Centre Of Mass Of Semicircular Ring - Practice Questions & MCQ

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 $x_{c m}=0$
And its $z_{c m}=0$ as z-coordinate is zero for all particles of semicircular ring.
Now, we will calculate its $y_{\mathrm{cm}}$ which is given by

$
y_{c m}=\frac{\int y \cdot d m}{\int d m}
$

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

Its angular width $\mathrm{d} \theta$
If the radius of the ring is R then its y coordinate will be $\mathrm{R} \sin \theta$
So, $d m=\frac{M}{\pi R} * R d \theta=\frac{M}{\pi} d \theta$

As, $y_{c m}=\frac{\int y \cdot d m}{\int d m}$

So, $y_{c m}=\frac{\int_0^\pi \frac{M}{\pi R} \times R \times R \sin \theta d \theta}{M}=\frac{R}{\pi} \int_0^\pi \sin \theta d \theta=\frac{2 R}{\pi}$