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
$\text{dm} = \lambda \text{dx} = \lambda _{0} \text{ xdx}$



$∴ \, \, \left(\text{x}\right)_{\text{cm}} = \frac{\displaystyle \int _{0}^{\text{L}} \text{xdm}}{\displaystyle \int \text{dm}} = \frac{\displaystyle \int _{0}^{\text{L}} \text{x} \left(\left(\lambda \right)_{0} \text{xdx}\right)}{\displaystyle \int _{0}^{\text{L}} \left(\lambda \right)_{0} \text{xdx}}$
$= \frac{\lambda _{0} \left[\frac{\text{x}^{3}}{3}\right]_{0}^{\text{L}}}{\lambda _{0} \left[\frac{\text{x}^{2}}{2}\right]_{0}^{\text{L}}} = \frac{\lambda _{0} \frac{\text{L}^{3}}{3}}{\lambda _{0} \frac{\text{L}^{2}}{2}} = \frac{\text{2}}{\text{3}} \text{L}$