Question

From a cirlce of radius $a$, an isosceles right angled triangle with the hypotenuse as the diameter of the circle is removed. The distance of the centre of gravity of the remaining position from the centre of the circle is :

• A. $3(\pi -1)a$
• B. $\frac{(\pi -1)a}{6}$
• C. $\frac{a}{3(\pi -1)}$
• D. $\frac{a}{3(\pi +1)}$

Answer 1

Answer: (C)

$\frac{a}{3(\pi -1)}$

COM of isosceles triangle is on median and h/3 above base.
so, COM of right triangle is r/3 cm from base and center.
For calculating COM of system. (hollow triangle is considered as -ve mass ) from the center.

Y co-ordinate of the com: $y=\frac{m_1{r}_1+m_2{r}_2}{m_1+m_2}$
$=\frac{(\rho \ast \pi \ast a\ast a)\ast 0-(\rho \ast \frac{1}{2}\ast a\ast 2a)\ast a/3}{\rho \ast \pi \ast a\ast a-\rho \ast \frac{1}{2}\ast a\ast 2a}$
$=-\frac{\frac{a}{3}}{(\pi -1)}=-\frac{a}{3(\pi -1)}$

Ignoring the sign co-ordinate of the com is $\frac{a}{3(\pi -1)}$