Question

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

• A. $\frac{3R}{4\pi }$
• B. $\frac{4R}{3\pi }$
• C. $\frac{4R}{2\pi }$
• D. $\frac{3R}{2\pi }$

Answer 1

The correct option is B $\frac{4R}{3\pi }$

Step 1: Draw a rough diagram of semi-circular plate.
Consider small element of semi ring shape of radius r whose centre of mass is at $\frac{2r}{\pi }$ from centre

Step 2: Find c.m. of semi circular disc
Formula used: surface density $=\frac{\text{Mass}}{\text{area}},y_{semiring}=\frac{2R}{\pi },y_{cm}=\frac{\int ydm}{\int dm}$
Surface density $=\frac{\text{Mass}}{\text{area}}=\frac{M}{\frac{\pi R^2}{2}}=\frac{2M}{\pi R^2}$
Here the elements is semicircular ring of radius $r$ and thickness of $drdA=\pi rdrdm=\frac{2M}{\pi R^2}\times \pi rdr=\frac{2M}{R^2}rdr$

The coordinates of the c.m. of element $y=\frac{2r}{\pi }{y}_{cm}=\frac{\int ydm}{\int dm}=\frac{{\int }_0^R\frac{2r}{\pi }\frac{2Mrdr}{R^2}}{M}\frac{4}{\pi R^2}{\int }_0^R{r}^{2}dr=\frac{4}{\pi R^2}{\left(\frac{r^3}{3}\right)}_0^R=\frac{4}{\pi R^2}\left(\frac{R^3}{3}\right)=\frac{4R}{3\pi }$