Question

From a uniform circular disc of radius R, acircular disc of radius R/6 and having a centre at a distance R/2 from the centre of disc is removed. determine the centre of mass of the remaining portion of the disc.

Answer 1

Suppose them mass per unit area of the disk be $m$

Mass of the disk

$M=\pi R^{2}m$

Mass of the portion removed,

$M^{\prime }=\pi {\left(\frac{R}{6}\right)}^{2}m=\frac{\pi R^{2}m}{36}$

Suppose M and M' are connected at the center O and O'

As the portion is removed taking M' as negative, center of moment of remaining portion is at a distance x from the O

$x=\frac{M{x}_1-M^{\prime }{x}_2}{M-M^{\prime }}$

$x=\frac{M\times 0-M^{\prime }\left(R/2\right)}{M-M^{\prime }}$

$x=-\frac{R}{70}$

Negative sign indicate that center of moment is lies left to the origin O.