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Mechanics

The Moment of Inertia of a Point Mass

A body, of mass, m, is moving with angular speed w, as shown below.

The K.E. possessed by the body is

K.E. = ½mv²

now, v = rw

so,

K.E. = ½(mr²)w²

We see that, for a point mass, moving in a circle of radius, r, the quantity mr² is the rotational equivalent of m.

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

I = mr²

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

Ibody =

Using this idea gives the following results:

M. of I. of a uniform rod of length l* I = (ml²)/3
M. of I. of a disc or cylinder** I = (mr²)/2
M. of I. of a hollow cylinder or ring I = mr²
M. of I. of a sphere** I = 2(mr²)/5

* rotating about one end

** of uniformly distributed mass

 

© David Hoult 2008