Question

Mass per unit area of the disc in first quadrant is $\sigma ,$ in the second quadrant is $4\sigma ,$ in the third quadrant is $5\sigma $ and in the fourth quadrant is $8\sigma $ . The coordinates of centre of mass of this system are :

• A. $\left(0 , \frac{- 16 R}{27 \pi }\right)$
• B. $\left(0 , \frac{- 2 R}{3 \pi }\right)$
• C. $\left(0 , \frac{- 16 R}{65}\right)$
• D. $\left(0 , \frac{- 4 R}{3 \pi }\right)$

Answer 1

Answer: (A)

$x_{cm}=\frac{m \left(x\right) + 4 m \left(- x\right) + 5 m \left(- x\right) - 8 m \left(x\right)}{m + 4 m + 5 m + 8 m}=0$
$y_{cm}=\frac{m \left(x\right) + 4 m \left(x\right) + 5 m \left(- x\right) + 8 m \left(- x\right)}{18 m}$
$=-\frac{8 m x}{18 m}=\frac{- 4}{9}x=\frac{- 4}{9}\times \left(\frac{4 R}{3 \pi }\right)=-\frac{16 R}{27 \pi }$