Question

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

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

Answer 1

Answer: (A)

$\frac{3L}{4}$

Answer 2

$dm = \frac{kx^2}{L} dx$ $x_{cm} = \frac{\int\limits_0^L xdm}{\int\limits_0^L dm} = \frac{\int\limits_0^L \frac{kx^3}{L} dx}{\int\limits_0^L \frac{kx^2}{L} dx} = \frac{\left(\frac{L^4}{4} \right)}{\left(\frac{L^3}{3} \right)} = \frac{3L}{4}$