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7. Problem 11.58P (HRW) Show that the rotational inertia of a solid cylinder of mass
M radius R about its central axis is equal to the rotational inertia of a thin
loop of mass M and radius
The radius k of the equivalent loop is called the radius of gyration of the given body. |
about
its central axis. (b) Show that the rotational inertia I of any given
body of mass M about any given axis is equal to the rotational inertia of an
equivalent loop about that axis, if the loop has the same mass M and radius
k given by
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Solution: Click For PDF Version (a) In problem 1 we have calculated expressions for rotational inertia of different geometrical bodies including that of a cylinder of mass M and radius R about its central axis. We use the result that the rotational inertia I of a cylinder about its central axis is
The rotational inertia of a thin loop of mass M and
radius k about its central axis
(b) By definition the radius of gyration k for a body with moment of inertia I about a given axis is the radius of a loop of the same mass M and having the same moment of inertia as that of the body about the same axis passing through its centre. Therefore,
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is
basically the definition of rotational inertia. We thus have the relation
