The moment of inertia $I$ is the rotational analogue of mass. It measures the resistance to angular acceleration.

$I = \sum m_i r_i^2 \quad [M^1L^2T^0] \quad \text{kg·m}^2$

Two key factors:

1. Total mass: Larger mass → larger I (for same geometry)
2. Mass distribution: Mass farther from axis contributes more ($r_i^2$ dependence)

Why $r^2$? In rotational motion, each mass element moves with speed $v_i = \omega r_i$. Its KE is $\frac{1}{2}m_i v_i^2 = \frac{1}{2}m_i r_i^2 \omega^2$. Summing: $KE = \frac{1}{2}\omega^2 \sum m_i r_i^2 = \frac{1}{2}I\omega^2$. So I naturally appears as the "effective mass" for rotation.

Radius of gyration: $K = \sqrt{I/M}$ is the RMS distance of mass from the axis. For a disc: $K = R/\sqrt{2} \approx 0.707R$. For a ring: $K = R$.

Key property: $I$ depends on the axis position. The same disc has $I = MR^2/2$ about its central perpendicular axis, $I = MR^2/4$ about a diameter, and $I = 3MR^2/2$ about a tangent perpendicular to the disc.