question_answer

A rod of length L rotates about an axis passing through one of its ends and perpendicular to its plane. If the linear mass density of the rod varies as \[\rho =(A{{r}^{3}}+B)\,kg/m,\]then the moment of inertia of the rod about the given axis of rotation is

A) \[\frac{{{L}^{3}}}{3}\left[ \frac{A{{L}^{3}}}{2}+B \right]\]

B)  \[\frac{L}{3}\left[ \frac{A{{L}^{2}}}{2}+B \right]\]

C)  \[\frac{{{L}^{3}}}{3}\left[ \frac{A}{2}+B \right]\]

D)  None of the above

Correct Answer: A

Solution :

 Consider a small element of the rod, having length \[dr,\]situated at a distance r from the axis. Mass of element, \[dm=\rho \,dr\] \[=(A{{r}^{3}}+B)dr\] Moment of inertia of the rod about the axis \[l=\int\limits_{0}^{L}{dm.{{r}^{2}}=\int\limits_{0}^{L}{(A{{r}^{3}}+B)}}\,{{r}^{2}}dr\] \[=\int\limits_{0}^{L}{A{{r}^{5}}dr}+\int\limits_{0}^{L}{B{{r}^{2}}dr}\] \[=A\left[ \frac{{{r}^{6}}}{6} \right]_{0}^{L}+B\left[ \frac{{{r}^{3}}}{3} \right]_{0}^{L}\] \[=\frac{A{{L}^{6}}}{6}+\frac{B{{L}^{3}}}{3}\] \[=\frac{{{L}^{3}}}{3}\left[ \frac{A{{L}^{3}}}{2}+B \right]\]