



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. 
