Question

From a circular disk of radius $R$, a triangular portion is cut. The distance of the centre of mass of the remainder from the centre of the disk is:

• A. $\frac{4R}{3(\pi -2)}$
• B. $\frac{2R}{3(\pi -2)}$
• C. $\frac{5R}{7(\pi -2)}$
• D. $\frac{R}{3(\pi -2)}$

Answer 1

The correct option is B $\frac{2R}{3(\pi -2)}$

Center of mass of triangle and disk is at c and O.

Taking mass of $\Delta$ positive.

Now mass of complete disk=$\sigma \left(\pi R^2\right)$

mass of triangle=$2\sigma R^2$ where $\Delta$=mass per unit area.

center of mass from c=$\frac{\sigma \left(\pi R^2\right)\times 0-2\left(\frac{\sigma }{2}{R}^2\right)\frac{R}{3}}{\sigma \left(\pi R^2\right)-2\sigma R^2}$

$y_{com}=\frac{-\frac{2R}{3}}{\pi -2}$

$=-\frac{2R}{3(\pi -2)}$

Distance from c =$\left|y\right|=\frac{2R}{3(\pi -2)}$