If the temperature of a uniform rod is slightly elevated by t , the increase in its moment of inertia I about a perpendicular bisector is

Step-by-step solution

The moment of inertia of a uniform rod of mass M and length L about its perpendicular bisector is given by I = ML^2 12 When the temperature is increased by t , the length of the rod increases due to thermal expansion. The new length is L' = L(1 + t) The new moment of inertia is I' = M(L')^2 12 = M[L(1 + t)]^2 12 I' = ML^2 12 (1 + t)^2 Since t is very small, we can use the binomial approximation (1 + x)^n 1 + nx I' I(1 + 2 t) I' = I + 2 I t The increase in the moment of inertia is I = I' - I = 2 I t