Question

What is the distance of the centre of mass from end $\text{P}$of a non-uniform rod of length $L$, whose mass per unit length $\lambda =\frac{K{x}^2}{L}$ ($K$ is a constant) and $x$ is the distance from end $\text{P}$ ?

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

Answer 1

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


Mass of the element $\text{PQ}$is $dm=\lambda dx=\frac{K{x}^2}{L}.dx$
So, position of centre of mass for non uniform rod is given by
$x_{com}=\frac{{\int }_0^{L}xdm}{{\int }_0^{L}dm}$
$=\frac{{\int }_0^L\frac{K{x}^3}{L}dx}{{\int }_0^L\frac{K{x}^2}{L}dx}=\frac{\left(\frac{L^4}{4}\right)}{\left(\frac{L^3}{3}\right)}=\frac{3L}{4}$ from end $\text{P}$