Unit 6 · Topic 08

Work and power in rotation

Rotational work accumulates torque over angle, and power measures instantaneous transfer rate.

Unit 6Work and powerOpen navigation

Concept 01

Signed area

Essential The minimum you should retain

For a fixed axis, W=∫τ_zdθ W=\int\tau_zd\theta .

UnderstandInterpret and connect

A τ-versus-θ graph gives work as algebraic area.

DeepenFormulation and conditions

For constant τ, W=τΔθ.

ExploreConnections for further study

Variable torque cannot use an arbitrary product instead of the integral.

Mathematical relation

Rotational work and power

W=∫τzdθ,P=τzω,Wnet=ΔKrot W=\int\tau_zd\theta,\quad P=\tau_z\omega,\quad W_{net}=\Delta K_{rot}
Represents

Energy transfer by torque.

Physical interpretation

Work is signed area under τ(θ).

DeepenVariables, conditions, and checks

Variables

W
work; usual unit: J
P
power; usual unit: W

Conditions of application

  • Fixed-axis scalar model.

Dimensional check

J and W.

Algebraic area under τ(θ) is W

A torque-versus-angle curve crosses the axis.sign changeP=τω
  • τ=8-2θ

Torque changes with angle; positive and negative regions contribute signed work.

Concept 02

Rate of doing work

Essential The minimum you should retain

In the axial case, P=τ_zω P=\tau_z\omega .

UnderstandInterpret and connect

Its sign indicates energy delivery or removal under the convention.

DeepenFormulation and conditions

Torque and power are not synonyms and use different units.

ExploreConnections for further study

A motor can trade high torque and low ω at the same power.

Worked example

Shaft power

A motor supplies 12 N·m while its shaft turns at 20 rad/s.

Given
  • τ=12 N·m
  • ω=20 rad/s
Target

Find P.

  1. Model

    P=τω.

  2. Substitution

    P=(12)(20).

  3. Result

    P=240 W.

  4. Signs

    Axial components have the same sign.

Conclusion

P=240 W.

Concept 03

Rotation states

Essential The minimum you should retain

Net work is ΔK_rot for the fixed-axis model.

UnderstandInterpret and connect

It may determine final ω without finding α(t).

DeepenFormulation and conditions

With translation or potential energy, include all relevant energies.

ExploreConnections for further study

Losses appear as nonconservative work.

Concept 04

Interpreting the result

Essential The minimum you should retain

Work uses J and torque uses N·m.

UnderstandInterpret and connect

Negative work reduces K_rot if no other transfer occurs.

DeepenFormulation and conditions

Negative P means torque removes energy at that instant.

ExploreConnections for further study

Keep the chosen orientation in τ, θ, and ω.

Concept review

Common errors

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

Using W=τΔθ with variable torque.

In general W=∫τdθ.

Confusing torque and power.

In the axial case, P=τω.