Question

The centre of mass of a non uniform rod of length L whose mass per unit length $\lambda$ varies as $\lambda ={kx}^2$ where k is a constant & x is the distance of any point on rod from left end A is (from the same end)

• A. at the centre of the rod.
• B. is at $x=\frac{3L}{4}$
• C. is at $x=\frac{4L}{5}$
• D. $x=\frac{5L}{6}$

Answer 1

The correct option is B is at $x=\frac{3L}{4}$

$OP=x$, $OQ=x+dx$

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

$x=\frac{{\int }_0^{L}xdm}{{\int }_0^{L}dm}=\frac{{\int }_0^{L}x\frac{k{x}^2}{L}dx}{{\int }_0^L\frac{k{x}^2}{L}dx}=\frac{L4/4}{L3/3}=\frac{3L}{4}$.