Question

A straight rod of length L has one of its ends at the origin and the other at $x=L$. If the mass per unit length of the rod is given by Ax where A is constant, where is its mass centre?

• A. $L/3$
• B. $L/2$
• C. $2L/3$
• D. $3L/4$

Answer 1

Answer: (B)

$L/2$

Let assume "A is a CONSTANT".

$x_c$ = coordinate of center of mass

M = total mass

$x_c=\int xdm/\int dm=\int xdm/M$

Given:

$m\left(x\right)=Ax\Rightarrow dm/dx=A\Rightarrow dm=Adx$

$\int xdm=\int xAdx=\frac{A{x}^2}{2}{|}_0^L=\frac{A{L}^2}{2}$

$x_c=\frac{\int xdm}{M}=\frac{\frac{A{L}^2}{2}}{M}=\frac{\frac{\frac{M}{L}{L}^2}{2}}{M}=\frac{L}{2}$