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COM of Center of Masses

A semicircula...

Question

A semicircular ring of mass $m$ and a semicircular disc of mass $2 m$ of same radius are joined together as shown. Find the distance of COM from their geometric centre.

\frac{2 R}{3 π}

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\frac{2 R}{9 π}

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\frac{5 R}{3 π}

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0

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Solution

The correct option is B $\frac{2 R}{9 π}$
Let their geometric centre be at origin.
COM of semicircular disc, $y_{1} = \frac{4 R}{3 π}$
COM of semicircular ring, $y_{2} = \frac{- 2 R}{π}$
So, COM of the combined body $= \frac{m_{1} y_{1} + m_{2} y_{2}}{m_{1} + m_{2}}$
$= \frac{(2 m) \frac{4 R}{3 π} + m \frac{(- 2 R)}{π}}{m + 2 m}$
$= \frac{2 R}{9 π}$

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