Question

The centre of mass of a non uniform rod of length $L$ whose mass per unit length $\lambda =\left(\frac{K{x}^2}{L}\right)$ where $K$ is the constant and $x$ in the distance from one end is :-

• A. $\frac{3L}{4}$
• B. $\frac{L}{8}$
• C. $\frac{L}{K}$
• D. $\frac{3L}{K}$

Answer 1

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

X-coordinate of Centre of mass
$x_{cm}=\frac{\int x.dm}{\int dm}$

Mass per unit length
$\frac{dm}{dx}=\lambda =\frac{K{x}^2}{L}$
$dm=\frac{K{x}^{2}dx}{L}$

$x_{cm}=\frac{\underset{0}{\overset{L}{\int }}x\left(\frac{K{x}^2}{L}\right)dx}{\underset{0}{\overset{L}{\int }}\frac{K{x}^2}{L}dx}=\frac{3L}{4}$