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 disk.

Answer 1

$2R=$ radius of largest disc

$R$ radius of smaller disc

smaller disc is cut out from the largest disc

$\begin{array}{c}{m}_1=\pi R^2\rho \\ x_1=R\\ y_1=0\\ m_2=\pi {\left(2R\right)}^2\rho \\ x_2=0\\ y_2=0\end{array}$

The position of center of mass $\begin{array}{c}=\left(\frac{-\pi R^2\rho R+0}{-\pi R^2\rho +\pi {\left(2R\right)}^2\rho R},\frac{0+0}{m_1+m_2}\right)\\ =\left(\frac{-\pi R^2\rho R}{3\pi R^2\rho },0\right)=\left(-\frac{R}{3},0\right)\end{array}$

therefore center of mass is at $X_{cm}=\frac{R}{3}$