Question

A cavity of radius $b$ is made in a disc of mass $M$, radius $R$, as shown in Figure. Find the new COM

• A. $\frac{-b^2}{R+b}$
• B. $\frac{-b^2}{R-b}$
• C. $\frac{-R}{2R+b}$
• D. $\frac{-R}{3R+b}$

Answer 1

The correct option is A $\frac{-b^2}{R+b}$

The areal density of the disc is $\frac{M}{\pi R^2}$
mass removed $m=\frac{M}{\pi R^2}\times \pi b^2=\frac{M{b}^2}{R^2}$
We will assume the mass of the cut portion as the negative mass and find the COM accordingly.
$\overline{x}=\frac{M\left(0\right)-\frac{M{b}^2}{R^2}(R-b)}{M-\frac{M{b}^2}{R^2}}=\frac{-b^2}{(R+b)}$