CBSE Class 11 Physics Chapter 6 Notes: Work, Energy and Power

Chapter Introduction

Work, Energy and Power is one of the most scoring chapters in CBSE Class 11 Physics (8–10 marks). It connects force to motion through the work-energy theorem, introduces the powerful principle of conservation of mechanical energy, and covers collisions — a topic that appears in every board and competitive exam. The concepts here are used throughout Class 12 (electrostatics, current electricity) and JEE/NEET. Master the work-energy theorem and collision formulas, and you can solve many problems faster than with Newton's laws alone.

What this guide covers: work done (by constant and variable force), kinetic energy, work-energy theorem, potential energy (gravitational and elastic), conservative and non-conservative forces, conservation of mechanical energy, power, collisions (elastic, inelastic, coefficient of restitution) — all with worked numericals.

Key Definitions

• Work (W): W = F · d = Fd cos θ. Scalar; SI unit joule (J). Positive if force has a component along displacement (0° ≤ θ < 90°), negative if against displacement (90° < θ ≤ 180°), zero if perpendicular.
• Kinetic energy (KE): Energy possessed by a body by virtue of its motion. KE = ½mv². Scalar; SI unit joule. Always positive or zero (never negative).
• Potential energy (PE): Energy possessed by a body by virtue of its position or configuration. Scalar; SI unit joule.
• Gravitational PE: PE = mgh (near Earth's surface). Reference level can be chosen arbitrarily; only changes in PE matter.
• Elastic PE (spring): PE = ½kx², where k = spring constant, x = extension/compression.
• Conservative force: A force for which work done depends only on the end points, not the path. Work done in a closed loop is zero. Examples: gravity, spring force, electrostatic force.
• Non-conservative force: A force for which work done depends on the path. Work done in a closed loop is non-zero. Examples: friction, viscous drag, air resistance.
• Mechanical energy: KE + PE. The sum of kinetic and potential energies.
• Power (P): Rate of doing work. P = W/t (average) or P = F · v (instantaneous). SI unit watt (W = J/s).
• Collision: An event in which two or more bodies exert strong forces on each other for a short time. Total momentum is always conserved.
• Coefficient of restitution (e): e = (v₂ − v₁) / (u₁ − u₂) = relative velocity of separation / relative velocity of approach.

Detailed Concepts

1. Work Done

General formula: W = Fd cos θ

Work done by variable force: W = ∫F dx (area under the force-displacement graph).

Work done by spring: W = ∫(−kx)dx from 0 to x = −½kx² (work done by the spring force when stretched by x; negative because force opposes extension).

Work done by gravity: W = mgh when a body falls through height h (positive, gravity does work). W = −mgh when raised (negative, work done against gravity).

Work done is zero when:

• Force is perpendicular to displacement (centripetal force, carrying a load horizontally)
• Displacement is zero (pushing a wall, no motion)
• Force is zero

2. Kinetic Energy and Work-Energy Theorem

KE = ½mv²

Work-energy theorem: The net work done on a body equals the change in its kinetic energy.

W_net = ΔKE = ½mv² − ½mu²

Derivation (from Newton's second law): F = ma = m(dv/dt) Multiply both sides by dx: F dx = m(dv/dt)dx = mv dv Integrate: ∫F dx = ∫mv dv = m[v²/2] = ½mv² − ½mu² Therefore, W_net = ΔKE

Steps to apply the work-energy theorem: 1. Identify all forces doing work on the body. 2. Calculate work done by each force. 3. Sum all work = change in KE. 4. Solve for the unknown.

Advantage over F = ma:

• The work-energy theorem is a scalar equation — no vector components to resolve.
• It directly relates force, distance, and speed without needing acceleration.
• It works even for variable forces (just integrate F dx).
• It gives a direct relation between the initial and final states.

3. Potential Energy and Conservative Forces

Conservative vs. Non-conservative Forces:

Gravitational PE: PE = mgh (near Earth's surface, h measured from chosen reference level).

Spring PE: PE = ½kx² (always positive, whether stretched or compressed).

PE and force relation: F = −dU/dx (force is the negative gradient of potential energy).

• Gravity: U = mgh → F = −dU/dh = −mg (downward, as expected)
• Spring: U = ½kx² → F = −dU/dx = −kx (restoring force, as expected)

PE curve and equilibrium:

• At a minimum of the PE curve: stable equilibrium (F = 0, small displacement → restoring force)
• At a maximum of the PE curve: unstable equilibrium (F = 0, small displacement → force away)
• At a flat region: neutral equilibrium

4. Conservation of Mechanical Energy

Principle: If only conservative forces act on a system, the total mechanical energy (KE + PE) is conserved.

KE₁ + PE₁ = KE₂ + PE₂

When non-conservative forces are present: KE₁ + PE₁ + W_nc = KE₂ + PE₂

where W_nc is the work done by non-conservative forces (usually negative, since friction removes energy).

Applications:

• A body dropped from height h: ½mv² = mgh → v = √(2gh)
• A body sliding down a smooth incline: ½mv² = mgH (H = vertical height of incline)
• A spring compressed by x and released: ½kx² = ½mv² → v = x√(k/m)
• A pendulum at different heights: ½mv² + mgh = constant

Steps to solve problems using energy conservation: 1. Identify initial and final positions. 2. Write KE + PE at both positions. 3. If only conservative forces, set them equal. If friction, include W_nc. 4. Solve for the unknown.

5. Power

P = W/t (average power over time t)

P = F · v = Fv cos θ (instantaneous power)

Note: 1 kWh is a unit of energy (not power). It is the energy consumed by a 1 kW device in 1 hour.

Instantaneous power from force: If a force F pushes an object at velocity v, the power delivered is P = Fv. Example: A car engine providing force F at speed v delivers power P = Fv.

6. Collisions

One-dimensional (head-on) collisions:

Elastic collision formulas (one dimension, two bodies):

• v₁ = ((m₁ − m₂)u₁ + 2m₂u₂) / (m₁ + m₂)
• v₂ = ((m₂ − m₁)u₂ + 2m₁u₁) / (m₁ + m₂)

Special cases:

• Equal masses (m₁ = m₂): v₁ = u₂, v₂ = u₁ (velocities are exchanged)
• m₂ at rest, m₁ = m₂: v₁ = 0, v₂ = u₁ (first body stops, second moves with first's velocity)
• m₂ >> m₁ (wall): v₁ ≈ −u₁ (first body rebounds with same speed)
• m₁ >> m₂: v₁ ≈ u₁ (first body barely affected), v₂ ≈ 2u₁

Coefficient of restitution: e = (v₂ − v₁) / (u₁ − u₂)

• e = 1: perfectly elastic; e = 0: perfectly inelastic
• For a ball dropped from height h and bouncing to height h', e = √(h'/h)

Completely inelastic collision (bodies stick together): v = (m₁u₁ + m₂u₂) / (m₁ + m₂)

KE lost in completely inelastic collision: ΔKE = ½(m₁m₂)/(m₁ + m₂) × (u₁ − u₂)²

Collisions in two dimensions: Momentum is conserved as a vector (x and y components separately). For elastic collisions in 2D, KE is conserved as a scalar. The collision is usually resolved along the line of impact and perpendicular to it.

Important Numericals

Numerical 1: Work Done by a Force at an Angle

A force F = 3î + 4ĵ N acts on a body which moves from (0, 0) to (2, 3) m. Find the work done.

Numerical 2: Work-Energy Theorem

A body of mass 2 kg moving at 10 m s^-1 is brought to rest by friction over 25 m. Find the friction force.

Numerical 3: Conservation of Energy

A body is dropped from a height of 20 m. Find its speed when it is 5 m above the ground. (g = 10 m s^-2)

Numerical 4: Elastic Collision

Two balls of masses 2 kg and 3 kg move at 10 m s^-1 and 5 m s^-1 respectively in the same direction. They collide elastically. Find their velocities after collision.

Numerical 5: Power

An engine pumps water at 10 kg per second from a well 20 m deep. Find the power of the engine. (g = 10 m s^-2)

Numerical 6: Completely Inelastic Collision

A 5 kg body moving at 8 m s^-1 collides with a 3 kg body at rest and they stick together. Find the final velocity and KE lost.

Numerical 7: Spring Energy

A spring of constant 200 N/m is compressed by 0.1 m. A 0.5 kg block is released from it on a smooth surface. Find the velocity of the block.

NCERT Exercise Reference

Common Mistakes

1. Forgetting cos θ in W = Fd cos θ: Work is not always Fd. If the force is at an angle to displacement, include the cosine factor. Centripetal force does zero work because θ = 90°. 2. KE is always positive: Unlike velocity, KE = ½mv² can never be negative, even if v is negative. 3. Confusing conservation of KE with conservation of momentum: Momentum is conserved in all collisions; KE is conserved only in elastic collisions. 4. Applying elastic collision formulas to inelastic collisions: The formulas for v₁ and v₂ apply only when e = 1. 5. Sign of work by friction: Friction does negative work (removes KE), not positive work. 6. PE reference level: You can choose any reference level for gravitational PE. Be consistent within a problem. 7. Spring PE is always positive: ½kx² is positive whether the spring is stretched (x > 0) or compressed (x < 0). 8. 1 kWh is energy, not power: 1 kWh = 3.6 × 10⁶ J. It is the energy used by a 1 kW appliance in 1 hour.

Exam Tips

• Work-energy theorem is the most frequently asked derivation (3–5 marks). State it clearly, derive it from F = ma, and apply it to a numerical.
• Collision problems (3–5 marks): Know the elastic collision formulas and the coefficient of restitution definition. Practise at least 5 problems.
• Conservation of energy is the simplest way to solve problems involving height and speed — use it instead of kinematic equations when possible.
• Power appears as 1–2 mark questions: know P = Fv and P = W/t.
• For derivations, start from the definition, show each step, and write the final formula clearly.
• In collision numericals, always verify momentum conservation after finding final velocities.
• Spring problems: Remember that spring PE = ½kx² and the maximum velocity occurs when all spring PE converts to KE (at equilibrium position).
• Variable force work: Use the area under the F-x graph. This is a common 2-mark question.

Quick Revision Notes

• Work: W = Fd cos θ. Positive, negative, or zero. Area under F-x graph = work.
• KE: ½mv². Always positive. Scalar. Proportional to v².
• Work-energy theorem: W_net = ΔKE. Works for all forces, scalar equation. Derived from F = ma.
• Gravitational PE: mgh; Spring PE: ½kx². Both are scalars.
• Conservative force: path-independent, closed-loop work = 0, PE definable. F = −dU/dx.
• Non-conservative force: path-dependent. Friction removes mechanical energy (converts to heat).
• Conservation of ME: KE + PE = constant (only conservative forces). With friction: W_friction = Δ(ME).
• Power: P = W/t = F · v. Unit: watt. 1 kWh = 3.6 × 10⁶ J (energy, not power).
• Elastic collision: e = 1, momentum + KE conserved. Equal masses → velocities exchange.
• Inelastic collision: e < 1, only momentum conserved. Completely inelastic (e = 0) → bodies stick together.
• Coefficient of restitution: e = (v₂ − v₁) / (u₁ − u₂). For bouncing ball: e = √(h'/h).
• KE lost in inelastic: ΔKE = ½m₁m₂(u₁−u₂)²/(m₁+m₂).