Moment of Inertia — Formulas and Examples

Moment of inertia measures how strongly an object resists changes in rotation about a chosen axis. In rotational dynamics, it plays a role similar to mass in straight-line motion.

If you searched for the formula, the key idea is simple: each part of the mass contributes according to its distance from the axis, and that distance is squared. That is why mass farther from the axis matters so much.

The axis is not optional. The same object can have different moments of inertia about different axes.

What moment of inertia depends on

For a set of point masses,

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

Here, $m_i$ is one piece of mass and $r_i$ is its distance from the axis. The $r^2$ term is the main reason the concept behaves the way it does. If you move the same mass twice as far from the axis, its contribution becomes four times larger.

For a continuous object, the same idea becomes

$$I = \int r^2\,dm$$

You do not need that integral for every problem, but it explains where standard shape formulas come from. The SI unit of moment of inertia is $\mathrm{kg \cdot m^2}$.

How moment of inertia affects angular acceleration

If a rigid body rotates about a fixed axis, rotational dynamics often uses

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

where $\tau_{net}$ is net torque and $\alpha$ is angular acceleration. For the same applied torque, a larger $I$ means a smaller angular acceleration.

That is why a figure skater spins faster when pulling arms inward. The total mass stays almost the same, but more of it moves closer to the axis, so the moment of inertia becomes smaller.

Common moment of inertia formulas

These formulas are standard only for the stated shape and axis.

• Point mass at distance $r$: $I = mr^2$
• Thin hoop or ring about its center: $I = MR^2$
• Solid disk or solid cylinder about its center: $I = \frac{1}{2}MR^2$
• Solid sphere about its center: $I = \frac{2}{5}MR^2$
• Thin rod of length $L$ about its center, axis perpendicular to the rod: $I = \frac{1}{12}ML^2$
• Thin rod of length $L$ about one end, axis perpendicular to the rod: $I = \frac{1}{3}ML^2$

If the axis changes, the formula can change too. That is one of the most common sources of errors.

Worked example: moving the same mass inward

Suppose two small masses of $2\ \mathrm{kg}$ each are attached to a light rod, with the rotation axis at the center.

Case 1: each mass is $0.50\ \mathrm{m}$ from the axis.

$$I = \sum m_i r_i^2 = 2(2)(0.50)^2 = 1.0\ \mathrm{kg \cdot m^2}$$

Case 2: each mass is moved inward so it is only $0.25\ \mathrm{m}$ from the axis.

$$I = 2(2)(0.25)^2 = 0.25\ \mathrm{kg \cdot m^2}$$

The total mass stayed the same, but the moment of inertia became four times smaller because the distance was cut in half.

Now suppose the same net torque of $2.0\ \mathrm{N \cdot m}$ acts in both cases. Then

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

so the angular accelerations are

$$\alpha_1 = \frac{2.0}{1.0} = 2.0\ \mathrm{rad/s^2}$$ $$\alpha_2 = \frac{2.0}{0.25} = 8.0\ \mathrm{rad/s^2}$$

The total mass did not change, but the angular acceleration became four times larger. That is the core intuition: moving mass inward makes rotation easier to change.

Common mistakes with moment of inertia

Forgetting to specify the axis

Moment of inertia is always about an axis. A disk about its center and the same disk about a tangent line do not have the same value.

Using the right formula for the wrong shape

$I = \frac{1}{2}MR^2$ is for a solid disk or solid cylinder about its center axis, not for a hoop. A hoop with the same $M$ and $R$ has $I = MR^2$.

Ignoring the squared distance

Students often notice that distance matters, but underestimate how strongly it matters. Because the formula contains $r^2$, small changes in radius can have a large effect.

Treating it as only a mass question

More mass often increases moment of inertia, but distribution matters too. A lighter object can still have a larger $I$ if more of its mass is far from the axis.

Where moment of inertia is used

Moment of inertia appears whenever rotation matters:

1. wheels, flywheels, and motors
2. rotating rods, disks, and pulleys
3. figure skating and diving
4. balancing torque and angular acceleration in mechanics problems
5. engineering designs where rotational response matters

In physics classes, it usually shows up together with torque, angular acceleration, angular momentum, and rotational kinetic energy.

Try a similar problem

Take the worked example and move each $2\ \mathrm{kg}$ mass to $0.40\ \mathrm{m}$ from the axis instead. Compute the new $I$, then predict how the angular acceleration changes under the same torque. That one variation is enough to make the $r^2$ dependence stick.