Q. 32

# Show that moment of inertia of a solid body of any shape changes with temperature as I = I1(1 + 2αθ), where I, is the moment of inertia at 0°C and α is the coefficient of linear expansion of the solid.

Given:
Moment of inertia at 0° C = I1
Say temperature changes to θ .
Change in temperature : Δ T = θ – 0 = θ
Formula Used:

We know that,
Where M is the mass of the body and R0 is the radius of gyration at 0° C and I1 is the initial moment of inertia at 0° .
When temperature of the solid body increases, the radius of gyration also changes due to thermal expansion.
Hence, formula for thermal expansion of radius of gyration is
Here R’ is the changed radius of gyration due to expansion.
Let I be the changed moment of inertia when temperature is θ .

Here, α2θ2is neglected, we get

Hence proved that the moment of inertia of a solid body of any shape changes with temperature as I = I1(1 + 2αθ).

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