Rotational Motion — Torque, Angular Momentum & Moment of Inertia

When something spins, a wheel, a disk, a skater, you analyze it with a procedure that mirrors linear mechanics but adds one decision linear problems never force: you must pick an axis first. With the axis chosen, three questions drive the rest. What is trying to turn the object? How hard is it to change that rotation? What rotational motion does it already have?

Those questions map to torque, moment of inertia, and angular momentum. A quick analogy helps: torque is the rotational version of force, moment of inertia of mass, and angular momentum of linear momentum. The analogy is only a starting point, because in rotation the choice of axis matters at every step.

When This Procedure Applies

Use this approach for any rigid body that turns about an axis: wheels, turbines, pulleys, motors, gyroscopes, even planets. The cleanest case is fixed-axis rotation, where the standard equations are

$$\tau_{net} = I\alpha, \qquad L = I\omega$$

These forms hold when the body stays with one fixed axis; in fully general rigid-body motion, angular momentum is not always parallel to angular velocity, so reserve them for fixed-axis problems.

The Procedure, Step By Step

1. Choose the axis

Fix the axis before anything else. The same object has different moments of inertia about different axes, so this choice shapes every later number. A long rod is easier to spin about its center than about one end, even though the rod has not changed.

2. Identify the turning cause

Find the net torque about the chosen axis. Torque measures the turning effect of a force, with magnitude

$$\tau = rF\sin\theta$$

where $r$ is the distance from the axis to the force, $F$ is its magnitude, and $\theta$ is the angle between $\vec{r}$ and $\vec{F}$. A large force near the axis produces little torque, which is why a door opens easily when you push far from the hinge and nearly perpendicular.

3. Find rotational inertia

Compute the moment of inertia about the chosen axis. For discrete particles,

$$I = \sum m_i r_i^2$$

and for a continuous body this becomes an integral. Mass farther from the axis counts more, because the contribution scales with distance squared.

4. Connect motion and cause

Link torque, inertia, and motion. Angular acceleration follows from $\tau_{net} = I\alpha$, and once the body spins it carries angular momentum $L = I\omega$. The deeper rule is

$$\tau_{net} = \frac{dL}{dt}$$

so if the net external torque about the axis is zero, angular momentum about that axis stays constant. That is why a skater who pulls in their arms reduces $I$ and speeds up: $\omega$ rises to keep $L$ fixed.

Full Worked Example: A Disk Under Constant Torque

Take a uniform solid disk of mass $M = 2.0\ \mathrm{kg}$ and radius $R = 0.50\ \mathrm{m}$ rotating about its central axis, with a constant net torque of $3.0\ \mathrm{N \cdot m}$, starting from rest.

Step 1 fixes the central axis. Step 3 gives the inertia for a uniform solid disk,

$$I = \frac{1}{2}MR^2 = \frac{1}{2}(2.0)(0.50)^2 = 0.25\ \mathrm{kg \cdot m^2}$$

Step 4 connects torque to acceleration,

$$\alpha = \frac{\tau_{net}}{I} = \frac{3.0}{0.25} = 12\ \mathrm{rad/s^2}$$

After $2.0\ \mathrm{s}$,

$$\omega = \omega_0 + \alpha t = 0 + (12)(2.0) = 24\ \mathrm{rad/s}$$

and the angular momentum is

$$L = I\omega = (0.25)(24) = 6.0\ \mathrm{kg \cdot m^2/s}$$

The full chain is visible: torque causes angular acceleration, the amount depends on the moment of inertia, and the spinning object then carries angular momentum.

Where Students Get Stuck, And How To Check

• Treating torque as just another word for force. They are related but not identical; torque depends on where and how the force is applied relative to the axis (step 2).
• Forgetting that moment of inertia depends on the axis. There is no single universal $I$; specify the axis in step 1 before computing it.
• Using $L = I\omega$ without checking the model. It works for common fixed-axis problems, but in general rigid-body motion angular momentum need not be parallel to angular velocity.
• Ignoring direction. Torque and angular momentum have direction; let the sign convention or right-hand rule handle it, and do not drop the sign too early.

Where Rotational Motion Shows Up

Rotational motion appears in wheels, turbines, pulleys, motors, planets, gyroscopes, and molecules. It also connects directly to linear mechanics: force pairs with torque, mass with moment of inertia, and momentum with angular momentum, so lining up the rotational and linear versions makes many problems easier.

Practice the Procedure

Keep the same disk but double the radius while holding the mass fixed. Because $I$ scales with $R^2$, the moment of inertia grows, so the same torque gives a smaller angular acceleration. Run steps 3 and 4 to compute the new $I$, then the new $\alpha$ and $L$ after the same $2.0\ \mathrm{s}$.