Power and Efficiency

Overview

This page develops two important practical ideas:

• Power: the rate of doing work or transferring energy
• Efficiency: how much of the input becomes useful output

These ideas are widely tested in mechanics, engines, motors, lifts, electrical devices and real-world systems.

This page extends the overview in Work, Energy and Power.

Why It Matters

Power and efficiency connect energy transfer to time and usefulness, which is essential for motors, machines, circuits, and real-world systems with losses.

Definition

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

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

or more generally:

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

The SI unit is the watt, where $1\text{W}=1{\text{J s}}^{-1}$.

Key Representations

$$P=\frac{W}{t}$$ $$P=\frac{E}{t}$$ $$P=\vec{F}\cdot \vec{v}$$ $$\eta =\frac{\text{useful output}}{\text{total input}}$$

Scalar and Vector Distinction

Be precise.

Vector Quantities

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

Scalar Quantities

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

Power and efficiency are scalar quantities.

What Is Power?

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

Average Power

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

Also:

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

where:

• $W$ = work done
• $E$ = energy transferred
• $t$ = time taken

Unit:

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

Meaning of Power

A high-power machine does not necessarily do more total work.

It does work faster.

Examples:

• two lifts raise same load
• higher-power lift reaches top sooner

Instantaneous Power

For a force acting on a moving object:

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

Magnitude form:

$$P=Fv\cos \theta$$

where $\theta$ is the angle between force and velocity.

Special Cases

Force Parallel to Motion

$$\theta =0^{\circ }$$ $$P=Fv$$

Examples:

• engine thrust
• towing force

Force Perpendicular to Motion

$$\theta =90^{\circ }$$ $$P=0$$

Example:

• centripetal force in uniform circular motion

Although force changes direction of motion, it does no work and transfers no power.

Mechanical Interpretation

Since:

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

and:

$$\vec{v}=\frac{d\vec{s}}{dt}$$

power is the time rate of work.

Efficiency

Efficiency compares useful output to total input.

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

Can be based on:

• energy
• power

As percentage:

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

Why Efficiency Is Less Than 100%

Real systems lose energy to:

• friction
• heating
• sound
• vibration
• air resistance
• electrical resistance

Hence:

$$\text{efficiency}<1$$

for real devices.

Common Applications

Engines

Chemical energy → kinetic + thermal losses

Electric Motors

Electrical input → mechanical output + heating

Lifts

Motor power raises load against weight.

Pumps

Mechanical input → fluid energy output

Worked Examples

Example 1: Average Power

A machine does $6000\text{ J}$ of work in $20\text{ s}$.

$$P=\frac{W}{t}$$ $$=\frac{6000}{20}$$ $$=300\text{ W}$$

Example 2: Constant-Speed Car

A car moves at constant speed $25{\text{ m s}}^{-1}$ against a resistive force of $800\text{ N}$.

At constant speed, driving force balances resistance.

$$P=Fv$$ $$=800(25)$$ $$=2.0\times 10^4\text{ W}$$ $$=20\text{ kW}$$

Example 3: Lift Motor

A lift raises total mass $1200\text{ kg}$ vertically at constant speed $2.0{\text{ m s}}^{-1}$.

Required force:

$$F=mg$$ $$=1200(9.81)$$

Power:

$$P=Fv$$ $$=1200(9.81)(2.0)$$ $$=2.35\times 10^4\text{ W}$$

Example 4: Efficiency

A motor receives $500\text{ W}$ electrical input and delivers $400\text{ W}$ useful mechanical output.

$$\text{efficiency}=\frac{400}{500}=0.80$$ $$=80\%$$

Example 5: Find Input Power

A pump gives useful output power of $900\text{ W}$ at $60\%$ efficiency.

$$0.60=\frac{900}{P_{\text{in}}}$$ $$P_{\text{in}}=1500\text{ W}$$

Power in Motion Problems

Constant Speed on Level Road

Driving force = resistive force

Use:

$$P=Fv$$

Accelerating Vehicle

Engine power contributes to:

• increasing kinetic energy
• overcoming resistive forces

Climbing Slope

Power used to:

• gain GPE
• overcome resistance

Relationship with Other Topics

Work

Power is rate of work done.

See Work, Energy and Power

Dynamics

Force balance often needed before using:

$$P=Fv$$

See Dynamics

Current Electricity

Electrical power also studied in circuits.

See Current Electricity Fundamentals

Common Exam Pitfalls

1. Confusing Energy with Power

• Energy in J
• Power in W

2. Forgetting Constant Speed Means Zero Resultant Force

Driving force may still be non-zero.

It balances resistance.

3. Wrong Force in $P=Fv$

Use force component in direction of velocity.

4. Percentage Error

$75\%=0.75$

5. Using Distance Instead of Speed

Formula is:

$$P=Fv$$

not $Fs$.

Summary

• Power measures how quickly work is done.

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

• Instantaneous power:

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

• Efficiency measures useful fraction of input.

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

• Real devices are always less than $100\%$ efficient.

Links

• Work, Energy and Power
• Kinetic Energy and Work-Energy Theorem
• Potential Energy and Conservative Forces
• Energy Forms and Conservation
• Dynamics
• Forces
• Current Electricity Fundamentals