First slide

Centre of mass(com)

Question

Two semicircular rings of linear mass densities $λ$ and 2 $λ$ and of radius 'R' each are joined to form a complete ring. The distance of the center of the mass of complete ring from its center is

Moderate

A

$\frac{3 R}{8 π}$
B

$\frac{2 R}{3 π}$
C

$\frac{4 R}{21 π}$
D

None of these

Solution

Let the two half rings be placed in left and right of y-axis with centre as shown in figure.
Then the coordinate of centre of mass of left and right half rings are $(- \frac{2 R}{π}, 0)$ and $(\frac{2 R}{π}, 0)$ .
$∴$ x-coordinates of centre of mass of complete ring is
$\frac{m (- \frac{2 R}{π}) + 2 m (\frac{2 R}{π})}{3 m} = \frac{2 R}{3 π}$

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