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Two sectors made of same material and same radius R but angle $120^{0}$ and $60^{0}$ are arranged as shown with centre as origin. The distance of center of mass from origin is ?

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a

$\frac{4 R (\sqrt{3} - 1)}{3 π}$

b

$\frac{R (\sqrt{3} - 1)}{3 π}$

c

$\frac{3 R (\sqrt{3} - 1)}{3 π}$

d

$\frac{2 R (\sqrt{3} - 1)}{3 π}$

answer is A.

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Detailed Solution

Centre of mass of arc is $\frac{2 R \sin α}{3 α}$
$m_{1} = 2 m, m_{2} = m$
Centre of mass $= \frac{\frac{2 m (2 R \sin (\frac{π}{3}))}{(3 \frac{π}{3})} + m (\frac{- 2 R \sin (\frac{π}{6})}{3 (\frac{π}{6})})}{3 m π} = \frac{2 R (\sqrt{3} - 1)}{3 π}$

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Two sectors made of same material and same radius R but angle 1200 and 600 are arranged as shown with centre as origin. The distance of center of mass from origin is ?