6.2 Torque and Work

Torque does work on a rotating system when it acts through an angular displacement, and you find that work with $W = \tau \Delta\theta$. The work equals the area under a torque-versus-angular-position graph, which connects rotational work to changes in rotational kinetic energy.

Why This Matters for the AP Physics 1 Exam

This topic is the rotational version of work that you already studied for linear motion, so it connects directly to energy ideas across the course. On both multiple-choice and free-response questions, you may need to calculate rotational work, read it off a graph, or explain how that work changes a system's rotational kinetic energy.

When you write justifications, naming the equation is not enough. You need to walk through the reasoning: how the torque, the angular displacement, and the direction lead to your claim about energy transfer or final speed. That style of clear, step-by-step explanation is exactly what the free-response section rewards.

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Key Takeaways

• Work by a torque is $W = \tau \Delta\theta$, with torque in N·m and angular displacement in radians, giving work in joules.
• A torque only does work when the system actually rotates through an angle; no rotation means no work.
• Torque in the same direction as the rotation does positive work and adds energy; torque opposing the rotation does negative work and removes energy.
• The area under a torque vs. angular position graph equals the work done over that angular displacement.
• For a constant torque the graph area is a rectangle; for a changing torque, break the area into rectangles and triangles or use a trapezoid.
• Rotational work changes rotational kinetic energy, $K = \frac{1}{2} I \omega^2$, just like linear work changes translational kinetic energy.

Work Done by Torque on Rigid Systems

Energy Transfer by Torque

A torque transfers energy into or out of a rigid system when it acts over an angular displacement. How much energy moves depends on both the size of the torque and how far the object rotates while the torque is applied.

• Torque in the same direction as the angular displacement transfers energy into the system.
• Torque opposite to the angular displacement transfers energy out of the system.
• If the system does not rotate, the torque does no work, no matter how large it is.

The Work-Torque Relationship

The work done by a torque is found with a clean relationship that lets you quantify the energy transferred during rotation.

• $W = \tau \Delta\theta$
• $W$ is work done (joules, J)
• $\tau$ is torque magnitude (newton-meters, N·m)
• $\Delta\theta$ is angular displacement (radians, rad)
• Doubling either the torque or the angular displacement doubles the work.

Notice that N·m and J are equivalent units here, just like in linear work. A small torque applied over a large angle can do the same work as a large torque over a small angle.

Torque-Angle Graphs for Work

A torque vs. angular position graph is the rotational version of a force vs. displacement graph, and the area under the curve gives the work done.

• Torque goes on the vertical axis; angular position goes on the horizontal axis.
• The area under the curve over an angular interval equals the work done during that interval.
• For a constant torque, the area is a rectangle (base × height).
• For a changing torque, split the area into rectangles and triangles, or use a trapezoid when the torque changes linearly.
• Area above the axis (torque and displacement in the same direction) is positive work, meaning energy goes into the system.
• Area below the axis (torque opposing the displacement) is negative work, meaning energy leaves the system.

How to Use This on the AP Physics 1 Exam

Problem Solving

• Check that your angular displacement is in radians before plugging into $W = \tau \Delta\theta$. Degrees will give wrong answers.
• Decide the sign first. Compare the torque direction to the rotation direction, then assign positive or negative work.
• Connect work to energy. If you find the rotational work, you can use it to predict the change in $K = \frac{1}{2} I \omega^2$ and solve for a final angular speed.

Free Response

• For a justification, do not stop at "by the work equation." Explain that the torque acts over the angular displacement, state whether that work is positive or negative, and then connect it to the change in rotational kinetic energy or the resulting motion.
• When a graph is given, state clearly that work is the area under the torque vs. angular position curve, then show the area calculation with units.

Common Trap

• Plugging in degrees instead of radians, or forgetting the negative sign when a torque opposes the rotation.

Practice Problem 1: Work by Constant Torque

Solution:

1. Identify the given quantities:

   • Torque: $\tau = 15$ N·m
   • Angular displacement: $\Delta\theta = 2\pi$ rad

2. Apply the work-torque equation: $W = \tau \Delta\theta$ $W = 15 \text{ N·m} \times 2\pi \text{ rad}$ $W = 15 \text{ N·m} \times 6.28 \text{ rad}$ $W = 94.2 \text{ J}$

3. The student performs 94.2 joules of work on the wheel.

Practice Problem 2: Work from Torque-Angle Graph

Solution:

1. Since the torque varies linearly, find the area under the line using the trapezoid formula: Area = (1/2) × (sum of parallel sides) × height

2. Identify the quantities:

   • Initial torque: 20 N·m
   • Final torque: 5 N·m
   • Angular displacement: 3 rad

3. Calculate the work: $W = \frac{1}{2} \times (20 \text{ N·m} + 5 \text{ N·m}) \times 3 \text{ rad}$ $W = \frac{1}{2} \times 25 \text{ N·m} \times 3 \text{ rad}$ $W = 12.5 \text{ N·m} \times 3 \text{ rad}$ $W = 37.5 \text{ J}$

4. The total work done is 37.5 joules.

Practice Problem 3: Energy Transfer Direction

Solution:

1. Identify the given quantities:

   • Torque: $\tau = 8$ N·m (clockwise)
   • Angular displacement: $\Delta\theta = 4\pi$ rad (counterclockwise)

2. Find the magnitude of the work: $|W| = \tau \Delta\theta = 8 \text{ N·m} \times 4\pi \text{ rad} = 32\pi \text{ J} \approx 100.5 \text{ J}$

3. Because the torque opposes the direction of rotation, the work is negative: $W = -100.5 \text{ J}$

4. The negative work means energy is transferred out of the wheel. This makes sense because the braking torque removes the wheel's rotational kinetic energy, converting it to heat in the brake pads.

Common Misconceptions

• A large torque always does a lot of work. Work depends on torque and how far the system rotates. If there is no angular displacement, the work is zero even for a huge torque.
• You can use degrees in $W = \tau \Delta\theta$. The angular displacement must be in radians for the work to come out in joules.
• Work from a torque is always positive. When the torque opposes the rotation, the work is negative and energy leaves the system.
• N·m and J are different things here. For rotational work they are equivalent units, just as a newton times a meter gives a joule for linear work.
• The area under a torque-angle graph is just a number. That area is the work done, and area below the axis counts as negative work.
• Rotational work has nothing to do with kinetic energy. The net work done by torques changes a system's rotational kinetic energy, $K = \frac{1}{2} I \omega^2$, the same way linear work changes translational kinetic energy.

Related AP Physics 1 Guides

• 6.3 Angular Momentum and Angular Impulse
• 6.4 Conservation of Angular Momentum
• 6.5 Rolling
• 6.6 Motion of Orbiting Satellites
• 6.1 Rotational Kinetic Energy