Question

The centre of mass of a non-uniform rod of length $L$ whose mass per unit length $\lambda =\frac{K x^{2}}{L}$ , Where k is a constant and $x$ is the distance from one end is:

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

Answer 1

Answer: (A)

$dm=\frac{k x^{2}}{L}dx$
$\therefore x_{c m}=\frac{\displaystyle \int _{0}^{L} x d m}{\displaystyle \int _{0}^{L} d m}=\frac{\displaystyle \int _{0}^{L} \frac{k x^{3}}{L} d x}{\displaystyle \int _{0}^{L} \frac{k x^{2}}{L} d x}=\frac{\left(\frac{L^{4}}{4}\right)}{\left(\frac{L^{3}}{3}\right)}=\frac{3 L}{4}$