Question

Show that the moment of inertia of a solid body of any shape changes with temperature as I = I₀ (1 + 2αθ), where I₀ is the moment of inertia at 0°C and α is the coefficient of linear expansion of the solid.

Answer 1

Given:
Coefficient of linear expansion of solid = α
Moment of inertia at 0 °C = I₀
If temperature changes to θ from 0 °C, then change in temperature, $\left(∆T\right)$ = $\theta$
Let I be the new moment of inertia attained due to rise in temperature.
Let R₀ be the radius of gyration at 0 °C.
We know that on heating, radius of gyration will change as
R = R₀(1 + αθ)
Here, R is the radius of gyration after heating.
I₀ = MR₀² , where M = mass of the body
Now, I = MR² = MR₀²(1 + αθ)²
Expanding binomially and neglecting the higher terms of order (αθ) that will be very small, we get
I = MR₀²(1 + 2 αθ)
So, I = I₀(1 + 2 αθ)
Hence, proved.