Question

Find coordinates of center of mass of a quarter ring of radius $r$ placed in the first quadrant of a Cartesian coordinate system, with centre at origin.

Answer 1

$X_{cm}=\frac{{\int }_0^{\pi /2}xdm}{{\int }_0^{\pi /2}dm}$

$=\frac{{\int }_0^{\pi /2}R\cos \theta \left(uRd\theta \right)}{{\int }_0^{\pi /2}uRd\theta }$

$=R\frac{{\int }_0^{\pi /2}\cos \theta d\theta }{{\int }_0^{\pi /2}d\theta }$

$=\frac{2R}{\pi }$

$Y_{cm}=\frac{{\int }_0^{\pi /2}ydm}{{\int }_0^{\pi /2}dm}$

$=\frac{{\int }_0^{\pi /2}R\sin \theta \left(uRd\theta \right)}{{\int }_0^{\pi /2}uRd\theta }$

$=R\frac{{\int }_0^{\pi /2}\sin \theta d\theta }{{\int }_0^{\pi /2}d\theta }$

$=\frac{2R}{\pi }$

$\therefore (X_{cm},Y_{cm})=(\frac{2R}{\pi },\frac{2R}{\pi })$