Question

The coordinates of the center of mass of the following quarter circular arc are

• A. $\left(\frac{r}{2} , \frac{r}{2}\right)$
• B. $\left(\frac{2 r}{3} , \frac{2 r}{3}\right)$
• C. $\left(\frac{2 r}{\pi } , \frac{2 r}{\pi }\right)$
• D. $\left(\frac{4 r}{\pi } , \frac{4 r}{\pi }\right)$

Answer 1

Answer: (C)

Consider a small element as shown in the diagram



$dm=\lambda Rd\theta =\frac{2 m}{\pi }d\theta $
$x_{c m}=\frac{1}{m}\displaystyle \int _{0}^{\frac{\pi }{2}} R cos \theta \frac{2 m}{\pi }d\theta $
$x_{c m}=\frac{2 R}{\pi }\displaystyle \int _{0}^{\frac{\pi }{2}} cos \theta d \theta =\frac{2 R}{\pi }\left[sin ⁡ \pi / 2 - sin ⁡ 0\right]=\frac{2 R}{\pi }$
Similarly
$y_{c m}=\frac{1}{m}\displaystyle \int _{0}^{\frac{\pi }{2}} R sin \theta \frac{2 m}{\pi }d\theta $
$y_{c m}=\frac{2 R}{\pi }\displaystyle \int _{0}^{\frac{\pi }{2}} sin \theta d \theta =\frac{2 R}{\pi }\left[- cos ⁡ \pi / 2 - \left(\right. - cos ⁡ 0 \left.\right)\right]=\frac{2 R}{\pi }$