Question

The centre of mass of a non uniform rod of length $L$ whose mass per unit length varies as $\rho =k{x}^2/L$ (where $k$ is a constant and $x$ is the distance measured from one end) is at what distance from the same end?

• A. $3L/4$
• B. $L/4$
• C. $2L/3$
• D. $L/3$

Answer 1

The correct option is A $3L/4$

Total mass, $M={\int }_0^L\rho dx={\int }_0^L\frac{k{x}^2}{L}dx=\frac{k{L}^3}{3}$

Centre of mass$=\frac{1}{M}{\int }_0^{L}x\rho dx=\frac{1}{M}{\int }_0^L\frac{k{x}^3}{L}dx=\frac{3}{k{L}^3}\times \frac{k{L}^4}{4}=\frac{3L}{4}$