Unit 4 · Topic 04

Power

Power distinguishes processes that transfer the same energy in different times. It measures a transfer rate, not a total amount of energy.

Unit 4PowerOpen navigation

Concept 01

Work per unit time

Essential The minimum you should retain

Average power is work divided by the elapsed time.

UnderstandInterpret and connect

Two machines can do the same work with different powers if their times differ.

DeepenFormulation and conditions

P_avg=ΔW/Δt and its SI unit is the watt, 1 W=1 J/s.

ExploreConnections for further study

A small power sustained for a long time can transfer a large amount of energy.

Mathematical relation

Average and instantaneous power

Pmed=ΔWΔt,P=dWdt=F→·v→ P_{med}=\frac{\Delta W}{\Delta t},\quad P=\frac{dW}{dt}=\vec F\cdot\vec v
Represents

Average or instantaneous rate of energy transfer through work.

Physical interpretation

The sign indicates instantaneous delivery or removal of mechanical energy.

DeepenVariables, conditions, and checks

Variables

P
power; usual unit: W
W
work; usual unit: J
t
time; usual unit: s

Conditions of application

  • For F·v, force and velocity refer to the same particle and instant.

Dimensional check

J/s=W.

Errors it helps prevent

  • Confusing power with energy.
  • Using P=Fv without the angle.
Two processes lift equal loads through the same height in different times.process Aprocess BloadloadhhΔt = 3 sΔt = 12 s

Both processes lift the same load through the same height and do equal work; the shorter time requires greater average power.

Concept 02

Power at an instant

Essential The minimum you should retain

Instantaneous power is the rate of work at a particular instant.

UnderstandInterpret and connect

For a force on a particle, P=F·v P=\vec F\cdot\vec v .

DeepenFormulation and conditions

It can also be written P=dW/dt; only the component parallel to v contributes.

ExploreConnections for further study

Even when |F| is constant, P can vary if speed or the angle between F and v changes.

Worked example

Instantaneous power with an angled force

A 120 N force acts on a cart moving at 2.5 m/s; the angle between force and velocity is 30°.

Given
  • F=120 N
  • v=2.5 m/s
  • θ=30°
Target

Calculate instantaneous power.

  1. Principle

    P=F·v P=\vec F\cdot\vec v =Fv cosθ.

  2. Substitution

    P=(120)(2.5)cos30°.

  3. Calculation

    P≈260 W.

  4. Interpretation

    Only the component of F parallel to v transfers energy at that instant.

Conclusion

Instantaneous power is approximately 260 W.

Concept 03

Delivering or removing energy

Essential The minimum you should retain

P>0 means that force delivers mechanical energy at that instant.

UnderstandInterpret and connect

P<0 means it removes energy, and P=0 means no instantaneous transfer by that force.

DeepenFormulation and conditions

The sign follows from the dot product F·v.

ExploreConnections for further study

A perpendicular force can change velocity direction without delivering kinetic energy.

Concept 04

Do not confuse energy and power

Essential The minimum you should retain

The joule measures energy and the watt measures energy per time.

UnderstandInterpret and connect

Multiplying power by time produces energy.

DeepenFormulation and conditions

A kilowatt-hour is an energy unit: power in kW multiplied by time in hours.

ExploreConnections for further study

The symbol W may denote the watt unit or the work variable; context and units remove ambiguity.

Concept review

Common errors

Each warning includes a concrete way to review the reasoning, not only an incorrect-answer marker.

Confusing power with energy.

Power is energy per unit time; watt and joule measure different quantities.

Using P=Fv while ignoring the angle.

In general P=F·v P=\vec F\cdot\vec v =Fv cosθ.