Centre Of Mass Of Semicircular Ring MCQ - Practice Questions & Answers

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Position of centre of mass for semicircular ring

Have a look at the figure of semicircular ring.

Since it is symmetrical about y-axis on both sides of the origin

So we can say that its $x_{cm} = 0$

And its $z_{cm} = 0$ as z-coordinate is zero for all particles of semicircular ring.

Now, we will calculate its $y_{cm}$ which is given by

$y_{cm} = \frac{\int y.dm}{\int dm}$

So , Take a small elemental arc of mass dm at an angle $\theta$ from the x-direction.

Its angular width d $\theta$

If the radius of the ring is R then its y coordinate will be Rsin $\theta$

So, $dm=\frac{M}{\pi R}*Rd\theta =\frac{M}{\pi }d\theta$

As, $y_{cm} = \frac{\int y.dm}{\int dm}$

So, $y_{cm}=\frac{\int_{0 }^{\pi}\frac{M}{\pi R}\times R\times Rsin\theta d\theta}{M}=\frac{R}{\pi }\int_{0 }^{\pi}sin\theta d\theta=\frac{2R}{\pi }$

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Centre Of Mass Of Semicircular Ring - Practice Questions & MCQ

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