Chapter 5: Work, Energy and Power
Original notes — ready CBSE (NCERT, English Medium) • Class 11 • Physics • Book: Physics — Part I
Work, Energy and Power — EG's own summary, in simple words
Book: Physics — Part I (CBSE, Class 11) | Chapter 5 | Medium: English
Work, energy and power are everyday words with precise meanings in physics. Work is done when a force moves something; energy is the capacity to do work; power is how fast work is done. Together they give a shortcut that often avoids difficult force-and-acceleration calculations.
Diagram (labels on the image): Work against gravity: lifting the barbell raises its gravitational potential energy (mgh). The higher and heavier the lift, the more work is done.
What you should learn
- Compute work as W = F s cos θ and give its sign for θ = 0°, 90° and 180°.
- Use kinetic energy ½mv² and the work–energy theorem for constant and variable forces.
- Find work from the area under an F–x graph, including for a spring.
- Apply conservation of mechanical energy only when non-conservative forces do no work.
- Classify collisions as elastic or inelastic and use momentum conservation in both.
1. What work means in physics
Work is done when a force causes displacement. For a constant force F acting at an angle θ to the displacement s, work W = F s cos θ. Work is a scalar, measured in joules (J). Only the component of force along the displacement does work: pushing a wall that does not move does no work, however tired you feel, and carrying a load horizontally does no work against gravity, because the force is vertical while the displacement is horizontal. Work can be positive (force along displacement), negative (force opposite displacement, like friction) or zero (force perpendicular to displacement).
2. Kinetic energy and the work-energy theorem
Kinetic energy is the energy of motion: K = ½mv² for a mass m moving at speed v. The work-energy theorem says the net work done on a body equals its change in kinetic energy. This is a powerful shortcut: if you know the work done by all forces, you know the change in speed without following the motion step by step. Stopping distance problems are the classic use — friction does negative work until the kinetic energy is used up.
3. Potential energy — stored energy
Potential energy is stored energy due to position or shape. Gravitational potential energy near the Earth is U = mgh, where h is height above a chosen reference level. Elastic potential energy is stored in a stretched spring or rubber band: U = ½kx², where k is the spring constant and x is the stretch. Potential energy only makes sense for conservative forces — forces like gravity and spring force, where the work done does not depend on the path taken. Friction is non-conservative: the work it does depends on the path, and the energy becomes heat.
4. Conservation of mechanical energy
When only conservative forces do work, total mechanical energy (kinetic + potential) stays constant. A falling ball loses potential energy and gains exactly the same amount of kinetic energy; a swinging pendulum swaps the two back and forth. This principle solves many problems in one line: equate total energy at the start and at the end. When friction acts, mechanical energy is not conserved — some becomes heat — but total energy, including heat, still is.
5. Power
Power is the rate of doing work: P = work ÷ time, measured in watts (W). One watt is one joule per second. Power can also be written as P = Fv for a force acting on a body moving at speed v. That is why a car engine's power matters for climbing hills and for top speed: at a given power, a larger force means a smaller speed. Efficiency compares useful output to total input — no machine is 100% efficient, because some energy is always lost, usually as heat.
Full coverage — every section of this chapter
§5.1 Introduction: Work, energy and power are everyday words with strict physics meanings. Energy methods often solve problems that force methods make messy, because energy is a scalar and only initial and final states matter.
§5.2 Notions of work and kinetic energy — the work-energy theorem: This section previews the chapter's spine: work done on a body changes its kinetic energy. The detailed statements follow in §§5.3–5.6.
§5.3 Work: For a constant force, W = F s cos θ, measured in joules. Work is positive when force has a component along displacement, negative when it opposes displacement (friction), and zero when force is perpendicular to displacement (a satellite's gravity, a porter carrying a load horizontally).
§5.4 Kinetic energy: K = ½mv² for a body of mass m at speed v. Kinetic energy is a scalar and depends on the frame of reference, since speed does. Doubling speed quadruples kinetic energy.
§5.5 Work done by a variable force: When force varies with position, work is the area under the F–x graph. For a spring the graph is a triangle, which is where ½kx² comes from.
§5.6 The work-energy theorem for a variable force: In general, the net work done by all forces equals the change in kinetic energy, W_net = K_f − K_i. Stopping-distance problems are the classic use: friction's negative work eats the kinetic energy to zero.
§5.7 The concept of potential energy: Potential energy is energy stored by position or configuration, defined only for conservative forces, whose work is path-independent. The force follows from the potential energy by F = −dU/dx, and the zero of potential energy is a free choice of reference level.
§5.8 The conservation of mechanical energy: When only conservative forces do work, K + U stays constant. A falling body trades potential energy for kinetic energy and back; friction breaks the conservation by converting mechanical energy into heat.
§5.9 The potential energy of a spring: For a spring of constant k stretched or compressed by x, U = ½kx², from the triangular area under the F–x graph. The same result follows from integrating the spring force.
§5.10 Power: Power is the rate of doing work, P = dW/dt, in watts (1 W = 1 J/s). Average power is total work over total time; instantaneous power for a force F on a body moving at velocity v is P = F·v. The commercial unit of energy, the kilowatt-hour, is 1 kW used for 1 hour, equal to 3.6 × 10⁶ J.
§5.11 Collisions: Momentum is conserved in every collision. Kinetic energy is conserved only in elastic collisions; in a perfectly inelastic collision the bodies stick together and kinetic energy loss is maximum. The coefficient of restitution e (1 for perfectly elastic, 0 for perfectly inelastic) measures the bounce. Collisions in two dimensions are handled component by component.
All eleven sections above are numbered sections of the rationalised chapter; this page carries no legacy topic.
Chart / Table — Energy forms — where the energy sits
| Form | Formula | Example |
|---|---|---|
| Kinetic | ½mv² | a moving car |
| Gravitational potential | mgh | water behind a dam |
| Elastic potential | ½kx² | a stretched spring |
| Heat (lost) | — | friction warming brakes |
Caption: Falling water loses gravitational potential energy and gains kinetic energy, which the turbine turns into useful work — the energy chain behind hydroelectric power.
Practice set — try it yourself
- A person pushes a box 5 m with a horizontal force of 20 N. How much work is done? Hint: Force and displacement are in the same direction.
- Why is no work done when you hold a heavy bag still, even though you get tired? Hint: Ask whether there is any displacement.
- A 2 kg ball falls from a height of 10 m. What is its speed just before hitting the ground? Take g = 10 m/s². Hint: Use conservation of mechanical energy instead of motion equations.
- A motor lifts 100 kg of water 10 m in 20 s. Find its power output. Take g = 10 m/s². Hint: Power = work/time; work = mgh.
- A car brakes and stops. Where has its kinetic energy gone? Hint: Think about what friction does to energy.
Key Terms / Important Terms
- Work — force × displacement in the direction of the force (W = Fs cos θ), in joules.
- Joule — the SI unit of work and energy; 1 J = 1 N × 1 m.
- Kinetic energy — energy of motion, ½mv².
- Potential energy — stored energy due to position or shape.
- Work-energy theorem — net work done = change in kinetic energy.
- Conservative force — a force whose work does not depend on the path (gravity, spring force).
- Non-conservative force — a force whose work depends on the path (friction).
- Mechanical energy — kinetic energy + potential energy.
- Power — rate of doing work, in watts.
- Watt — 1 joule per second.
- Efficiency — useful energy output ÷ total energy input.
- Variable force — a force whose magnitude or direction changes with position, so its work must be found from the area under the F–x graph rather than F s cos θ.
- Coefficient of restitution (e) — the ratio of the speed of separation to the speed of approach in a collision: 1 for perfectly elastic, 0 for perfectly inelastic.
- Kilowatt-hour — the commercial unit of energy: 1 kW of power used for 1 hour, equal to 3.6 × 10⁶ J.
Remember
- Work needs displacement in the direction of the force — no movement, no work.
- Work-energy theorem: net work = change in kinetic energy.
- Mechanical energy is conserved only when non-conservative forces do no work.
- Power = work/time = Fv; more power means more work per second.
- Friction turns mechanical energy into heat; total energy is still conserved.
- The work–energy theorem uses NET work by ALL forces — forgetting friction's negative work is the commonest error in stopping-distance problems.
Exam Tip / Common Mistake
Common mistake: counting work done against gravity when carrying a load horizontally. The lifting force is vertical and the displacement is horizontal, so cos 90° = 0 and the work against gravity is zero. Also, watch signs: friction does negative work because it opposes displacement, and that sign decides whether kinetic energy falls or rises.
Questions with Answers — full answers
- Define work and state when it is zero even though a force acts. Answer: Work = force × displacement in the direction of the force. It is zero when there is no displacement, or when the force is perpendicular to the displacement, as when carrying a load horizontally.
- A 1 kg ball moves at 10 m/s. What is its kinetic energy? If a force does 30 J of work on it, what is its new kinetic energy? Answer: Initial K = ½ × 1 × 100 = 50 J. By the work-energy theorem, new K = 50 + 30 = 80 J.
- Distinguish conservative and non-conservative forces with examples. Answer: A conservative force's work does not depend on the path — gravity and spring force. A non-conservative force's work depends on the path — friction, which turns energy into heat along the way.
- A pendulum swings. Explain its energy changes at the highest and lowest points. Answer: At the highest point it is momentarily at rest, so energy is all potential. At the lowest point it is fastest, so energy is all kinetic. Total mechanical energy stays constant (ignoring air resistance).
- An engine does 5000 J of work in 10 s. What is its power in watts? Answer: Power = 5000 ÷ 10 = 500 W.
Extension Questions with Answers — go further
- A force varies with position as shown by an F–x graph that is a triangle of base 0.4 m and height 20 N. How much work is done? Answer: Work is the area: ½ × 0.4 × 20 = 4 J.
- A spring of constant 200 N/m is stretched by 0.1 m. How much elastic potential energy is stored? Answer: U = ½kx² = ½ × 200 × 0.01 = 1 J.
- Two identical balls collide elastically head-on, one at rest. What happens? Answer: They exchange velocities: the moving ball stops dead and the resting ball moves off with the first ball's velocity.
- A motor lifts 50 kg through 10 m in 5 s (g = 10 m/s²). What is its average power output? Answer: Work = mgh = 5000 J; power = 5000/5 = 1000 W.
- A bullet embeds itself in a wooden block. Is momentum conserved? Is kinetic energy conserved? Answer: Momentum is conserved (no external impulse during the impact); kinetic energy is not — much of it becomes heat and deformation in this perfectly inelastic collision.
Board-pattern Questions with Answers
- Find the work done when a force of 10 N pushes a body 5 m (a) along the force, (b) at right angles to it, (c) opposite to it. Answer: (a) 50 J; (b) 0 J; (c) −50 J.
- A car's speed doubles. By what factor does its kinetic energy change? Answer: Four times, because K = ½mv² depends on v².
- Prove the work–energy theorem for a constant force. Answer: W = Fs = ma × s, and with v² = u² + 2as, Fs = ½mv² − ½mu², which is the change in kinetic energy.
- A ball is dropped from a height of 20 m (g = 10 m/s²). With what speed does it hit the ground? Answer: By conservation of mechanical energy, ½mv² = mgh, so v = √(2 × 10 × 20) = 20 m/s.
- A spring of constant 100 N/m is compressed by 0.2 m. Find its stored potential energy. Answer: U = ½ × 100 × 0.04 = 2 J.
Recap — the chapter in five lines
- Work is F s cos θ; its sign is decided by the angle, not by effort.
- Net work equals change in kinetic energy — for variable forces, read it as area under F–x.
- Potential energy exists only for conservative forces; its zero is your choice.
- Mechanical energy is conserved only while non-conservative forces do no work.
- Momentum survives every collision; kinetic energy survives only elastic ones.