A uniform circular plate of radius 2R has a circular hole of radius R cut out of it. The centre of the smaller circle is at a distance of R/3 from the centre of the larger circle. What is the position of the centre of man of the plate

Moderate

A

Along the line joining the centers and lies at a distance 4R/9 left side, from the center of the hole
B

Along the line joining the centers and lies at a distance 4R/9 right side, from the center of the hole
C

Along the line joining the centers and lies at a distance R/9 left side, from the center of the hole
D

Along the line joining the centers and lies at a distance R/9 right side, from the center of the hole

Solution

$\begin{array}{l} M = ρ (2 R)^{2} π; m = M = ρ (R)^{2} π \\ {CMx}_{cm} = \frac{M (0) - m (x_{2})}{M - m}; x_{2} = x = R / 3; ∵ m = M / 4 \end{array}$
$x_{Cm} = \frac{- x}{3} = \frac{- R}{9}$ (from the centre of the large disc)
i.e. 4R/9 from centre of the hole.

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