CBSE · Class 9 · Science

Chapter 7: Work, Energy, and Simple Machines

Work, kinetic and potential energy, conservation of mechanical energy, power, pulley, inclined plane and lever.

Notes

Open each topic to read it.

Work done by a constant force

Work done on an object by a constant force = force × displacement in the direction of the force: W = F × s.
The SI unit is the joule (J): 1 J = 1 N × 1 m = 1 kg m² s^-2.
Work has magnitude but no direction; it can be positive or negative.
For a force-displacement graph, the work done equals the area under the graph (the shaded rectangle for a constant force).
Always state which force does the work and on which object.

When is work zero? Positive and negative work

Work is zero if F = 0 or if the displacement s = 0 (pushing a rigid wall, holding a barbell steady).
A force perpendicular to the displacement does no work (a girl carrying a box horizontally; the upward force does zero work).
Positive work: displacement in the same direction as the force (pushing a wheelchair, lifting a bucket).
Negative work: displacement opposite to the force (a goalkeeper stopping a ball, friction on a sliding object, lowering a dumbbell).

Energy and the work-energy theorem

Energy is the capacity to do work. Its SI unit is the joule.
Work-energy theorem: work done on an object = change in its energy.
Positive work increases the energy of the object; negative work decreases it.
Energy can be transferred by work, by heat (hot to cold), by radiation, and by sound waves.

Forms of energy

Mechanical (motion or position), thermal, light, sound, electrical, nuclear and chemical energy. Energy changes from one form to another: a bulb changes electrical energy to light and heat, food gives chemical energy for muscles, a bell changes mechanical energy into sound, a solar panel changes light into electrical energy, photosynthesis stores light as chemical energy.

Kinetic energy

Energy due to motion. Work done on an object of mass m that changes its speed from u to v: W = ½m(v² − u²).
Starting from rest:
K = ½ m v²
Kinetic energy has no direction. If the speed doubles, K becomes four times.
Kinetic energy does not depend on position if the speed stays constant.

Potential energy

Energy stored in an object due to its deformation (stretched rubber band, compressed spring, bent bow) or the relative positions of objects that interact by forces (magnets, charges, Earth and ball).
Gravitational potential energy of a mass m at a small height h:
U = m g h
Activity 7.1: a ball dropped from a greater height makes a deeper depression in sand.
The expression mgh is valid only near the surface of the Earth. Horizontal motion at constant height does not change it.

Conservation of mechanical energy

Mechanical energy = kinetic energy + potential energy.
If no external force (such as friction) acts, the mechanical energy is conserved. For a freely falling object, the loss in potential energy equals the gain in kinetic energy, and the total remains mgh.
A pendulum bob swings to almost the same height on the other side. In real life it slows and stops because of friction at the support and air resistance.
Sliding from height h without friction: v = √(2gh), independent of mass and the shape of the slide.

Power

Power is the rate of doing work: P = W / t.
The SI unit is the watt (W): 1 W = 1 J s^-1. Named after James Watt.
1 horsepower = 746 W.
Doing the same work in less time needs more power.

Simple machines and mechanical advantage

Simple machines make work easier by changing the magnitude or direction of the force. They do not reduce the total work.
Effort is the force we apply; load is the force to be overcome.
Mechanical advantage = load / effort
Machines do not create energy.

Pulley and inclined plane

A fixed pulley only changes the direction of the force. Its mechanical advantage is 1. Movable pulleys or systems of pulleys can have MA greater than 1.
An inclined plane lets a smaller force move a load to a height over a longer distance. F′ × L = mgh, so MA = L / h (greater than 1).
A longer, gentler slope needs a smaller effort but the effort acts over a larger distance. This is why hill roads wind around the slopes.

Lever

A lever is a rigid bar that rotates about a fixed point, the fulcrum. It has a load, an effort, a load arm and an effort arm.
Balance:
effort × effort arm = load × load arm (F₁d₁ = F₂d₂)
MA = effort arm / load arm. A longer effort arm needs less effort, but it moves through a larger distance, so the total work is the same.
Classes: Class I fulcrum in between (scissors, crowbar, seesaw, balance); Class II load in between (wheelbarrow, lemon squeezer, bottle opener); Class III effort in between (tweezers, broom, tongs, hammer).

At a glance

Work is done when a force displaces the object in the direction of the force.
Kinetic energy ½mv²; potential energy mgh; power W/t.
Machines change the size or direction of a force, but cannot reduce the total work.

Numericals

Worked solutions for the examples and calculation questions in this chapter.

EXAMPLE 7.2 · GOALKEEPER (NEGATIVE WORK)

W = F × s = 200 N × (−0.15 m) = −30 J

EXAMPLE 7.4 · VELOCITY DOUBLED

K₂ = ½m(2v)² = 4 × ½mv² = 4K₁

The kinetic energy becomes 4 times.

EXAMPLE 7.5 · FAST BOWLER

154.8 km h⁻¹ = 43 m s⁻¹, m = 0.2 kg.

K = ½mv² = ½ × 0.2 × 43² = 184.9 J

EXAMPLE 7.6 · AIRCRAFT ON A CARRIER

−½ × 15000 × v² = −(367500 × 100)

v² = 4900 → v = 70 m s⁻¹ = 252 km h⁻¹

EXAMPLE 7.7 · POTENTIAL ENERGY OF A BALL

mgh = 0.2 × 10 × 10 = 20 J

EXAMPLE 7.8 · CHILD ON A SLIDE

½mv² = mgh → v = √(2gh)

The velocity depends only on h, not on the mass or the shape of the slide.

EXAMPLE 7.9 · ESCAPE RAMP

v = 72 km h⁻¹ = 20 m s⁻¹; initial energy = ½ × 10000 × 20² = 2 000 000 J.

Final energy = mg(d/2) = 50 000 × d

−50 000 d = 50 000 d − 2 000 000 → d = 20 m

EXAMPLE 7.10 · POWER OF A WEIGHTLIFTER

W = mgh = 75 × 10 × 2 = 1500 J

P = W/t = 1500 / 5 = 300 W

EXAMPLE 7.11 · POWER OF A CAR ENGINE

v = 20 m s⁻¹, W = ½ × 1000 × 20² = 200 000 J

P = 200 000 / 10 = 20 000 W

EXAMPLE 7.12 · INCLINED RAMP

Height = 30 cm, width = 40 cm, so length = 50 cm.

MA = L/h = 50/30 = 1.67

EXAMPLE 7.13 · SEESAW

15 × 2 = 30 × L → L = 1 m

The 30 kg child sits at seat D (1 m from the fulcrum, on the opposite side).

REVISE, REFLECT, REFINE 5 · STUDENT AND BUILDING

(i) U = mgh = 50 × 10 × 72.5 = 36 250 J

(ii) Same height, so the gain is also 36 250 J

(iii) The potential energy depends only on the initial and final heights, not on the path taken.

REVISE, REFLECT, REFINE 6 · CRANE

10th floor: energy E = mgh₁ in time t. 20th floor: 2E in time 2t.

P₁ = E/t, P₂ = 2E/2t = E/t

The energy is doubled (the extra energy is another E); the power is the same.

REVISE, REFLECT, REFINE 8 · SCOOTER

Day 1 mass = 100 + 60 = 160 kg; day 2 = 100 + 60 + 40 = 200 kg

Fuel energy ∝ ½mv²: 160 : 200 = 4 : 5

REVISE, REFLECT, REFINE 10 · BALL THROWN UP

Initial K = ½ × 2 × 20² = 400 J

Final U at 19.4 m = 2 × 10 × 19.4 = 388 J

Work by air resistance = 388 − 400 = −12 J

REVISE, REFLECT, REFINE 11 · VARIABLE FORCE

Work = area = ½(1)(50) + (2)(50) + ½(1)(50) = 150 J

(i) ½(10)v² = 180 → v = 6 m s⁻¹

(ii) K = 180 + 150 = 330 J → v = √66 ≈ 8.1 m s⁻¹

REVISE, REFLECT, REFINE 12 · MOON

h = v²/(2g); with g/6, h becomes 6 × 8 = 48 m

REVISE, REFLECT, REFINE 13 · CAR BRAKING

(ii) K at A = ½ × 1000 × 35² = 612 500 J

(iii) Work by brakes = 0 − 612 500 = −612 500 J

REVISE, REFLECT, REFINE 14 · ENERGY GRAPH

Total mechanical energy = 30 J (at O, v = 0). m = 0.5 kg.

P: U = 20 J, K = 10 J, v = √(2×10/0.5) = √40 ≈ 6.3 m s⁻¹

Q: U = 30 J, K = 0, v = 0

R: U = 40 J is more than the total energy, so the ball cannot reach R.

REVISE, REFLECT, REFINE 15 · COCONUT

(i) v = √(2gh) = √(2 × 10 × 10) ≈ 14.1 m s⁻¹

(ii) K = mgh = 1.5 × 10 × 10 = 150 J; 3000 × d = 150 → d = 0.05 m = 5 cm

NCERT Q&A

Answers to the Pause and Ponder questions and the Revise, Reflect, Refine exercises. Numerical working is in the Numericals tab.

Pause and Ponder

PAUSE AND PONDER 1

Is the weightlifter doing any work on the barbell while holding it steady?

Answer: No. The barbell has no displacement, so the work done is zero.

PAUSE AND PONDER 2

Is the work done by friction on the stack of coins that travels on a rough surface positive, negative or zero?

Answer: Negative. Friction acts opposite to the displacement of the coins.

PAUSE AND PONDER 3

When you pedal a bicycle on a flat road, in what forms does your muscular energy appear?

Answer: As kinetic energy of you and the bicycle, and as thermal energy (heating of tyres, chain, road and air due to friction and air resistance) and some sound.

PAUSE AND PONDER 4

Two objects A and B of masses m and 4m have the same kinetic energy. What is the ratio of their velocities?

Answer: ½mv_A² = ½(4m)v_B² gives v_A² = 4v_B², so v_A : v_B = 2 : 1.

PAUSE AND PONDER 5

Does the kinetic energy of an object moving with constant velocity change with its position?

Answer: No. Kinetic energy depends only on mass and speed, and these do not change.

PAUSE AND PONDER 6

Does the potential energy near the Earth change if an object moves horizontally with constant velocity? What if it is raised vertically?

Answer: It does not change for horizontal motion at the same height. When raised gradually vertically, it increases (U = mgh).

PAUSE AND PONDER 7

For Fig. 7.19, find the mechanical energy of the ball just before it hits the ground.

Answer: At the ground h′ = 0, so U = 0. Its speed is v = √(2gh), so K = ½mv² = mgh. The mechanical energy is mgh, the same as at the start.

PAUSE AND PONDER 8

In the ball roller coaster, how do the kinetic and potential energy change at A, B and C? Why are later points lower?

Answer: Going down a slope, potential energy changes to kinetic energy; going up, kinetic energy changes back to potential energy. Later peaks are lower because some mechanical energy is lost as heat and sound due to friction and air resistance.

PAUSE AND PONDER 9

Why are roads on hills built to wind around gentle slopes?

Answer: A gentle, longer slope gives a larger mechanical advantage (L/h), so the force (and engine effort) needed to climb is smaller. The distance is larger, but the work done against gravity remains the same.

PAUSE AND PONDER 10

Why is climbing an inclined ladder easier than climbing a vertical one?

Answer: The inclined ladder works like an inclined plane. The longer path means a smaller force is needed at each step to reach the same height.

PAUSE AND PONDER 11

Why is it easier to open the lid of a can with a spoon?

Answer: The spoon acts as a lever. The rim of the can is the fulcrum, the long handle gives a long effort arm and the lid a short load arm, so a small effort is enough.

PAUSE AND PONDER 12

Why do you push the object closer to the fulcrum of scissors when cutting a hard object?

Answer: Moving the object closer to the fulcrum shortens the load arm. The mechanical advantage (effort arm / load arm) increases, so less effort is needed.

PAUSE AND PONDER 13

Why do all real machines eventually slow down and stop?

Answer: Friction and air resistance do negative work and convert mechanical energy into heat and sound. Machines cannot create energy, and the lost energy cannot be reused, so without a fuel or power supply they stop.

Revise, Reflect, Refine

QUESTION 1

State True or False.

Answer: (i) False: no displacement means no work. (ii) True. (iii) True. (iv) False: a motionless stretched rubber band has potential energy. (v) True.

QUESTION 2

Fill in the blanks.

Answer: (i) force × displacement. (ii) 1. (iii) ½mv². (iv) mgh. (v) rate.

QUESTION 3

At the highest point of a ball thrown upward, which statements are correct?

Answer: (iii) and (iv). The force is not zero (gravity acts) and the acceleration equals g; but the velocity is zero, so kinetic energy is zero, and the height is greatest, so potential energy is maximum.

QUESTION 4

Identify the energy transformation in each situation.

Answer: (i) Truck uphill: chemical energy of fuel to kinetic and potential energy (and heat). (ii) Unwinding watch spring: elastic potential energy to kinetic energy. (iii) Photosynthesis: light energy to chemical energy. (iv) Water flowing from a dam: potential to kinetic energy. (v) Burning matchstick: chemical to heat and light energy. (vi) Fire cracker: chemical to heat, light, sound and kinetic energy. (vii) Microphone: sound to electrical energy. (viii) Glowing bulb: electrical to light and heat energy. (ix) Solar panel: light to electrical energy.

QUESTION 5

A 50 kg student is lifted in an elevator, and later climbs the stairs, to the top of a 72.5 m building. Find the gain in potential energy and the dependence on path.

Answer: (i) 36 250 J. (ii) 36 250 J. (iii) Potential energy depends only on the height gained, not on the path.

QUESTION 6

A crane lifts mass m to the 10th floor in time t and to the 20th floor in time 2t. How much more energy and power?

Answer: The energy is doubled (an extra mgh₁ is needed), but the power is the same, because both the energy and the time are doubled.

QUESTION 7

Which factors determine the energy to raise a flag with a pulley? Does speed change the work? If the speed doubles, how does power change?

Answer: The mass of the flag and the height. The work (mgh) does not depend on the speed. If the flag is raised at double the speed, the time is halved, so the power doubles.

QUESTION 8

A scooter of mass 100 kg is ridden by a 60 kg man, and next day with his 40 kg son as well, reaching the same speed in the same time. Find the fuel ratio.

Answer: 4 : 5 (masses 160 kg and 200 kg).

QUESTION 9

Draw a seesaw balanced with an adult twice as heavy as the child.

Answer: The adult sits at distance d from the fulcrum, and the child on the other side at distance 2d, since F₁d₁ = F₂d₂. Diagram: a seesaw with the fulcrum at the centre, the adult at half the distance of the child.

QUESTION 10

A 2 kg ball is thrown up with 20 m s⁻¹. Find the sign of the work done by gravity and the work done by air resistance if it reaches 19.4 m.

Answer: (i) Negative during the upward motion and positive during the downward motion. (ii) −12 J.

QUESTION 11

A 10 kg block has 180 J kinetic energy at 0 m. Using the force-displacement graph in Fig. 7.37, find its speed at 0 m and 4 m, and whether the acceleration is ever negative.

Answer: (i) 6 m s⁻¹. (ii) The work done is 150 J, so K = 330 J and v ≈ 8.1 m s⁻¹. The acceleration is never negative, as the force is always in the direction of motion.

QUESTION 12

The Moon's gravity is 1/6 that of the Earth. A ball goes 8 m up on Earth. How high on the Moon with the same upward velocity?

Answer: 48 m.

QUESTION 13

A 1000 kg car moves at constant speed, then brakes to a stop (Fig. 7.38). Describe the motion and find the kinetic energy at A and the work by the brakes.

Answer: (i) Between A and B the car moves with constant speed (35 m s⁻¹). (ii) 612 500 J. (iii) −612 500 J (negative work). (iv) The kinetic energy changes into thermal energy (and some sound) through friction in the brakes and between the tyres and the road.

QUESTION 14

From Fig. 7.39, calculate the velocity of the 0.5 kg ball at P, Q and R.

Answer: Total mechanical energy = 30 J. At P: 6.3 m s⁻¹. At Q: 0. At R: the potential energy (40 J) exceeds 30 J, so the ball cannot reach R.

QUESTION 15

A 1.5 kg coconut falls 10 m on to wet sand, with an average resistive force of 3000 N. Find its speed just before impact and the depth of the depression.

Answer: (i) About 14.1 m s⁻¹. (ii) 0.05 m (5 cm).