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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.
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Solution
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, (ΔT) =θ
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.
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