Question

The centre of mass of a non uniform rod of length L whose mass per unit length varies as $\rho =$ $k{x}^2/L$ (where $k$ is a constant and $x$ is the distance measured from one end) is at the following distance from the same end

• A. $3L/4$
• B. $L/4$
• C. $2L/3$
• D. $L/3$

Answer 1

The correct option is A $3L/4$

Given,

density of rod is $\rho =\frac{k{x}^2}{L}$

Consider small distance $ds$ centre of mass is $dm$

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

Now centre of mass od non uniform rod is ,

$dm=\rho dx\int dm=\frac{{\int }_0^{L}xdm}{{\int }_0^{L}dm}$

Substitute value of $dm$ in above equation therefore,

$M=\frac{{\int }_0^{L}x\frac{k{x}^2}{L}dx}{{\int }_0^L\frac{k{x}^2}{L}dx}M=\frac{{\int }_0^L{x}^{3}dx}{{\int }_0^L{x}^{2}dx}M=\frac{{\left[\frac{x^4}{4}\right]}_0^L}{{\left[\frac{x^3}{3}\right]}_0^L}M=\frac{3L}{4}$

Distance from the centre of mass is $3L/4$

Hence, correct option is $\left(A\right)$