Question

Find the expression for moment of inertia of a body and its radius of gyration.

Answer 1

A rigid body can be supposed to be made of many small particles. The moment of inertia of the rigid body is obtained by the sum of moment of inertias of individual particles i.e.,

I = I₁ + I₂ + I₃ + ......... + I_n

where m₁, m₂, m₃, ....., m_n are the masses of the particles of rigid body and their distances from the axis of rotation are r₁, r₂, r₃, ...., r_n respectively as shown in fig.

Moment of inertia of a rigid body can be obtained by

I = m₁r₁² + m₂r₂² + m₃r₃² + ..... + m_nr_n² 

This method is pure theoretical but practically it is very difficult, say rather impossible. Therefore to obtain moment of inertia of a rigid body, we consider a point in the body where all the mass of the body is considered centralized such that the square of its distance from the axis of rotation when multiplied with the mass of the body, then moment of inertia is obtained. This distance is called the ‘radius of gyration’ and it is denoted by K.

∴ I = MK²

^{or \(K = \sqrt{\frac{I}{M}}\ ...(1)\)} 

If the body is supposed to be made of n particles, then

I = m₁r₁² + m₂r₂² + m₃r₃² + .......... + m_nr_n²

And M = m₁ + m₂ + m₃ + ....... + m_n

_{\(\therefore\ K = \sqrt{\frac{m_1r_1^2 + m_2r_2^2 + ....+m_nr_n^2}{m_1 + m_2 + ...+m_n}}\ ...(2)\)} 

If mass of each particle be m, i.e.,

m₁ = m₂ = m₃ = .......... = m_n

Then M = m + m + .......... + m(n times)

or M = mn ..........(3)