Moment of Inertia — Definition

What moment of inertia is

In linear motion, mass measures resistance to acceleration: $a = F/m$.

In rotational motion, the corresponding quantity is moment of inertia $I$: $\alpha = \tau/I$.

A body with larger $I$ is harder to spin up or slow down — it has more rotational inertia.

For a system of point masses:

$I = \sum m_i r_i^2$

For a continuous body:

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

where $r$ is the perpendicular distance of each mass element from the rotation axis.

Why $r^2$ and not $r$?

The $r^2$ dependence comes from the dynamics. Consider a point mass $m$ at distance $r$ from the axis, pulled by tangential force $F_t$:

$F_t = ma_t = m(r\alpha)$

Torque: $\tau = F_t \cdot r = mr^2\alpha$

So: $\tau = (mr^2)\alpha$

The quantity $mr^2$ is the moment of inertia — the coefficient relating torque to angular acceleration for a single particle.

For a system: $\tau = \sum m_i r_i^2 \cdot \alpha = I\alpha$

Key properties

$I$ depends on the axis. The same body has different moments of inertia about different axes. Always specify the axis.

$I$ depends on mass distribution, not just total mass. Two bodies of equal mass can have very different $I$ if mass is distributed differently relative to the axis.

Mass farther from axis contributes more. Due to $r^2$, mass at distance $2r$ contributes 4 times as much as mass at distance $r$.

Units

$[I] = \text{kg} \cdot \text{m}^2$

Radius of gyration $k$

The radius of gyration $k$ is defined by:

$I = Mk^2 \implies k = \sqrt{\frac{I}{M}}$

$k$ is the distance from the axis at which all the mass could be concentrated to give the same moment of inertia. It characterises the "effective" radius of the mass distribution.