Question

A non-uniform thin rod of length $L$ is placed along the $x$ -axis such that one of its ends is at the origin. The linear mass density of the rod is $\lambda ={\lambda }_0\text{ x}$ . Find the distance of the centre of mass of the rod from the origin.

• A. $\frac{L}{2}$
• B. $\frac{2L}{3}$
• C. $\frac{L}{4}$
• D. $\frac{L}{5}$

Answer 1

Answer: (B)

The mass of considered element is
$\text{dm}=\lambda \text{dx}={\lambda }_0\text{ xdx}$



$\therefore {\left(\text{x}\right)}_{\text{cm}}=\frac{{\int }_0^{\text{L}}\text{xdm}}{\int \text{dm}}=\frac{{\int }_0^{\text{L}}\text{x}\left({\left(\lambda \right)}_0\text{xdx}\right)}{{\int }_0^{\text{L}}{\left(\lambda \right)}_0\text{xdx}}$
$=\frac{{\lambda }_0{\left[\frac{{\text{x}}^3}{3}\right]}_0^{\text{L}}}{{\lambda }_0{\left[\frac{{\text{x}}^2}{2}\right]}_0^{\text{L}}}=\frac{{\lambda }_0\frac{{\text{L}}^3}{3}}{{\lambda }_0\frac{{\text{L}}^2}{2}}=\frac{\text{2}}{\text{3}}\text{L}$