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Rotation and moment of inertia

The moment of inertia is rotation's counterpart to mass: it tells you how hard it is to change a body's angular velocity, and depends both on the mass and on how it is distributed relative to the rotation axis. The farther the mass lies from the axis, the larger the moment of inertia.

I = \sum mr^2

moment of inertia, general

I = \tfrac12MR^2

solid cylinder about its central axis

I = \tfrac25MR^2

solid sphere about its central axis

Symbols

$I$	moment of inertia	kg·m²
$M$	total mass	kg
$R$	radius	m

Example

A solid cylinder with $M = 4.0$ kg and $R = 0.20$ m has $I = \tfrac12\cdot 4.0\cdot 0.20^2 = 0.080$ kg·m².

A thin hoop has

I = MR^2

, a solid cylinder only

\tfrac12MR^2

– mass close to the axis contributes less.

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