Question

Find the centre of mass of a non uniform rod of length $L$, whose mass per unit length varies as $\rho =\frac{k\cdot x^2}{L}$ (where $k$ is a constant and $x$ is the distance of any point from one end) from the same end.

• A. $\frac{3}{4}L$
• B. $\frac{1}{4}L$
• C. $\frac{1}{6}L$
• D. $\frac{2}{3}L$

Answer 1

The correct option is A $\frac{3}{4}L$

Since the density varies with position(distance), integration has to be used.

Take a small element $dx$ at a distance $x$ from the left end.
$dm=$(mass per unit length) (Length)

$dm=\frac{k{x}^2}{L}dx$

$X_{com}=\frac{{\int }_0^{L}xdm}{{\int }_0^{L}dm}$

$X_{COM}=\frac{{\int }_0^{L}x\frac{k{x}^2}{L}dx}{{\int }_0^L\left(\frac{k{x}^2}{L}\right)dx}=\frac{3}{4}L$