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Stretch Rule

Note that the moment of inertia does not change when masses are moved along a vector parallel to the axis of rotation (see, e.g., Eq.$ \,$(B.9)). Thus, any rigid body may be ``stretched'' or ``squeezed'' parallel to the rotation axis without changing its moment of inertia. This is known as the stretch rule, and it can be used to simplify geometry when finding the moment of inertia.

For example, we saw in §B.4.4 that the moment of inertia of a point-mass $ m$ a distance $ R$ from the axis of rotation is given by $ I=mR^2$. By the stretch rule, the same applies to an ideal rod of mass $ m$ parallel to and distance $ R$ from the axis of rotation.

Note that mass can be also be ``stretched'' along the circle of rotation without changing the moment of inertia for the mass about that axis. Thus, the point mass $ m$ can be stretched out to form a mass ring at radius $ R$ about the axis of rotation without changing its moment of inertia about that axis. Similarly, the ideal rod of the previous paragraph can be stretched tangentially to form a cylinder of radius $ R$ and mass $ m$, with its axis of symmetry coincident with the axis of rotation. In all of these examples, the moment of inertia is $ I=mR^2$ about the axis of rotation.

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Parallel Axis Theorem