Question

A rod of length $3m$ has its mass per unit length directly proportional to distance $x$ from one of its ends then its centre of gravity from that end will be at

• A. $1.5m$
• B. $2m$
• C. $2.5m$
• D. $3.0m$

Answer 1

The correct option is B $2m$

Let $\lambda$ is mass per unit length , $\lambda =kx$ where $k$ is constant and $x_{cg}$ is centre of gravity from the one end.

Mass of the rod $M=\int dm=\int \lambda dx=\int kxdx=\frac{k{L}^2}{2}$

Centre of the gravity , $x_{cg}=\frac{\int xdm\times g}{Mg}=\frac{\int kg{x}^{2}dx}{Mg}=\frac{\frac{kg{L}^3}{3}}{\frac{kg{L}^2}{2}}=\frac{2L}{3}$

$\Rightarrow x_{cm}=2m$