1. Introduction

In translational motion, forces perform work and transfer energy.

In rotational motion, torques perform work and transfer energy.

The concepts of work and power in rotational motion are direct analogs of their translational counterparts and play an important role in rotational dynamics and energy conservation.

2. Review of Translational Work

For a constant force acting through a displacement:

W=F⃗⋅Δr⃗W = \vec{F} \cdot \Delta \vec{r}W=F⋅Δr

or

$$W=Fd\cos \theta$$

where:

• $W$ = work
• $F$ = force
• $d$ = displacement
• $\theta$ = angle between force and displacement

Work measures the transfer of energy.

3. Rotational Work

Definition

Rotational work is the work done by a torque as an object rotates through an angular displacement.

For a constant torque:

$$W=\tau \Delta \theta$$

where:

• $W$ = rotational work
• $\tau$ = torque
• $\Delta \theta$ = angular displacement in radians

Important Note

Angular displacement must be measured in radians.

This equation is valid only when torque remains constant.

4. Derivation of Rotational Work

Recall that:

$$dW=Fds$$

For rotational motion:

$$ds=rd\theta$$

and

$$\tau =rF$$

Substituting:

$$dW=(rF)d\theta$$ $$dW=\tau d\theta$$

Integrating:

$$W=\int \tau d\theta$$

For constant torque:

$$W=\tau \Delta \theta$$

5. Work Done by a Variable Torque

If torque changes during rotation:

$$W=\int \tau d\theta$$

This equation is the rotational equivalent of:

$$W=\int Fdx$$

The work equals the area under a torque-versus-angle graph.

6. Rotational Work–Energy Theorem

The rotational version of the Work–Energy Theorem states:

Wnet=ΔKrotW_{\text{net}} = \Delta K_{\text{rot}}Wnet​=ΔKrot​

where:

Krot=12Iω2K_{\text{rot}} = \frac12 I\omega^2Krot​=21​Iω2

Thus:

Wnet=12Iωf2−12Iωi2W_{\text{net}} = \frac12 I\omega_f^2 – \frac12 I\omega_i^2Wnet​=21​Iωf2​−21​Iωi2​

Interpretation

Net work done by torques changes the rotational kinetic energy of an object.

7. Positive and Negative Work

Positive Work

If torque and angular displacement have the same direction:

$$W>0$$

Rotational kinetic energy increases.

Angular speed increases.

Negative Work

If torque opposes the rotation:

$$W<0$$

Rotational kinetic energy decreases.

Angular speed decreases.

8. Power in Rotational Motion

Definition

Power is the rate at which work is done.

Average power:

Pavg=WΔtP_{\text{avg}} = \frac{W}{\Delta t}Pavg​=ΔtW​

where:

• $P$ = power
• $W$ = work
• $\Delta t$ = time interval

9. Instantaneous Rotational Power

Starting with:

P=dWdtP = \frac{dW}{dt}P=dtdW​

and using:

$$dW=\tau d\theta$$

gives:

P=τdθdtP = \tau \frac{d\theta}{dt}P=τdtdθ​

Since:

ω=dθdt\omega = \frac{d\theta}{dt}ω=dtdθ​

we obtain:

$$P=\tau \omega$$

This is one of the most important power equations in rotational mechanics.

10. Rotational–Linear Analogy

These relationships show the strong parallel between translational and rotational motion.

11. Example: Rotational Work

A constant torque of:

$$10N\cdot m$$

rotates a wheel through:

$$4rad$$

Find the work done.

Solution

Using:

$$W=\tau \Delta \theta$$ $$W=(10)(4)$$ $$W=40J$$

12. Example: Rotational Power

A motor produces:

$$\tau =20N\cdot m$$

while spinning at:

$$\omega =50rad/s$$

Find the power output.

Solution

Using:

$$P=\tau \omega$$ $$P=(20)(50)$$ $$P=1000W$$

13. Energy in Rotating Machines

Many machines convert energy through rotational motion.

Examples include:

• electric motors
• turbines
• engines
• generators
• flywheels

In these systems:

$$P=\tau \omega$$

is frequently used to determine power output.

14. Rolling Objects and Work

For rolling objects:

Ktotal=12mv2+12Iω2K_{\text{total}} = \frac12 mv^2 + \frac12 I\omega^2Ktotal​=21​mv2+21​Iω2

Work done on the object may change:

• translational kinetic energy
• rotational kinetic energy
• both

Energy methods are often easier than force methods for these problems.

15. Common AP Physics C Mistakes

Mistake 1

Using degrees instead of radians in:

$$W=\tau \Delta \theta$$

Angular displacement must be measured in radians.

Mistake 2

Confusing angular velocity with rotational power.

Remember:

$$P=\tau \omega$$

Power depends on both torque and angular velocity.

Mistake 3

Ignoring rotational kinetic energy when applying energy conservation.

Always include:

12Iω2\frac12 I\omega^221​Iω2

when an object is rotating.

Summary

Rotational work:

$$W=\tau \Delta \theta$$

Variable torque:

$$W=\int \tau d\theta$$

Rotational kinetic energy:

Krot=12Iω2K_{\text{rot}} = \frac12 I\omega^2Krot​=21​Iω2

Rotational Work–Energy Theorem:

Wnet=ΔKrotW_{\text{net}} = \Delta K_{\text{rot}}Wnet​=ΔKrot​

Rotational power:

$$P=\tau \omega$$

Key ideas:

• torque does work during rotational motion
• work changes rotational kinetic energy
• power measures the rate of rotational energy transfer
• rotational equations closely parallel translational equations

Work and Power in rotational motion provide the connection between torque, energy, and angular motion, making them essential tools in AP Physics C Mechanics.