Question

A disc of radius ${}^{\prime }{R}^{\prime }$ is given: its surface are density varies with the radius: $\sigma =2r$, find $COM$ of the system.

Answer 1

$dx=\sigma dA$

$=2^{-1}\times \pi r^{2}dr$

$=2\pi r^{2}dr$

$y=\frac{2r}{\pi }$

$y_{cm}=\frac{\int \left(dm\right)y}{\int dm}$

$=\frac{{\int }_0^{R}2\pi r^{2}dr\times \frac{2r}{\pi }}{{\int }_0^{R}2\pi r^{2}dr}$

$=\frac{4\times \frac{R^4}{4}}{2\pi \frac{R^3}{3}}=\frac{3R}{2\pi }$