

So E = V/d

Problem:



Electricity

Electric Current:

Qualitatively, an electric current is a flow of charge or charged particles.

In a metal conductor, the charged carries are free electrons and so, an electric current is usually a flow

of electrons.

Qualitatively, an electric current is defined as the rate of flow of charge, i.e.

Current =

I =

I = ∆

∆

SI unit: Ampere (A)

1 A = 1 Cs‐1

Charge is always transferred discretely.

Charge of an electron = 1.6 10

Since fraction of electron cannot be transferred, it is not possible for any amount of charge to be

transferred, i.e. the amount of charge must always be whole number multiple of electronic charge.

How many electrons have to moved to transfer 0.5C charge?

Q = ne

N = =.

.

therefore n = 3.125 x 1018

To make the free electron in a metal flow, a push is required. Qualitatively, this electrical push is called

e.m.f. or electromotive force. Usually a cell or a battery or a generator provides this electrical push.

Electromotive Force:

Quantitatively, e.m.f. is defined as the amount of work done or energy transfer to move one coulomb

charge in a complete loop of an electrical circuit.

E.m.f. of a cell is the amount of chemical energy transferred to electrical energy per coulomb charge

by the cell.

E.m.f. = /

E = or



W =

Unit: JC‐1 or Volt (V)

1 V = 1 JC‐1

Terminal Potential Difference:



E.m.f T + lost voltage VT = V1 + V2 + V3 VT = terminal voltage E = E1 + E2 + E3 + Elost The potential difference between the two points is an electrical circuit is the amount of work done to

flow 1 C of charge between the points.

Potential difference =

Terminal potential difference is the potential difference between the positive and the negative

terminals of the cell, when connected in an electrical circuit.

Some voltage is always lost due to the internal resistance of the supply itself.

Vlost = Ir r = internal resistance

E = VT + Vlost

∴ E = VT + Ir

E = e.m.f.

If the cell does not have any internal resistance,

E = VT

Therefore, e.m.f. is the total energy transferred by the cell per coulomb charge and terminal potential

difference is the total energy used up per coulomb charge by all the components in the external circuit.

To measure the e.m.f. ( ) and the Internal Resistance (r) of a cell roughly

When the switch is opened, the voltmeter reading is recorded. This reading is an approximate value

for the e.m.f. of the cell.

When the switch is closed, the voltmeter reading is again recorded. This reading is the terminal

potential difference, Vt. also the ammeter reading, I is recorded.



VT + Vlost

VT + r

r = – VT

∴ r =

Ohm’s Law:

The current in a conductor is directly proportional to the potential difference between two points of

the conductor, provided the temperature is constant.

∝ or, ∝

∴

∴ [R is the resistance of the conductor]

SOMETHING MISSING ABSENT WRITTEN IN COPY

Ohm’s Law:

Under constant physical conditions, the resistance of a conductor is constant, i.e. R is independent

of or or their directions



Obeying Ohm’s Law

Experiment: To investigate Ohm’s Law using a coil of wire and to measure its resistance

A set of values for are obtained by altering the total resistance of the circuit with the help of

the variable resistor. The fixed resistor is used to limit the maximum current of the circuit within a

small value, like not more than 1 (A). Otherwise, with greater current, the coil would become hot,

causing a change in resistance. If the wire is thin, then it might burn or melt from overheating.

The whole experiment is repeated by reversing the polarity of the battery.

A graph is plotted with . The best fit line is a straight line through the origin.



R = gradient of the line.

∝

Advantages of Graphical Calculation of Resistance over a tabular method:

i. The graphical method averages the set of good readings, avoiding the wrong values.

ii. Any systematic error is eliminated in the graphical methods.

Current in Microscopic View: ∆

∆ (Macroscopic view)

= carrier density, i.e. the number of free electrons in unit volume. [unit: m‐3]

= area of cross‐section of the conductors. [unit: m2]

electronic charge = 1.6 10

drift velocity of the charge carriers. [unit: ms‐1]

According to conductivity, materials are of three types:

i. conductors

ii. Semi‐conductors

iii. Insulator

In conductors, there are always a remarkable number of free electrons, i.e. carrier density is large.

The change in temperature does not affect their carrier density.

Semi‐conductors have very few electrons at normal temperature. As temperature increases, number

of free electrons, i.e. the carrier density increases. So, at low temperatures, semi‐conductors are like

insulators, and at high temperatures, they behave like good conductors. Example: silicon, graphite and

etc.

Insulators do not have any free electrons whatever the temperature is.



Random Speed and Mean free path:

The mean free path is the average distance travelled by free electrons between two successive

collisions.

The free path is travelled by the electron at a large speed, around 106 ms‐1. This speed is called the

random speed.

Drift Velocity:

When a potential difference is applied between the ends of the conductor, free electrons drift slowly

in the opposite direction of the convectional current. This average displacement of the charge carriers

in 1 second is called their drift velocity (typical value: 10‐3m or 104m)

Volume covered by the charged particles in 1s is ( )

∴ Number of free electrons passing through any cross‐section in 1s is .

∴ Charge transferred in 1s is .

1 1 1.6 10 10

Calculate the drift velocity

1 10

∴ 1 10 10

V = .



∴ V = 6.25 10

The length of the wire connecting the lamp with the battery is 1m. How long will it take for a free

electron to move from the power to the lamp?

Time = .

1.6 10

When the switch is closed, it takes a long time for the free electrons to move from the power supply

to the filament of the lamp. But practically the lamp lights instantly as the switch is closed. This is

because, when the switch is closed, all the charge carriers in the circuit start to move together at a

time, and the electrical effect travels around the circuit at the speed of light in the form of

electromagnetic wave.

Application of I = nAqv to interpret some cases:

Case 1:

I is constant because wires are connected in series

A is same because material is same

q is constant because it is electronic charge



Experiment: To compare the Drift Velocity of Charge Carriers in two wires

∴

∴ ∝ [I, n & q are constant]

Thin wire has a smaller area of cross‐section. So, drift velocity v in the thin wire is greater than in the

thick wire.

Filament diameter 1cm

Area of cross‐section A of the wire is 106 times that of the area of cross‐section of the filament.

∴ Drift velocity of the charge carriers in the filament is 106 times in the wire.

As calculated before, in wire 6.25 10

∴ In filament, 6.25 10 10

⇒ 6.25 10

∴ 625



Case 2:

Experiment: Compare the Drift Velocities of the charged Carriers in the two materials

∴

I, A & q are constant

∴ ∝

For conductor, 10

For semi‐conductor, 10

∴ For conductor, is 105 times that for semi‐conductor.

Since ∝ , in semi‐conductor, the charge carriers move 105times faster than in the conductor.



V‐I & I‐V Graphs for some non‐ohmic Conductors or Electrical Components:

Filament of a Lamp:

With increasing temperature, resistance of a conductor increases. At greater current, the filament is

heated up more; its temperature increases, so does its resistance. As a result, the gradient of V‐I graph

increases and the gradient of the I‐V graph decreases.

Microscopic View:

At higher temperature, , does not change because of the increase in internal energy, the

fixed ions in the metal vibrate more vigorously. The free electrons collide with fixed ions more

frequently. As a result, the passage of charge carriers is hindered more, their drift velocity v decreases.

Hence, current decreases and resistance R increases.

Thermistor:

A temperature‐sensitive resistor is called a thermistor.

There are two types of thermistors:

I. NTC (Negative Temperature Coefficient) Thermistor

Resistance of this types of thermistors decreases with the increase in temperature.

II. PTC (Positive Temperature Coefficient) Thermistor

Resistance of this types of thermistors increases with the increase in temperature.



For NTC Thermistor:

At lower temperature, these materials have very small carrier density . As temperature increases,

more electrons become free from the atoms. So, carrier density increases significantly and drift

velocity decreases slightly. Consequently, the current increases i.e. resistance decreases.



Semi‐Conducting Diode:

An electrical component, which allows current to flow only in one direction, is known as a diode.

The diode in this series is in forward bias. When voltage across the diode is very small no current flows

through the circuit. On increasing the voltage, the current very slowly increases. On reaching a certain

voltage (0.1V), the current starts to increase significantly. At this voltage, the carrier density

increases remarkably. So, increases and hence R decreases.



In this diagram, the diode is in reverse bias. Here the current reading remains zero even if the voltage

is increased too high. This is because in reverse bias, carrier density remains zero or negligible

whatever the voltage is.

Resistivity:

At constant temperature, resistance of a conductor depends on two factors:

i. Its length

∝ ‐‐‐‐‐‐‐(i) [when area of cross‐section A is constant]

ii. Its area of cross‐section (A)

∝ ‐‐‐‐‐‐‐‐(ii) [when is constant]



Combining (i) and (ii):

∝

⇒ ƍ

Where ƍ is a constant called the resistivity of the material of the conductor.

ƍ

⇒

Unit:

If A = 1 m2 & 1

ƍ

∴ ƍ = R

Resistivity:

The resistivity of the material of a conductor is its resistance per unit length and unit area of cross‐

section.

If temperature changes, resistance also changes. This is why resistivity changes.

Experiment: To measure the resistivity of the material of a coil of wire

Measurement of R:

With the help of the variable resistor, a set of values for are recorded. graph is plotted. The best‐fit line is drawn through the points.

R = gradient of the line

Precautions:



The current should be kept small, within 1A because at a grater current, the wire will be heated up,

changing its resistance. For this purpose, the fixed resistance has been connected in series.

Measurement of l:

Number of turns in the coil is recorded N. Diameter of the coil is measured D using a slide calipers.

Length of wire,

Measurement of A:

Diameter of the wire is measured with a micrometer screw gauge.

Using the measured values, resistivity is calculated.

Internal Resistance & Short‐Circuit Current:

⇒

∴

∴



Experiment: To measure E and r

By changing the total resistance of the circuit with the variable resistor, a set of values for terminal

voltage and current in the circuit are recorded. A graph is plotted . It will be a straight line, having y‐intercept and a negative gradient.

Short Circuit:

If the two terminals of a cell or a battery or any other power supply are directly connected with a very

low resistance wire (i.e. without any load), it is called a short circuit. During short circuit, a large current

flow, which is called a short circuit current.

⇒

⇒

∴

In case of a short circuit, R = 0

∴



i.e. Short

Short circuit current is the maximum possible current in a circuit and the internal resistance of the

supply limits the current (to the maximum value).

Typical values of Internal Resistance for different Power Source:

Power Source Internal Resistance ( )

AA Cell 1.5 0.5 (Ishort.

.3 )

Car Battery 12 0.05 (Ishort .240 )

E.h.t. Supply (Extra high tension) 5 1 (Ishort 5 )

Combinations of Resistors:

Series Combination:

In series combination same current flows through all the resistors, but the total voltage is divided

across the resistor. According to Ohm’s Law,

V1=IR1

V2=IR2

V3=IR3

If the effective resistance is R3, V=IRs



According to Kirchhoff’s Second Law,

V = V1 + V2 + V3

∴ IRs = IR1 + IR2 + IR3

∴ Rs = R1 + R2 + R3

Parallel Combination:

In parallel combination, voltage drop across the resistors remains same but the total current in the

circuit is divided in the resistors.

Using Ohm’s Law,

I1 =

I2 =

I3 =

In the effective resistance is Rp

I =



According to Kirchhoff’s 1st Law,

I = I1 + I2 + I3

∴

∴

Potential Divider:

A potential divider is an arrangement of resistor or resistance wires, connected in series, which is used

to divide the voltage from a source in proportion to the resistances.

Vout = IR2

I = in

∴ Vout = 2



∴ Vout = Vin

Vin → VT

Vout = Vin

In a potential divider circuit, when a component of low resistance is connected across the output, the

voltage drops remarkably from the expected voltage. However, if a very high resistance component is

connected across nearly same as the expected voltage.

Temperature Sensitive Potential Divider:

An NTC thermistor can be as a temperature as a temperature sensitive potential divider.



V = Vin

As temperature increases, resistance of NTC thermistor RT decreases.

∵ Vout = Vin

∴ When RT decreases, Vout increases.

Therefore, an increase in voltmeter reading indicates an increase in temperature.

Circuit 2:

If temperature increases, resistance RT decreases. So, the voltage across thermistor decreases.

Therefore, a decrease in voltmeter reading indicates an increase in temperature

Light Sensitive Potential Divider:

A light‐dependent resistor (LDR) can be used as a light‐sensitive potential divider. Resistance of an

LDR decreases with increase in the intensity of light falling on it.

Resistance is high in the dark and low in the bright light.



VR = Vin

Circuit 1:

When light intensity increases, RL decreases.

Vout = Vin

Therefore Vout increases. So, increase in voltmeter reading indicates an increase in light intensity.

Circuit 2:

VR= Vin

as light intensity increases, RL decreases.

∴ Vout decreases. So, decrease in voltmeter reading indicates an increase in light intensity.

This light‐sensitivity potential divider can be used as a light meter.



Another types of light‐meter:

[i]

[ii]

Current is not zero, since resistance is not infinite.

Potentiometer:

A potentiometer can provide a continuous variable potential difference from zero to full voltage of

the supply.

At position A of the sliding contact, the lamp gets the full voltage of the supply, and so the current in

the lamp is maximum.



As the contact is moved down, voltage across the lamp decreases. At position C, the voltage across

the lamp becomes zero, and so the current in the lamp also becomes zero.

Rotation type Potentiometer:

Vout=

Vin

As the knob is rotated from A to C, the voltage reading decreases from full voltage (of the supply) to

zero.

In parallel connection, R=maximum

V= 0

Rheostat:

At A, the voltage across the lamp = full voltage of the supply

When the sliding contact is moved towards C, the voltage becomes minimum, but not zero.



A rheostat is a variable resistor, by means of which, voltage across another component in the circuit

can be varied from full voltage of the supply down to a minimum value, which is not zero.

Error in Ammeter and Voltmeter Readings while measuring Resistance:

An ideal ammeter should have zero resistance and an ideal voltmeter should have infinite resistance.

But in practice, meters are not ideal, so they give rise to error.

In circuit (i), V R

In this circuit, there is no error in the voltmeter reading. Since the voltmeter does not have infinite

resistance, some current flows through it. So the ammeter cannot read the current passing only

through the resistor. As a result, the ammeter reading has error. So, the calculated resistance value

will be less than the original value. The percentage error is more if the resistance of the resistor is

more.

However, in this circuit, error will be more prominent if is large. To measure small resistance, this

circuit is suitable.

In circuit (ii), A R

In circuit (ii), the ammeter reading does not have any error. Since the ammeter does not have zero

resistance, some voltage is dropped across the ammeter. So the voltmeter cannot measure the voltage

dropped only across the resistor. In this circuit, the smaller the resistance of , the more prominent is

the error. This circuit is suitable for measuring large resistances.



Solar Cells:

Symbol:

A solar cell is a semiconducting device which can convert light energy into DC electricity. It doesn’t

produce any harmful gas like CO2 (which increases global warming) and SO2 (which produces acid

rain), so it is eco‐friendly.

Problems regarding solar cell: Solar cell is very inefficient in terms of converting sun light into

electricity. At low sunlight it produces high current but low voltage. So overall power output of solar

cell is low. On the other hand at high solar radiation the output voltage becomes high but the current

supply falls and eventually the power output drops. The maximum power is therefore obtained for a

certain value of solar radiation. Since intensity (brightness) of solar radiation varies over the day, so

does the power output of the cell.

Kirchhoff’s Law

Kirchhoff’s laws are used to find the currents in different parts of an electrical circuit that consists of

different electrical components.

Kirchhoff’s 1st Law

Kirchhoff’s first law refers to any point in an electrical circuit. It states that the total current arriving at

any junction or at any point in a circuit is equal to the total current leaving the junction or point.



According to Kirchhoff’s 1st law,

At point B:

Total incoming current = total outgoing current

I1 + I2 = I

Current is taken to be positive when it flows to a point and negative when it flows away from a point.

The first law is a consequence of the law of conservation of charge. That is, electrical charges are neither

produced nor destroyed or accumulated at any point of an electrical circuit.

Kirchhoff’s 2nd Law

Kirchhoff’s second law relates to the total Ԑ.m.f in a closed loop with the total potential drop in the

loop. It states that, round a closed loop, the algebraic sum of the Ԑ.m.f is equal to the sum of the

potential difference.

An Ԑ.m.f is considered positive if its direction is the same as that of the current, and negative if its

direction is opposite to that of the current.

Applying Kirchhoff’s 2nd law along ABEF,

6V = (I1 + I2)8 + (I1)3

= I1(8+3) + I2(8)

∴ 6V = 11I1 + 8I2 ________ (1)

Along CDEF,

4V = (I1 + I2)8 + I2(2)

∴ 6V = 8 I1 +10 I2 ________ (2)

∴ I1 = 0.61 A & I2 = ‐0.09 A



The second law is a consequence of law of conservation of energy. That is, energy transferred per

coulomb charge by the cells is equal to energy consumed by all the resistors in a closed loop.

Potentiometer to find EMF of unknown batteries



Electrical Height:

A to B → water flows due to difference in height

C to D → no flow, since no difference in height

AF and XY are two identical resistance wires.

C = midpoint of AF

D = midpoint of XY

Current flows between two points when there is a potential difference. Current flows from a higher

potential to lower potential. In the circuit, A is a higher potential and D is a lower potential. Current

flows from A to D. as the contact is moved from A to C, current decreases because potential difference



between the contact and D decreases. At C, the ammeter reading becomes zero because there is no

potential difference between C and D.

As the contact is moved from C towards F, current reading starts to increase, but this time current

flows in the opposite direction through the ammeter because all the points between C and F are at a

lower potential than D. This potential difference is called electrical height.

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CIE A Level Notes