CBSE Class 9 Science Ch 7 Work Energy Notes

Lift a 5 kg wheat bag 1 m off the floor, and you would say you “did work”. Lift three such bags to the same height, and you have done three times as much work. Lift one bag three times as high instead, and you have again done three times as much work. Both observations point to the same rule.

Learn by heartDefinition 1

Work done by a constant force $F$ on an object that is displaced a distance $s$ in the direction of the force is $W = F \times s$ .

Formula

W = F \times s

F

in newtons,

s

in metres,

W

in joules.

1\ \mathrm{J} = 1\ \mathrm{N}\times 1\ \mathrm{m} = 1\ \mathrm{kg\ m^2\ s^{-2}}

Exam Tip

Always name both the agency doing the work and the object it is done on. “Work done by the boy on the wheelchair” is a complete answer; “work was done” is not.

Figure 1 · On a force-displacement graph, work done equals the area under the graph, even when the force is not constant — **Figure 1** · On a force-displacement graph, work done equals the area under the graph, even when the force is not constant

1.1 When is work zero?

✓ Work is done when

A force acts on the object
The object is displaced
That displacement has a component in the direction of the force

✗ Work is zero when

No force acts ( $F=0$ )
No displacement occurs ( $s=0$ ), like pushing a rigid wall
The force is perpendicular to the displacement, like a girl carrying a box she also walks with

Common Mistake

Pushing a wall makes your muscles tired, but if the wall does not move, you have done zero work on it in the scientific sense. Feeling tired and doing work are not the same thing.

1.2 Positive and negative work

Positive work

Force and displacement point the same way
e.g. pushing a wheelchair forward
The object gains energy

Negative work

Force and displacement point opposite ways
e.g. a goalkeeper stopping a ball
The object loses energy

Solved Example 1

Q. While exercising, a girl lifts a dumbbell and lowers it back down. When does she do positive work, and when negative?

Lifting: force and displacement both point up, so this is positive work.
Lowering: her hand still pushes up to hold the weight, but the displacement is downward, so this is negative work.

Solved Example 2

Q. A goalkeeper’s hand moves back 15 cm while stopping a ball with a force of 200 N. Find the work done on the ball.

The displacement is opposite to the applied force, so it is taken as negative:
$W = 200\ \mathrm{N}\times(-0.15\ \mathrm{m}) =$ $-30$ J

2 The Work-Energy Theorem

A thrown cricket ball can knock over a wicket; a flowerpot dropped from a height can damage whatever it lands on. Both have gained the capacity to do work, which is what it means to possess energy.

Learn by heartDefinition 2

The work-energy theorem: the work done on an object equals the change in its energy. $\text{work done on an object} = \text{change in its energy}$

This holds for a system of objects too, and even when the force is not constant. Energy shares work’s SI unit, the joule (J).

Did you know?

The joule is named after James Prescott Joule, who studied how mechanical and thermal energy convert into each other. Mechanical work is only one way to transfer energy; it also moves as heat, as radiation (like sunlight reaching Earth), as sound, through electric circuits, and in nuclear reactions.

Solved Example 3

Q. In carrom, a striker hits a white coin, which then hits the black coin. Who does work, and what happens to the energy?

The striker does positive work on the white coin (force along its displacement), gaining it energy. By Newton’s third law, the white coin does negative work back on the striker. The white coin then does positive work on the black coin, and receives negative work in return, in the same pattern.

3 Forms of Energy

Energy shows up as mechanical, thermal, light, sound, chemical, electrical and nuclear energy, and constantly converts between forms: electrical energy becomes light in a bulb and heat in a water heater; chemical energy in food becomes the mechanical energy of your muscles; a ringing bell turns mechanical energy into sound.

Forms of Energy

MechanicalDue to motion or position

ThermalMakes things warm or hot

ChemicalStored in bonds between atoms, as in fuel and food

ElectricalDue to position or motion of charges

NuclearStored in the nuclei of atoms

4 Mechanical Energy

Mechanical energy is the energy an object has because of its motion or its position. That splits it into two parts.

4.1 Kinetic energy

Learn by heartDefinition 3

Kinetic energy is the energy an object possesses because it is moving. An object at rest has zero kinetic energy.

Starting from rest, the work-energy theorem says the work a force does on an object equals the kinetic energy it gains. Combining $W = Fs$ with $F = ma$ and the kinematic equation $v^2 = u^2 + 2as$ gives:

Formula

K = \tfrac{1}{2}mv^2

m

= mass (kg),

v

= speed (m/s),

K

in joules. Kinetic energy has no direction.

Solved Example 4

Q. If a vehicle’s velocity doubles, what happens to its kinetic energy?

$K_{\text{new}} = \tfrac12 m(2v)^2 = 4\times\left(\tfrac12 mv^2\right) =$ 4 times the original

Solved Example 5

Q. A 0.2 kg cricket ball is bowled at 154.8 km/h (= 43 m/s). Find its kinetic energy.

$K = \tfrac12(0.2)(43)^2 =$ 184.9 J

Given

15,000 kg jet, arrestor wire force 367,500 N, stopping distance 100 m

To find

Landing velocity

Solved Example 6

Work done by the wire (force opposite to motion) $= -367500 \times 100 = -36{,}750{,}000$ J
By the work-energy theorem, this equals the change in kinetic energy: $0 – \tfrac12(15000)v^2 = -36{,}750{,}000$
Solving, $v =$ 70 m/s (252 km/h)

4.2 Potential energy

A stretched slingshot, a drawn bow, a compressed spring: release any of them and they set something else moving, giving it kinetic energy it did not have before. That energy had to come from somewhere, and it came from the work done earlier to deform the band, bow or spring.

Learn by heartDefinition 4

Potential energy is energy stored by an object due to its deformation, or by a system of objects due to their relative positions.

Separated magnets, separated charges and a ball lifted away from the Earth all store energy this way too: work done against an internal force (gravitational, electric or magnetic) is stored, and released as kinetic energy when the objects are allowed to come back together.

Common Mistake

Not every internal force stores energy this way. Work done against friction does not become potential energy; it is lost, mostly as heat.

Since the Earth is far more massive than any object near it, the Earth barely moves, so the stored energy of an Earth-object system is simply called the object’s gravitational potential energy.

Formula

U = mgh

m

= mass,

g \approx 9.8\ \mathrm{m/s^2}

h

= height above the reference level (usually the ground)

Solved Example 7

Q. A fielder throws a 200 g ball 10 m straight up in celebration. Find its potential energy at the top ( $g=10\ \mathrm{m/s^2}$ ).

$U = mgh = 0.2\times 10\times 10 =$ 20 J

4.3 Conservation of mechanical energy

Learn by heartDefinition 5

Mechanical energy is the sum of kinetic and potential energy: $ME = K + U$ . When only gravity acts, mechanical energy stays constant: this is the conservation of mechanical energy.

Drop an object from height $h$ : at the top, $ME = 0 + mgh = mgh$ . Partway down, some height is lost and some speed is gained, but the two changes exactly cancel, so $ME$ stays $mgh$ throughout the fall. A swinging pendulum shows the same thing: released from height $h$ on one side, it has only potential energy; at the bottom it has only kinetic energy; on the far side it climbs back to almost the same height, converting that kinetic energy back to potential. In real life it slowly loses height because friction and air resistance drain away a little mechanical energy on every swing.

Exam Tip

Conservation of mechanical energy is often a shortcut. It can hand you a final speed or position directly, without working through every intermediate step of the motion the way Newton’s laws would require.

Solved Example 8

Q. Find the speed of a child at the bottom of a slide of height $h$ , ignoring friction.

All the potential energy converts to kinetic energy: $mgh = \tfrac12 mv^2 \Rightarrow v = \sqrt{2gh}$

The mass cancels out, so the speed depends only on the height, not on the child’s mass or the slide’s shape.

Given

10,000 kg truck at 72 km/h (=20 m/s), 30° escape ramp, sand resists with 50,000 N

To find

Minimum ramp length $d$ to stop the truck (rises 1 m for every 2 m travelled)

Solved Example 9

Initial KE $= \tfrac12(10000)(20)^2 = 2{,}000{,}000$ J. Height gained $= d/2$ , so PE gained $= 10000\times10\times(d/2) = 50000d$ .
Work-energy theorem: loss in KE = work done against sand + gain in PE
$2{,}000{,}000 = 50000d + 50000d = 100000d \Rightarrow d =$ 20 m

5 Power

Carrying a bag up the stairs in one minute feels very different from carrying it up slowly in five, even though the work done is identical. That difference is power.

Learn by heartDefinition 6

Power is the rate at which work is done: $P = \dfrac{W}{t}$ . Its SI unit is the watt (W), where $1\ \mathrm{W} = 1\ \mathrm{J\ s^{-1}}$ .

Did you know?

The watt is named after James Watt, who built an efficient steam engine. Horsepower (hp), still used for car engines and pumps, equals 746 W; it comes from early engineers comparing their new engines to the real horses they were replacing.

Solved Example 10

Q. A weightlifter raises 75 kg by 2 m in 5 s. Find the power required ( $g=10\ \mathrm{m/s^2}$ ).

$W = mgh = 75\times10\times2 = 1500$ J $P = \dfrac{1500}{5} =$ 300 W

Solved Example 11

Q. A 1000 kg car accelerates from rest to 20 m/s in 10 s. Find the engine’s power.

$W = \tfrac12(1000)(20)^2 – 0 = 200{,}000$ J $P = \dfrac{200000}{10} =$ 20,000 W

6 Simple Machines

A machine cannot reduce the total work a task needs, but it can make that work easier by changing the magnitude or direction of the force you apply.

Learn by heartDefinition 7

The force you apply is the effort; the force you are working against is the load. Mechanical advantage $=\dfrac{\text{load}}{\text{effort}}$ .

6.1 Pulley

Figure 2 · A fixed pulley only changes the direction of the effort, not its size, so its mechanical advantage is 1 — **Figure 2** · A fixed pulley only changes the direction of the effort, not its size, so its mechanical advantage is 1

A fixed pulley lets you pull down instead of lifting straight up, which is often more convenient, but the effort still equals the load: mechanical advantage $= 1$ . A movable pulley or a system of pulleys can give mechanical advantage greater than 1, lifting a heavy load with a much smaller effort. This is why elevators and cranes rely on pulley systems.

6.2 Inclined plane

Pushing a box up a shallow ramp needs less force than lifting it straight up, but you push it a longer distance to reach the same height. Work stays the same either way.

Formula

\text{mechanical advantage}=\frac{mg}{F'}=\frac{L}{h}

L

= length of the incline,

h

= height raised,

F'

= effort needed along the incline

Exam Tip

Since $L > h$ on any real ramp, $F' < mg$ , and mechanical advantage is always greater than 1. A longer, shallower ramp needs even less force. This is exactly why hill roads wind gradually instead of climbing straight up, and why an inclined ladder is easier to climb than a vertical one.

Solved Example 12

Q. A ramp raises an object over a 30 cm step; the ramp’s horizontal width is 40 cm. Find its mechanical advantage.

The ramp length (hypotenuse) $= \sqrt{30^2+40^2} = 50$ cm.
$\text{MA} = \dfrac{L}{h} = \dfrac{50}{30} =$ 1.67

6.3 Lever

Figure 3 · A lever balances when effort × effort arm = load × load arm; a longer effort arm needs less effort — **Figure 3** · A lever balances when effort × effort arm = load × load arm; a longer effort arm needs less effort

Formula

F_1 \times d_1 = F_2 \times d_2 \qquad \text{mechanical advantage} = \frac{\text{effort arm}}{\text{load arm}}

A lever trades a smaller effort moving a larger distance for a larger force moving a smaller distance

Solved Example 13

Q. On a seesaw, a 15 kg child sits 2 m from the fulcrum. Where must a 30 kg child sit to balance it?

$15\times 2 = 30\times L \Rightarrow L =$ 1 m from the fulcrum

Class	Arrangement	Examples
Class I	Fulcrum in between load and effort	Scissors, seesaw, pliers, crowbar
Class II	Load in between fulcrum and effort	Wheelbarrow, bottle opener, lemon squeezer
Class III	Effort in between fulcrum and load	Tweezers, broom, oar, fishing rod

Common Mistake

A lever, ramp or pulley reduces the effort needed, never the total work done. Machines redirect and rescale force; they never create energy.

Did you know?

A traditional Himalayan watermill, the gharat or panchakki, converts a stream’s potential energy into kinetic energy as it drops down a pipe, spinning a wheel that drives a grinding stone. Modern hydroelectric dams work on exactly the same principle, at a much larger scale.

Quick Revision: read this the night before the exam

Work $W = F\times s$ , only counting displacement along the force’s own direction.
Work is zero if there is no force, no displacement, or the force is perpendicular to the displacement.
The work-energy theorem: work done on an object equals the change in its energy.
Kinetic energy $K=\tfrac12 mv^2$ ; potential energy (gravitational) $U = mgh$ .
Mechanical energy $ME = K+U$ stays constant when only gravity does work.
Power $P = W/t$ , measured in watts; $1\ \mathrm{hp} = 746\ \mathrm{W}$ .
Mechanical advantage = load ÷ effort; simple machines change force, never total work.
A fixed pulley only changes direction (MA = 1); a longer incline or effort arm reduces the effort needed.
Levers fall into three classes, based on where the fulcrum, load and effort sit relative to each other.

Practice Questions: 1 mark

1 markDefine work done by a constant force.
1 markWrite the formula for kinetic energy.
1 markWhat is the SI unit of power?
1 markState the work-energy theorem.
1 markWhat is mechanical advantage?
1 markGive one example each of a Class I and a Class II lever.
1 markWhat is the mechanical advantage of a fixed pulley?
1 markIs work done when a coolie carries a box horizontally on his head? Why?

Practice Questions: 2 and 3 marks

2 marksA weightlifter holds a barbell steady above her head. Is she doing any work on it? Explain.
3 marksA 0.2 kg ball is thrown 10 m straight up. Find its potential energy at the top ( $g=10\ \mathrm{m/s^2}$ ).
3 marksIf a vehicle’s speed is tripled, by what factor does its kinetic energy increase? Show your working.
2 marksWhy do hill roads wind around in gentle slopes instead of climbing straight up?
3 marksA weightlifter raises 75 kg by 2 m in 5 s. Find the work done and the power required.
2 marksState two examples each of positive and negative work done in daily life.
3 marksExplain, with an example, why work done against friction is not stored as potential energy.
3 marksA ramp raises an object over a 30 cm step and has a horizontal width of 40 cm. Find its mechanical advantage.
2 marksWhat is the difference between a fixed pulley and a movable pulley in terms of mechanical advantage?
3 marksOn a seesaw, a 15 kg child sits 2 m from the fulcrum. Where should a 30 kg child sit to balance it?

Practice Questions: 5 marks

5 marksDerive the expression for kinetic energy $K=\tfrac12 mv^2$ using the work-energy theorem and the kinematic equations.
5 marksDerive the expression $U = mgh$ for gravitational potential energy, and state the conservation of mechanical energy with an example.
5 marksA 15,000 kg jet lands and is stopped in 100 m by an arrestor wire exerting 367,500 N. Find its landing speed.
5 marksExplain the three classes of levers with one labelled diagram and one real-life example for each class.
5 marksA 1000 kg car accelerates from rest to 72 km/h in 10 s. Find the work done by the engine and the power required.

Case Study

A 10,000 kg truck moving at 72 km/h (20 m/s) has its brakes fail. The driver steers onto a 30° escape ramp, where sand exerts a constant force of 50,000 N opposing the truck’s motion. For a 30° incline, the truck rises 1 m vertically for every 2 m travelled along the ramp.

1 markWhat is the truck’s initial kinetic energy?
2 marksIf the truck travels a distance $d$ along the ramp, write an expression for the height gained.
2 marksUsing the work-energy theorem, find the minimum ramp length needed to stop the truck.

Multiple Choice

1 markThe SI unit of work and energy is:
(a) newton(b) watt(c) joule(d) pascal
1 markWork done is zero when:
(a) force and displacement are parallel(b) force is perpendicular to displacement(c) force and displacement are both large(d) never
1 markIf the height of a falling object doubles, its potential energy at that height:
(a) stays the same(b) doubles(c) quadruples(d) halves
1 markA fixed pulley has a mechanical advantage of:
(a) 0(b) 1(c) greater than 1(d) less than 1
1 markScissors are an example of a lever of:
(a) Class I(b) Class II(c) Class III(d) none of these
1 mark1 horsepower equals:
(a) 100 W(b) 746 W(c) 1000 W(d) 550 W
1 markA pendulum bob at its lowest point has:
(a) only potential energy(b) only kinetic energy(c) zero mechanical energy(d) neither kinetic nor potential energy

Assertion and Reason

Choose: (a) both A and R true, R explains A; (b) both true, R does not explain A; (c) A true, R false; (d) A false, R true.

Assertion (A): A person pushing against a rigid wall does zero work on it.
Reason (R): The wall does not undergo any displacement.

Assertion (A): The mechanical energy of a freely falling object remains constant.
Reason (R): Its kinetic energy and potential energy both stay constant during the fall.

Assertion (A): An inclined plane has a mechanical advantage greater than 1.
Reason (R): The length of the incline is greater than the height raised.

Answer Key

MCQ 27 to 33(c) · (b) · (b) · (b) · (a) · (b) · (b)

A and R1. (a) · 2. (c), KE and PE both change, only their sum stays constant · 3. (a)

Case study24. 2,000,000 J · 25. h = d/2 · 26. d = 20 m