# Mass moment of inertia

## Engineering Fundamentals

#### Introduction

The mass moment of inertia measures the extent to which an object resists rotational acceleration about a particular axis, and is the rotational analogue to mass. The torque needed to achieve an angular acceleration is defined by:

$$T=I\dot{\omega}$$

The mass moment of inertia for several shapes is shown in the table below.

FigureDescriptionMass moment of inertia
Point mass$$I=mr^2$$
Rod of length l and mass mWith l>>r
$$I_{z1}=\frac{1}{12}ml^2$$
$$I_{z2}=\frac{1}{3}mL^2$$
Solid cylinder$$I=\frac{1}{2}mr^2$$
Cylindrical shellWith r>>t
$$I=mr^2$$
Sphere$$I=\frac{2}{5}mr^2$$
Block$$I_{z1}=\frac{1}{12}m\left(w^2+h^2\right)$$
$$I_{z2}=\frac{1}{3}m\left(w^2+h^2\right)$$

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