Work, Energy, and Power

Overview

This chapter connects ideas from Forces, Dynamics, Kinematics and Vectors.

Instead of analysing motion only through forces and acceleration, many problems can be solved efficiently using energy methods.

Core ideas:

• Work is energy transferred by a force acting through a displacement.
• Energy is a scalar quantity associated with motion, position or configuration.
• Power is the rate of energy transfer.
• Efficiency measures useful output compared with total input.

This page is the main revision hub for the chapter.

Core Ideas

Scalar and Vector Distinction

Be precise.

Vector quantities

• Force: $\vec{F}$
• Displacement: $\vec{s}$
• Velocity: $\vec{v}$
• Acceleration: $\vec{a}$

Scalar quantities

• Work: $W$
• Energy: $E$
• Power: $P$
• Speed
• Mass

Although work depends on vectors, work itself is scalar.

What Is Work?

Work done by a constant force is defined using the dot product:

$$W=\vec{F}\cdot \vec{s}$$

Magnitude form:

$$W=Fs\cos \theta$$

where:

• $F$ = magnitude of force
• $s$ = magnitude of displacement
• $\theta$ = angle between $\vec{F}$ and $\vec{s}$

Unit:

$$1\text{ J}=1\text{ N m}$$

Positive, Negative and Zero Work

Positive Work

Force has a component in direction of motion.

Examples:

• pulling a trolley forward
• gravity acting on a falling object

$$0^{\circ }\le \theta <90^{\circ }$$

Negative Work

Force opposes motion.

Examples:

• friction
• drag
• braking force

$$90^{\circ }<\theta \le 180^{\circ }$$

Zero Work

Force perpendicular to displacement.

Examples:

• centripetal force in uniform circular motion
• carrying a bag horizontally at constant height

$$\theta =90^{\circ }$$

One-Dimensional Signed Forms

After choosing a positive direction, scalar signs may be used:

$$W=Fs$$

where $F$ and $s$ may be positive or negative according to direction.

However, remember that force and displacement are fundamentally vectors.

Work as Energy Transfer

Work done on a system transfers energy to the system.

Negative work transfers energy from the system.

Examples:

• Engine does work on car → kinetic energy increases
• Friction does negative work → mechanical energy decreases

Kinetic Energy Overview

Energy due to motion:

$$E_k=\frac{1}{2}m{v}^2$$

where:

• $m$ = mass
• $v$ = speed

Key facts:

• scalar quantity
• depends on speed squared
• always non-negative

See: Kinetic Energy and Work-Energy Theorem

Potential Energy Overview

Potential energy is associated with position or configuration.

Gravitational Potential Energy (near Earth’s surface)

$$E_p=mgh$$

where:

• $h$ measured relative to chosen reference level

Elastic Potential Energy

For a Hooke’s law spring:

$$E_p=\frac{1}{2}k{x}^2$$

where:

• $k$ = spring constant
• $x$ = extension or compression

See: Potential Energy and Conservative Forces

Work-Energy Theorem Overview

Net work done on an object equals change in kinetic energy:

$$\Delta E_k=W_{\text{net}}$$

Equivalent form:

$$W_{\text{net}}=E_{k,f}-E_{k,i}$$

Useful when force acts over displacement.

See: Kinetic Energy and Work-Energy Theorem

Conservation of Energy Overview

Total energy of an isolated system remains constant.

Energy may transfer between stores:

• kinetic
• gravitational potential
• elastic potential
• thermal/internal
• electrical

Mechanical Energy

If non-conservative forces are negligible:

$$E_k+E_p=\text{constant}$$

If friction or drag acts:

Some mechanical energy is transferred to thermal/internal energy.

See: Energy Forms and Conservation

Power Overview

Power is rate of doing work or transferring energy.

$$P=\frac{W}{t}$$

Also:

$$P=\frac{E}{t}$$

Instantaneous mechanical power:

$$P=\vec{F}\cdot \vec{v}$$

Magnitude form:

$$P=Fv\cos \theta$$

Unit:

$$1\text{ W}=1{\text{ J s}}^{-1}$$

See: Power and Efficiency

Efficiency Overview

$$\text{efficiency}=\frac{\text{useful output}}{\text{total input}}$$

As percentage:

$$\text{efficiency}\times 100\%$$

Can use energy or power values consistently.

Real systems are always less than 100% efficient.

Short Worked Examples

Example 1: Work Done by Pulling Force

A force of $20\text{ N}$ pulls an object through $5.0\text{ m}$ in same direction.

$$W=Fs=20(5.0)=100\text{ J}$$

Example 2: Negative Work by Friction

Friction force $6.0\text{ N}$ acts opposite motion over $3.0\text{ m}$.

$$W=Fs\cos 180^{\circ }=-18\text{ J}$$

Example 3: Gain in Kinetic Energy

Net work done on object is $40\text{ J}$.

$$\Delta E_k=40\text{ J}$$

So kinetic energy increases by $40\text{ J}$.

Example 4: Power

A motor does $2400\text{ J}$ of work in $8.0\text{ s}$.

$$P=\frac{2400}{8.0}=300\text{ W}$$

Formula Summary

Work

$$W=\vec{F}\cdot \vec{s}$$ $$W=Fs\cos \theta$$

Kinetic Energy

$$E_k=\frac{1}{2}m{v}^2$$

Gravitational Potential Energy

$$E_p=mgh$$

Elastic Potential Energy

$$E_p=\frac{1}{2}k{x}^2$$

Work-Energy Theorem

$$\Delta E_k=W_{\text{net}}$$

Power

$$P=\frac{W}{t}$$ $$P=\frac{E}{t}$$ $$P=\vec{F}\cdot \vec{v}$$

Efficiency

$$\text{efficiency}=\frac{\text{useful output}}{\text{total input}}$$

Exam Relevance

1. Forgetting Work Is Scalar

Do not write vector arrows on $W$ or energy terms.

2. Wrong Angle in $W=Fs\cos \theta$

Angle must be between force and displacement.

3. Confusing Speed and Velocity

Use speed in:

$$E_k=\frac{1}{2}m{v}^2$$

4. Assuming Mechanical Energy Always Conserved

Not true if friction/drag present.

5. Mixing Energy and Power

• Energy in J
• Power in W

6. Wrong Height Reference

Only differences in gravitational potential energy matter.

Problem-Solving Strategy

Use:

Forces + Dynamics when asked about acceleration

See Dynamics

Energy methods when asked about speed, height, compression, losses

Power methods when asked about rate or engine output

Links

• Kinetic Energy and Work-Energy Theorem
• Potential Energy and Conservative Forces
• Energy Forms and Conservation
• Power and Efficiency
• Forces
• Dynamics
• Kinematics
• Circular Motion