What Is Moment of Inertia and What Does It Depend On?

Moment of inertia is a measure of how strongly a body resists angular acceleration about a specified axis. It depends on both the body’s mass and how that mass is distributed relative to the axis of rotation.

This is an HL-only concept in A.4 Rigid body mechanics. For a system of point masses, moment of inertia is

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

where $m_i$ is each point mass and $r_i$ is its perpendicular distance from the axis. Its SI unit is $\text{kg m}^2$.

Moment of inertia is often described as the rotational analogue of mass. However, unlike mass, it is not fixed for a body because choosing a different axis changes the distances of its particles from that axis.

The squared distance is crucial: moving mass farther from the axis increases $I$ substantially. Newton’s second law for rotation is

$$\tau=I\alpha$$

so, for the same resultant torque $\tau$, a larger $I$ produces a smaller angular acceleration $\alpha$.

For example, two point masses of $2.0\,\text{kg}$ are each $0.50\,\text{m}$ from an axis:

$$I=2(2.0)(0.50)^2=1.0\,\text{kg m}^2$$

If both are moved to $1.0\,\text{m}$, then

$$I=2(2.0)(1.0)^2=4.0\,\text{kg m}^2$$

Doubling the distance makes the moment of inertia four times larger. This fourfold increase means that the same torque would produce only one-quarter of the original angular acceleration, since $\alpha=\tau/I$.

A common misconception is that moment of inertia is a property of the body alone. It is a property of the body and the chosen axis.

In an IB exam, identify the axis first, use perpendicular distances, and show each term in $\sum m_i r_i^2$. For explanation questions, link greater mass distribution away from the axis to larger $I$ and therefore smaller $\alpha$ for a fixed torque.