A solid body of any shape possesses a moment of inertia I₀ at 0^ C . If represents the coefficient of linear expansion of the solid, determine the expression for its moment of inertia I at a temperature .

Step-by-step solution

The moment of inertia of a solid body is given by I₀ = m_i r_i^2 . When the temperature of the body increases by , the distance of each mass element from the axis of rotation increases due to thermal expansion. The new distance r_i' is given by: r_i' = r_i(1 + ) where is the coefficient of linear expansion. The new moment of inertia I at temperature is: I = m_i (r_i')^2 I = m_i [r_i(1 + )]^2 I = (1 + )^2 m_i r_i^2 I = I₀ (1 + )^2 Since is very small, we can use the binomial approximation (1 + x)^n 1 + nx for x 1 an