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  • What is the position of the center of mass of a semicircular ring shown below - <img alt="" src="https://cdn.entrance360.com/media/uploads/2019/08/28/a1.PNG" style="height:101px; width:15

What is the position of the center of mass of a semicircular ring shown below -

Option: 1

\left ( 0,\frac{2R}{3\pi } \right )      


Option: 2

\left ( \frac{2R}{3\pi },0 \right )


Option: 3

\left ( \frac{2R}{\pi },0 \right )


Option: 4

\left (0, \frac{2R}{\pi }\right )


Answers (1)

best_answer

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

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 }

Now, the co-ordinate of centre of mass is (0,\frac{2R}{\pi})

 

 

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Riya

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