RADIUS OF GYRATION

Suppose the rigid body of mass M consists of n particles each of mass m.

Therefore             M = mn.

The moment of inertia of the body about a given axis,

I = m r₁ ² + m r₂ ²  +.………….+ m r_n ²    = m [ r₁ ² + r₂ ²  +.………….+ r_n ² ] = mK²  

where, K is called the radius of gyration corresponding to the given axis and is the mean of the squares of perpendicular distances of the particles of the body from the given axis.

Moment of inertia and radius of gyration for some symmetric bodies

	Body	Axis	I	K
1	Thin rod of length L	Passing through its center and Perpendicular to its length	1 ML2 12	L 2 Ö3
2	Ring of radius R                               → Thin-walled hollow cylinder of radius R	Passing through its center and Perpendicular to its plane Geometric axis	MR 2	R
3	Ring of radius R                               → Circular disc of radius R                  → Solid cylinder of radius R                 →	Any diameter Passing through its center and Perpendicular to its plane Geometric axis	1 MR 2 2	R Ö2
4	Circular disc of radius R                  →	Any diameter	1 MR 2 4	R 2
5	Thin-walled hollow sphere of radius R	Any diameter	2 MR 2 3	Ö2/3 R
6	Solid sphere of radius R                  →	Any diameter	2 MR 2 5	Ö2/5 R
7	Solid right circular cone of radius R→	Geometric axis	3 R 2 10	Ö3/10 R