Moment of Inertia — Formulas and Examples

Use moment of inertia whenever rotation matters and you need to know how strongly an object resists changes in spinning about a chosen axis. In rotational dynamics it plays the role mass plays in straight-line motion: the larger it is, the harder it is to start, stop, or speed up the rotation for a given torque. The defining feature is that each piece of mass contributes according to its distance from the axis squared, which is why mass far from the axis matters so much, and why the axis is never optional.

Step 1: Choose The Axis First

Moment of inertia is always defined about a specific axis, so the same object can have different values for different axes. A disk about its center and the same disk about a tangent line do not share a value. Fix the axis before anything else.

Step 2: Pick The Right Model

Match the situation to one of three forms:

• Point masses: $I = \sum m_i r_i^2$, where $m_i$ is one piece of mass and $r_i$ its distance from the axis.
• A continuous body: $I = \int r^2\,dm$, which is where standard shape formulas come from.
• A standard shape formula, used only when both its shape and its axis match:

The SI unit is $\mathrm{kg \cdot m^2}$.

Step 3: Respect The Squared Distance

The distance from the axis enters as $r^2$, so doubling the distance makes that mass contribute four times as much. This single fact drives most of the intuition, including why a figure skater spins faster when pulling arms inward: the total mass barely changes, but more of it moves close to the axis, so $I$ drops.

Step 4: Connect It To Motion

For a rigid body about a fixed axis, combine the result with

so that for the same applied torque, a larger $I$ means a smaller angular acceleration $\alpha$.

Full Example: Moving The Same Mass Inward

Two small masses of $2\ \mathrm{kg}$ each sit on a light rod, with the rotation axis at the center.

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

Case 2, each mass moved inward to $0.25\ \mathrm{m}$:

The total mass stayed the same, but $I$ became four times smaller because the distance was halved. Now apply the same net torque of $2.0\ \mathrm{N \cdot m}$ in both cases, using $\alpha = \tau_{net}/I$:

Same mass, same torque, yet four times the angular acceleration. Moving mass inward makes rotation easier to change.

Where Each Step Tends To Trip You Up

• Choosing the axis: forgetting to specify it is the most common error. A value without a stated axis is meaningless.
• Picking the model: using the right formula for the wrong shape. $I = \tfrac{1}{2}MR^2$ is a solid disk or cylinder about its center, not a hoop; a hoop with the same $M$ and $R$ has $I = MR^2$.
• The squared distance: students notice distance matters but underestimate how much. Because the formula contains $r^2$, small radius changes have a large effect.
• Connecting to motion: treating it as only a mass question. A lighter object can still have a larger $I$ if more of its mass is far from the axis.

A self-check at the end: redo the worked example with each $2\ \mathrm{kg}$ mass at $0.40\ \mathrm{m}$ from the axis. Compute the new $I$, then predict the angular acceleration under the same torque. If the $r^2$ dependence is sticking, you should see $I$ land between the two cases above. Moment of inertia usually shows up alongside torque, angular acceleration, angular momentum, and rotational kinetic energy.

Frequently Asked Questions