Moment of inertia
For rotation about an axis, plays the role of a rotational mass: and .
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Consider a rotating hoop of radius . We have
Hence .
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Consider a rotating rod of length about an axis through an endpoint.
Then
Hence .
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Consider a rotating disc of radius about an axis through its centre.
Then
Hence .
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Consider a rotating sphere of radius about an axis through its centre.
Then
Hence .
Definition
A particle at perpendicular distance from the axis of rotation contributes kinetic energy . In a rigid body all particles rotate with the same angular velocity, so
which defines the moment of inertia of the rigid body about that axis,
At large the particles are densely spaced and the sums are approximated by integrals against the density of mass:
where is the perpendicular distance from to the axis; typically the density is uniform, .
The bigger is, the harder it is to rotate the body: it is effectively a rotational mass.
Angular momentum and torque along the axis
Only the component of the angular momentum along the axis of rotation is considered, so
If the torque is also along the axis of rotation, , then dotting with gives
Hence acts like a rotational force, causing change in the angular velocity.
Dependence on the axis
depends on the choice of axis of rotation, since the distances do. The parallel axis theorem makes this precise: , about the axis through the centre of mass, is lower than about any parallel axis.
Related
Stated in
- Example 5.2 (Moment of Inertia of Rigid Bodies With Uniform Density)§5.2 Moment of Inertia