Class 11 Physics Work, Energy, and Power Notes

Chapter Overview

Work, energy and power connect force and motion. Work measures energy transfer by a force, energy represents the capacity to do work, and power measures how rapidly work is done or energy is transferred. The same ideas lead directly to the work–energy theorem, conservation of mechanical energy, conservative forces, and the analysis of collisions.

Main Quantities

Work W, kinetic energy K, potential energy U, mechanical energy E, and power P.

Core Principles

Work–energy theorem, conservation of energy, behavior of conservative forces, conservation of momentum in collisions, and conservation of kinetic energy in elastic collision.

5.1 Work Done by a Force

Work Done by a Constant Force

If a constant force F acts on an object and produces displacement s, while the angle between force and displacement is θ, then the work done is the scalar product:

W = F s cosθ = F · s

The SI unit of work is the joule (J).

1 J = 1 N·m

Diagram 1 — Work by a Constant Force at an Angle

Only the component of force parallel to the displacement contributes to work.

Positive, Negative and Zero Work

Case	Angle θ	Sign of work	Example
Force has component along displacement	0° ≤ θ < 90°	Positive	Pulling a trolley forward
Force is perpendicular to displacement	90°	Zero	Centripetal force in uniform circular motion
Force opposes displacement	90° < θ ≤ 180°	Negative	Friction on a sliding body

Work Done by a Variable Force

When force varies with position, we cannot generally use a single value of F. For a small displacement dx, the small work is dW = F(x)dx. Therefore:

W = ∫_x₁^x₂ F(x) dx

Graphically, work equals the signed area under the force–displacement graph.

Diagram 2 — Work by a Variable Force

The area between the F–x curve and the x-axis gives the work done.

Exam tip: Work is a scalar quantity. A force may act on an object but still do zero work if the displacement is zero or the force is perpendicular to displacement.

5.2 Work–Energy Theorem

The work–energy theorem states that the net work done on an object equals the change in its kinetic energy.

W_net = ΔK = K_f − K_i

Proof for Constant Net Force in One Dimension

For mass m, Newton’s second law gives:

F = ma

Using the kinematic relation v² − u² = 2as:

a = (v² − u²)/(2s)

Therefore:

W = Fs = mas = (1/2)mv² − (1/2)mu²

Hence: W_net = ΔK

Diagram 3 — Work–Energy Theorem

Positive net work increases kinetic energy; negative net work decreases it.

5.3 Kinetic Energy and Potential Energy

Kinetic Energy

The energy possessed by an object due to its motion is called kinetic energy.

K = (1/2)mv²

Kinetic energy is a scalar quantity.
It cannot be negative in classical mechanics.
For fixed mass, K ∝ v².

Potential Energy

Potential energy is energy associated with the position or configuration of a system. Near Earth’s surface, gravitational potential energy relative to a chosen zero level is:

U = mgh

Diagram 4 — Gravitational Potential Energy and Kinetic Energy

As an object falls under gravity, gravitational potential energy can be converted into kinetic energy.

Feature	Kinetic Energy	Potential Energy
Cause	Motion	Position or configuration
Common formula	K = ½mv²	U = mgh near Earth’s surface
Depends on	Mass and speed	System configuration and chosen reference
Can be transformed?	Yes	Yes

5.4 Principle of Conservation of Energy

The principle of conservation of energy states that energy can neither be created nor destroyed; it can only be transformed from one form to another. The total energy of an isolated system remains constant.

Conservation of Mechanical Energy

If only conservative forces perform work, the sum of kinetic and potential energies remains constant:

K + U = constant

K_i + U_i = K_f + U_f

Example: Freely Falling Body

Consider a body released from rest at height H, neglecting air resistance. At the top, its energy is entirely gravitational potential energy:

E = mgH

At an intermediate height h, potential energy is mgh and the remaining energy is kinetic. At the bottom, the potential energy relative to the ground is zero and the kinetic energy becomes mgH.

Diagram 5 — Conservation of Mechanical Energy in Free Fall

Potential energy decreases while kinetic energy increases, but their sum remains constant when only gravity acts.

5.5 Conservative and Non-Conservative Forces

Conservative Force

A force is conservative if the work done between two points is independent of the path followed. Equivalently, the work done around a closed path is zero.

∮ F · dr = 0

For a conservative force, potential energy can be defined and:

W_conservative = −ΔU

Examples include gravitational force and ideal spring force.

Non-Conservative Force

A non-conservative force does path-dependent work. Its work over a closed path is generally not zero, and mechanical energy is typically transformed into thermal or other forms.

Common examples include kinetic friction and air resistance.

Diagram 6 — Path Independence and Path Dependence

For conservative forces, work depends only on the endpoints; for non-conservative forces, the path matters.

Property	Conservative Force	Non-Conservative Force
Work between two points	Path independent	Path dependent
Closed-path work	Zero	Generally nonzero
Potential energy	Can be defined	No single-valued potential energy in the same sense
Mechanical energy	Conserved if only conservative forces act	May be transformed into thermal/internal energy
Examples	Gravity, ideal spring force	Friction, air resistance

Power

Power is the rate at which work is done or energy is transferred.

Average power: P_avg = W/Δt

Instantaneous power: P = dW/dt = F · v = Fv cosθ

The SI unit of power is the watt (W).

1 W = 1 J s⁻¹

Diagram 7 — Same Work, Different Power

If the same amount of work is completed in less time, the power is greater.

5.6 Elastic and Inelastic Collisions

Collision

A collision is a short interaction in which bodies exert large forces on one another. For an isolated system, total linear momentum is conserved during the collision.

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

Elastic Collision

In an elastic collision, both total momentum and total kinetic energy are conserved.

(1/2)m₁u₁² + (1/2)m₂u₂² = (1/2)m₁v₁² + (1/2)m₂v₂²

Inelastic Collision

In an inelastic collision, total momentum is conserved in an isolated system, but total kinetic energy is not. Some kinetic energy is transformed into deformation, sound, heat, or internal energy.

Perfectly Inelastic Collision

In a perfectly inelastic collision, the bodies stick together and move with a common final velocity v.

v = (m₁u₁ + m₂u₂)/(m₁ + m₂)

Diagram 8 — One-Dimensional Elastic and Inelastic Collisions

Elastic collision conserves both momentum and kinetic energy; a perfectly inelastic collision conserves momentum but the bodies stick together.

One-Dimensional Elastic Collision: Useful Results

For two bodies moving along the same straight line:

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

u₁ − u₂ = −(v₁ − v₂)

The second relation states that, for a perfectly elastic head-on collision, relative speed of approach equals relative speed of separation.

v₁ = [(m₁−m₂)/(m₁+m₂)]u₁ + [2m₂/(m₁+m₂)]u₂

v₂ = [2m₁/(m₁+m₂)]u₁ + [(m₂−m₁)/(m₁+m₂)]u₂

Special case: If two equal masses collide elastically in one dimension and the second mass is initially at rest, they exchange velocities.

Feature	Elastic Collision	Inelastic Collision
Total momentum	Conserved in isolated system	Conserved in isolated system
Total kinetic energy	Conserved	Not conserved
Deformation/heat/sound	Ideally no permanent loss to these forms	Some kinetic energy transforms into these forms
Bodies stick together?	No requirement	Only in perfectly inelastic case

Formula Summary

Concept	Formula	Meaning / Condition
Constant-force work	W = Fs cosθ	Force F, displacement s, angle θ
Variable-force work	W = ∫F(x)dx	Area under F–x graph
Kinetic energy	K = ½mv²	Energy due to motion
Gravitational potential energy	U = mgh	Near Earth’s surface
Work–energy theorem	W_net = ΔK	Net work changes kinetic energy
Mechanical energy	E = K + U	Constant if only conservative forces act
Conservative force	W = −ΔU	Path-independent work
Average power	P = W/Δt	Work per unit time
Instantaneous power	P = F·v = Fv cosθ	For force acting on moving body
Momentum conservation	m₁u₁+m₂u₂=m₁v₁+m₂v₂	Isolated collision
Elastic collision	K_before=K_after	Kinetic energy also conserved
Perfectly inelastic final speed	v=(m₁u₁+m₂u₂)/(m₁+m₂)	Bodies stick together

Solved Numerical Examples

Example 1 — Work at an Angle

Question: A 50 N force pulls an object through 8 m at 60° to the displacement. Find the work done.

W = Fs cosθ = (50)(8)cos60° = 400 × 0.5 = 200 J

Answer: 200 J.

Example 2 — Work–Energy Theorem

Question: A 2 kg body speeds up from 3 m s⁻¹ to 7 m s⁻¹. Find the net work done.

W_net = ½m(v²−u²) = ½(2)(49−9) = 40 J

Answer: 40 J.

Example 3 — Conservation of Mechanical Energy

Question: A 1 kg object falls freely from a height of 20 m. Find its speed just before reaching the ground. Take g = 9.8 m s⁻².

mgh = ½mv² ⇒ v = √(2gh) = √(2×9.8×20) ≈ 19.8 m s⁻¹

Answer: approximately 19.8 m s⁻¹.

Example 4 — Power

Question: A machine does 12,000 J of work in 30 s. Find its average power.

P = W/t = 12000/30 = 400 W

Answer: 400 W.

Example 5 — Perfectly Inelastic Collision

Question: A 2 kg cart moving at 6 m s⁻¹ collides with a 4 kg cart at rest. They stick together. Find their common speed.

v = (m₁u₁+m₂u₂)/(m₁+m₂) = [(2)(6)+(4)(0)]/(2+4) = 2 m s⁻¹

Answer: 2 m s⁻¹.

Example 6 — Equal-Mass Elastic Collision

Question: A 1 kg ball moving at 5 m s⁻¹ collides elastically head-on with an identical stationary ball. Find the final velocities.

In a one-dimensional elastic collision of equal masses where the second is initially at rest, the bodies exchange velocities.

v₁ = 0, v₂ = 5 m s⁻¹

Answer: first ball stops; second ball moves at 5 m s⁻¹.

Important Exam Questions

Short-Answer Questions

Define work. When is work positive, negative, and zero?
Why does centripetal force do no work in uniform circular motion?
How is work by a variable force obtained from an F–x graph?
State the work–energy theorem.
Define kinetic energy and gravitational potential energy.
State the principle of conservation of energy.
What is a conservative force? Give two examples.
Differentiate conservative and non-conservative forces.
Define power and write its SI unit.
Differentiate elastic and inelastic collisions.
What is a perfectly inelastic collision?
What quantities are conserved in an elastic collision?

Long-Answer Questions

Explain work done by a constant force and a variable force with suitable diagrams.
State and prove the work–energy theorem.
Establish the expressions for kinetic energy and gravitational potential energy.
State and explain the principle of conservation of energy using a freely falling body.
Explain conservative and non-conservative forces with examples and closed-path work.
Define power and derive P = F·v.
Differentiate elastic and inelastic collisions and discuss the conservation laws involved.
Explain a one-dimensional elastic collision and derive the relation between relative speeds of approach and separation.

Derivations to Practice

W = Fs cosθ using scalar product.
Work–energy theorem: W_net = ΔK.
K = ½mv².
Conservation of mechanical energy for a freely falling body.
P = F·v.
One-dimensional elastic collision relations.

Numerical Questions

A 20 N force acts at 30° to the direction of motion and moves an object 5 m. Find the work done.
A 4 kg object accelerates from 2 m s⁻¹ to 6 m s⁻¹. Find the net work done.
A 2 kg body is raised vertically through 10 m. Calculate the gain in gravitational potential energy using g = 9.8 m s⁻².
A motor performs 36 kJ of work in 20 s. Find its average power.
A 3 kg object moving at 8 m s⁻¹ sticks to a stationary 5 kg object. Find their common speed.
Two equal masses collide elastically in one dimension; one is initially at rest. Show how their velocities change.

Diagram Questions

Draw a force acting at angle θ to displacement and label the component responsible for work.
Draw an F–x graph and shade the area representing work.
Draw a falling body at three positions to illustrate conservation of mechanical energy.
Draw two different paths between the same points to explain conservative force.
Draw before-and-after diagrams for elastic and perfectly inelastic collisions.

One-Minute Revision

Work by a constant force is W = Fs cosθ.
Work is positive for θ < 90°, zero for θ = 90°, and negative for θ > 90°.
Variable-force work equals the area under the F–x graph.
The work–energy theorem is W_net = ΔK.
Kinetic energy is K = ½mv².
Near Earth’s surface, gravitational potential energy is U = mgh.
If only conservative forces act, K + U remains constant.
Conservative-force work is path independent.
For a conservative force, W = −ΔU.
Friction is a common non-conservative force.
Power is the rate of doing work: P = W/t.
Instantaneous mechanical power can be written P = F·v.
Momentum is conserved in an isolated collision.
Elastic collision conserves both momentum and kinetic energy.
In a perfectly inelastic collision, bodies stick together after impact.

Diagram Practice

Draw and label a block displaced by a force making angle θ with the displacement.
Draw a variable-force F–x graph and mark the work as area under the curve.
Draw the work–energy theorem situation showing initial and final speeds.
Draw a ball at height h and near the ground to show potential-to-kinetic energy conversion.
Draw top, middle, and bottom stages of a falling body with K, U, and total energy labels.
Draw two paths from A to B to compare conservative and non-conservative work.
Draw an illustration showing identical work completed in different times to explain power.
Draw elastic and perfectly inelastic one-dimensional collision diagrams.

Syllabus Coverage Checklist

NEB/CDC Chapter 5 scope	Covered
5.1 Work done by a constant force and a variable force	Yes
5.2 Work–energy theorem	Yes — statement and proof
5.3 Kinetic energy and potential energy; formulae	Yes
5.4 Principle of conservation of energy	Yes
5.5 Conservative and non-conservative forces	Yes
Power and its mechanical interpretation	Yes
5.6 Elastic and inelastic collisions	Yes
One-dimensional elastic collision	Yes
Numerical problems and conceptual questions	Yes