Rotational Motion — Torque, Angular Momentum & Moment of Inertia

Rotational motion is motion around an axis. To understand it quickly, ask three questions: What is trying to turn the object? How hard is it to change that rotation? What rotational motion does the object already have?

Those three questions lead to the three core ideas: torque, moment of inertia, and angular momentum. Torque measures the turning effect of a force. Moment of inertia measures how strongly the object resists angular acceleration about a chosen axis. Angular momentum measures rotational motion and stays constant when the net external torque is zero.

For many intro problems, this quick analogy helps:

• torque is the rotational version of force
• moment of inertia is the rotational version of mass
• angular momentum is the rotational version of linear momentum

That analogy is only a starting point. In rotational motion, the choice of axis matters at every step.

The Fixed-Axis Model

If a rigid body rotates about a fixed axis, the standard starting equation is

$$\tau_{net} = I\alpha$$

Here $\tau_{net}$ is the net external torque about the axis, $I$ is the moment of inertia about that axis, and $\alpha$ is the angular acceleration.

In the same fixed-axis setting, the angular momentum about that axis is

$$L = I\omega$$

where $\omega$ is angular speed, with direction handled by your sign convention or right-hand rule. This form is not the most general rule for every rigid body, so use it when the problem clearly stays with one fixed axis.

Torque: Force With Leverage

Torque measures the turning effect of a force about an axis. A force can be large and still produce little torque if it acts close to the axis or points almost through it.

Its magnitude is

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

where $r$ is the distance from the axis to the point where the force acts, $F$ is the force magnitude, and $\theta$ is the angle between $\vec{r}$ and $\vec{F}$.

This is why a door opens easily when you push far from the hinge and nearly perpendicular to the door. The same force near the hinge produces much less torque.

Moment Of Inertia: Where The Mass Is

Moment of inertia tells you how the mass is distributed relative to the axis. Mass farther from the axis contributes more strongly, which is why the quantity depends on distance squared.

For discrete particles,

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

and for a continuous body the idea becomes an integral. The main practical point is simpler: the same object can have different moments of inertia about different axes.

That is why a long rod is easier to spin about its center than about one end, even though the rod itself has not changed.

Angular Momentum: What Stays Constant

Angular momentum describes rotational motion in a way that becomes especially powerful when torque is small or zero.

The most important rule is

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

So if the net external torque about an axis is zero, angular momentum about that axis stays constant.

That conservation idea explains many familiar effects. A skater who pulls in their arms reduces $I$, so $\omega$ increases if external torque is negligible and angular momentum stays the same.

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. A constant net torque of $3.0\ \mathrm{N \cdot m}$ acts on it. Assume it starts from rest.

For a uniform solid disk about its center,

$$I = \frac{1}{2}MR^2$$

So

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

Now use

$$\tau_{net} = I\alpha$$

to find the angular acceleration:

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

After $2.0\ \mathrm{s}$, the angular speed is

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

Then the angular momentum is

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

This example shows the full chain:

1. torque causes angular acceleration
2. the amount of acceleration depends on the moment of inertia
3. once the object is spinning, it has angular momentum

Common Mistakes In Rotational Motion

Treating torque as just another word for force

Force and torque are related, but they are not the same quantity. Torque depends on where and how the force is applied relative to the axis.

Forgetting that moment of inertia depends on the axis

There is no single universal $I$ for an object. You must specify the axis before choosing or calculating the moment of inertia.

Using $L = I\omega$ without checking the model

That form works cleanly in common fixed-axis problems. In more general rigid-body motion, angular momentum is not always parallel to angular velocity.

Ignoring the direction of torque and angular momentum

These quantities have direction. In many class problems the sign convention or right-hand rule handles that direction, so dropping the sign too early can reverse the answer.

Where Rotational Motion Shows Up

Rotational motion appears in wheels, turbines, pulleys, motors, planets, gyroscopes, and molecules. In engineering and physics, it is the natural language whenever turning, spinning, or orbit-related effects matter.

It also connects directly to linear mechanics. A lot of rotation problems become easier once you line up the rotational and linear versions of the same idea:

• force $\leftrightarrow$ torque
• mass $\leftrightarrow$ moment of inertia
• momentum $\leftrightarrow$ angular momentum

Try A Similar Problem

Keep the same disk, but double the radius while keeping the mass the same. Because $I$ changes with $R^2$, the moment of inertia becomes larger, so the same torque produces a smaller angular acceleration.

Try that version yourself: compute the new $I$, then find the new $\alpha$ and the new $L$ after the same $2.0\ \mathrm{s}$.