11.6: New Page

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For a rigid body rotating about a fixed axle, the moment of inertia is

\[I=\sum_{i} M_{i} d_{i}^{2}\label{11.23}\]

where \(\mathrm{d}_{\mathrm{i}}\) is the perpendicular distance of the \(\text { ith }\) particle from the axle. Equations (11.16)-(11.18) are valid for a rigid body consisting of many particles. Furthermore, the moment of inertia is constant in this case, so it can be taken out of the time derivative:

\[\tau=\frac{d I \omega}{d t}=I \frac{d \omega}{d t}=I \alpha \quad(\text { fixed axle, constant } I)\label{11.24}\]

The quantity \(a=d \omega / d t\) is called the angular acceleration.

The sum in the equation for the moment of inertia can be converted to an integral for a continuous distribution of mass. We shall not pursue this here, but simply quote the results for a number of solid objects of uniform density:

For rotation of a sphere of mass M and radius R about an axis piercing its center: \(I=2 M R^{2} / 5\).
For rotation of a cylinder of mass M and radius R about its axis of symmetry: \(I=M R^{2} / 2\).
For rotation of a thin rod of mass M and length L about an axis perpendicular to the rod passing through its center: \(\mathrm{I}=\mathrm{ML}^{2} / 12\).
For rotation of an annulus of mass M, inner radius R_a, and outer radius R_b about its axis of symmetry: \(I=M\left(R_{a}^{2}+R_{b}^{2}\right) / 2\).