Question

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

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

Answer 1

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

consider an elemental particle of mass $dm$ and length $dx$ at distance $x$ from one end,

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

By definition
$X_{cm}=\frac{\int xdm}{\int dm}$
$=\frac{{}_0\stackrel{L}{\int }\frac{k{x}^3}{L}dx}{{}_0\stackrel{L}{\int }k\frac{x^2}{L}dx}$
$=\frac{\frac{L^4}{4}}{\frac{L^3}{3}}$
$X_{cm}=\frac{3L}{4}$