Question

Find center of mass of thin, uniform semicircular wirs of radius $R$

• A. $\frac{R}{\pi }$
• B. $\frac{3R}{\pi }$
• C. $\frac{2R}{\pi }$
• D. $\frac{3R}{2\pi }$

Answer 1

The correct option is C $\frac{2R}{\pi }$

Step 1: Draw a rough diagram of element of semi ring

It is a uniform mass distributed semi ring

Consider an element $dm$ having length $Rd\theta$ as shown in figure
As body is symmetric about $y$ - axis hence we will find only $y$ co-ordinet
For small element $y=R\sin \theta$

Step 2: Find c.m. of the body

Formula used: $y_{cm}=\frac{\int ydm}{\int dm}dm=\frac{m}{\pi R}Rd\theta =\frac{md\theta }{\pi }{y}_{cm}=\frac{\int ydm}{\int dm}={\int }_0^{\pi }\frac{R\sin \theta \frac{m}{\pi }d\theta }{m}=\frac{R}{\pi }{\int }_0^{\pi }\sin \theta d\theta =\frac{R}{\pi }(-cos\theta {)}_0^{\pi }=\frac{R}{\pi }(-\cos \left(\pi \right)-(-\cos 0)\left)\right)=\frac{R}{\pi }[1+1]=\frac{2R}{\pi }$