Show that moment of inertia of a solid body of any shape changes with temperature as `I=I_0(1+2 alphatheta).` Where `I_0` is the moment of inertia at `0^@C ` and `alpha ` is the coefficient of linear expansion of the solid.

Explanation

Given, `I_0= moment of inertia at 0^@C`
` alpha= Coefficient of inertia `
expansion
` To prove I=I_0 (1+2 alpha theta)
Let the temperature changes to `theta` from `0^@C`
`Delta T= theta`
` Let R_0 be the radius of gyration `
` Now R_0=R(1+ alpha theta)`
`I_0=MR^2 where M is the mass. `
` Now I=MR^2=MR^2 ((1+alpha theta)^2)`
` [by binominal expansion and neglecting
alpha_2 theta_2 which is a very small value]`
`=MR^2(1+2 alpha theta)`
` So, I=I_0(1+2 alpha theta) .........(proved).`