question_answer

The mass per unit length of a non-uniform rod \[OP\] of length L varies as \[m=k\frac{x}{L}\] where \[k\]  is a constant and \[x\] is the distance of any point on the rod from end \[O\] The distance of the centre of mass of the rod from end \[O\] is -

A) \[\frac{L}{3}\]

B) \[\frac{2L}{3}\]

C) \[\frac{L}{2}\]

D) \[\frac{2L}{\sqrt{3}}\]

Correct Answer: B

Solution :

Consider an element AB of thickness dx at distance x

Mass of element AB is \[dm=\frac{k}{L}(xdx)\]

Formula for center of mass coordinate \[={{x}_{com}}=\frac{\int{(dm)x}}{\int{(dm)x}}=\frac{\frac{k}{L}\int\limits_{0}^{L}{{{x}^{2}}dx}}{\frac{k}{L}\int\limits_{0}^{L}{x\,dx}}=\frac{2L}{3}\]