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The Moment of Inertia of a Point Mass

A body, of mass, m, is moving with angular speed ω, as shown here.
The kinetic energy K possessed by the body is

and since v = rω, we have

We see that, for a point mass, moving in a circle of radius, r, the quantity mrČ is the rotational equivalent of m.

We therefore define the moment of inertia of a point mass to be given by

and the moment of inertia of any body can be found by adding together the moments of inertia of all its component particles.
This is often written as

where i (standing for integer) simply means the number of the particle; i=1 for the first particle to be added, i=2 for the next etc
Using this idea (and the mathematical process of integration) gives the following results
 Body Moment of Inertia Uniform rod of length L (rotating about the centre) Uniform rod of length L (rotating about one end) circular disc (or cylinder) of radius r (rotating about the centre) thin circular ring (or hollow cylinder) of radius r (rotating about the centre) thin hollow sphere of radius r (rotating about the centre) solid sphere of uniform mass distribution and radius r (rotating about the centre)

Notice that for the thin ring or cylinder or sphere, the moment of inertia is the same as for a point mass because in these cases, all the mass is essentially at the same distance from the centre.
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