A disc has mass M and radius R. If temperature coefficient of linear expansion of material of disc is a, then find the increase in the moment of inertia of disc when heated through θ°C.

Solution

The moment of inertia of disc at low temperature is,

straight I subscript straight o space equals space 1 half MR squared

When the disc is heated its mass remains same but radius will increase.
Let R' be the radius of disc at temperature θ.
That is,

straight R apostrophe space equals space straight R left parenthesis 1 plus αθ right parenthesis

Therefore the moment of inertia of disc at high temperature is,

straight I space equals 1 half MR apostrophe squared space space space equals space 1 half straight M left square bracket straight R left parenthesis 1 plus αθ right parenthesis right square bracket squared space

equals straight I subscript straight o left parenthesis 1 plus αθ right parenthesis squared space space equals space straight I subscript straight o left parenthesis 1 plus 2 αθ right parenthesis

Increase in moment of inertia of disc is,

straight I minus straight I subscript straight o space equals space 2 straight I subscript straight o αθ

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