Question

A disc of radius R is cut out from a larger disc of radius 2R in such a way that the edge of the hole touches the edge of the disc.Locate the centre of mass of the residual disc.

Answer 1

Step 1, Given data

R=radius of the smaller disc

2R=radius of the bigger disc

Let '0' be the origin of the system.

The smaller disc is cut out from the bigger disc.

From the figure

Step 2, Finding the COM

From the above figure we can write mass of the smaller and bigger discs

$m_1=\pi R^{2}T\rho ,x_1=0,y_1=0$

$m_2=\pi (2R{)}^{2}T\rho ,x_2=0,y_2=0$

Now we can find the position of the center of mass by using the formula

$\therefore$ Position of C.M.

$=(\frac{\pi R^{2}T\rho R+0m_2}{-\pi R^{2}T\rho +\pi (2R{)}^{2}T\rho },\frac{0+0}{m_1+m_2})$

$=(\frac{-\pi R^{2}T\rho R}{3\pi R^{2}T\rho },0)=\frac{R}{3}$

Hence, C.M. is at $\frac{R}{3}$ from the centre of bigger disc away from centre of the hole.