Welcome to Unit 6.2: Torque and Work

In previous units, we learned that a force acting over a displacement does work on an object, changing its translational kinetic energy. But what happens when we apply a "twist" (a torque) to an object and it rotates through an angle? Just as linear forces do work, torques do work too! In this chapter, we will bridge the gap between rotational dynamics and energy. By the end of these notes, you'll see that the math for rotating systems is beautifully parallel to the math for linear systems.

1. Defining Rotational Work

In linear motion, work is defined as the integral of force over a path: \( W = \int \vec{F} \cdot d\vec{r} \). In rotational motion, we follow the exact same logic. If a torque \( \tau \) is applied to an object and causes it to rotate through an angular displacement \( d\theta \), the work done by that torque is:

\( W = \int_{\theta_i}^{\theta_f} \tau \, d\theta \)

Breaking it down:

  • \( W \): The work done on the system (measured in Joules, \( J \)).
  • \( \tau \): The torque applied (measured in Newton-meters, \( N \cdot m \)).
  • \( \theta \): The angular position (measured in radians).

The "Constant Torque" Shortcut:
If the torque is constant (it doesn't change as the object spins), the math becomes much simpler. The integral reduces to:
\( W = \tau \Delta \theta \)

Analogy: Pushing a box across the floor for 5 meters requires work (\( F \times d \)). Pushing a heavy revolving door through a half-circle (\( \pi \) radians) also requires work (\( \tau \times \theta \)). It's the same physical "effort," just in a circle!

Quick Tip: Always make sure your angles are in radians! The AP exam will often give you rotations in "revolutions" or "degrees" to trick you. Remember: \( 1 \text{ rev} = 2\pi \text{ radians} \).

2. The Work-Energy Theorem (Rotational Version)

Recall from Unit 3 that the net work done on an object equals its change in kinetic energy (\( W_{net} = \Delta K \)). This fundamental law applies to rotation as well! The net work done by all torques on a rigid body is equal to the change in its rotational kinetic energy.

\( W_{net} = \Delta K_{rot} \)

\( W_{net} = \frac{1}{2} I \omega_f^2 - \frac{1}{2} I \omega_i^2 \)

Where:

  • \( I \): The rotational inertia of the object.
  • \( \omega \): The angular velocity (initial and final).

Why this is useful: If you know how much work a motor did on a flywheel, you can immediately calculate how much faster that flywheel is spinning without needing to calculate acceleration or time!

3. Power in Rotating Systems

Sometimes we don't just care about how much work is done, but how fast it is being done. This is Power (\( P \)).

In linear terms, \( P = \vec{F} \cdot \vec{v} \).
In rotational terms, the instantaneous power delivered by a torque is:

\( P = \frac{dW}{dt} = \tau \omega \)

Did you know? This is how car engines are rated. When you see a "torque curve" for a high-performance car, the horsepower (power) is directly calculated by multiplying the torque the engine produces by how fast the engine is spinning (\( \omega \)).

4. Calculus Connection: Solving Problems

Since AP Physics C is calculus-based, you might encounter a scenario where the torque is a function of the angle, such as \( \tau(\theta) = k\theta \). In these cases, you must use the integral form.

Step-by-Step for Variable Torque:

  1. Identify the expression for torque \( \tau \) in terms of \( \theta \).
  2. Set up your integral: \( W = \int \tau(\theta) \, d\theta \).
  3. Plug in your limits of integration (starting angle to ending angle).
  4. Integrate and evaluate.

Example: If a torque \( \tau = 3\theta^2 \) is applied to a disk from \( \theta = 0 \) to \( \theta = 2 \) radians:
\( W = \int_{0}^{2} 3\theta^2 \, d\theta = [\theta^3]_{0}^{2} = 2^3 - 0 = 8 \text{ Joules} \).

5. Sign Conventions and Common Pitfalls

Don't worry if signs get confusing; just keep these rules in mind:

  • Positive Work: If the torque and the angular displacement are in the same direction (e.g., both counter-clockwise), the torque is doing positive work and the object will speed up.
  • Negative Work: If the torque opposes the motion (like friction in a bearing), the work is negative and the object will slow down.
  • The "Perpendicular" Rule: Just like a force perpendicular to motion does no work, a force that points directly at the axis of rotation produces zero torque and therefore does zero rotational work.

Key Takeaway Summary

1. Work Formula: \( W = \int \tau \, d\theta \). For constant torque: \( W = \tau \Delta \theta \).
2. Energy Connection: Net rotational work equals the change in rotational kinetic energy (\( \frac{1}{2} I \omega^2 \)).
3. Power Formula: \( P = \tau \omega \).
4. Units: Always use radians for \( \theta \) and \( \omega \) to keep your Joules and Watts accurate!

Next up in Unit 6: We will explore how these rotating systems carry Angular Momentum!