Question

A non-uniform thin rod of lenrth $l$ lies along the axis with one end at the origin. It has a liner mass density $\lambda ={\lambda }_0(1+\frac{x}{l}).$ Find the center of mass of the rod.

Answer 1

Let us take an element of length $dx$ and mass $dm$ at distance $x$ from origin.

$dm=\lambda dx={\lambda }_0(1+\frac{x}{l})dx$

Now, Total mass $M={\int }_0^{l}dm={\lambda }_0{\int }_0^l(1+\frac{x}{l})dx$

$⟹M={\lambda }_{0}l+{\lambda }_o\frac{l}{2}=\frac{3{\lambda }_{0}l}{2}$

$X_{CM}=\frac{\int xdm}{M}=\frac{{\lambda }_o{\int }_0^l(x+\frac{x^2}{l})dx}{\frac{3{\lambda }_{o}l}{2}}$

$X_{CM}=\frac{5l}{9}$