Define moment of inertia and radius of gyration. If a body consists of n particles of equal masses then show that radius of gyration is equal to rms of the distance of constituent particles from axis of rotation.

Moment of inertia is the ability of body by virtue of which the body remains in the state of rest or uniform circular motion unless external torque is applied on the body.

Consider a rigid body consisting of n particles of masses m₁ m₂, m₃ .......... m_n situated at distances r₁, r₂, r₃ ............. r_n respectively from the axis of rotation AB as shown in figure.
                        
The moment of inertia of rigid body about AB axis is,
 
 

Radius of gyration of a body about an axis is the distance, at which whole of mass of the body is supposed to be concentrated, so that it would have the same moment of inertia as that of body.
It is denoted by K.
If M is the mass of body, then

I =MK²                               ...(2) 

From (1) and (2), we get 
 
If all the particles are identical having each mass m, then
         M = nm
 
Hence the result