Question

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 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. $\frac{2L}{\sqrt{3}}$

Answer 1

The correct option is B $\frac{2L}{3}$

Consider a rod $OP$ of length $L$ lying along the $x-$ axis with $O$ as the origin. Consider a small element $AB$ of length $dx$ at a distance $x$ from $O$.

Mass of element $AB(=dm)=mdx=\frac{k}{L}\left(xdx\right)$
The distance of the centre of mass from O is given by
$x_{CM}=\frac{\int \left(dM\right)x}{\int \left(dM\right)}=\frac{\frac{k}{L}{\int }_0^L{x}^{2}dx}{\frac{k}{L}{\int }_0^{L}xdx}$
$=\frac{{\left|\frac{x^3}{3}\right|}_0^L}{{\left|\frac{x^2}{2}\right|}_0^L}=\frac{\frac{L^3}{3}}{\frac{L^2}{2}}=\frac{2L}{3}$
The correct choice is (b).