Question

The coordinates of centre of mass of the following quarter circular arc is -

• 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)

$dm=\lambda Rd\theta \, =\frac{2 m}{\pi }d\theta $
$\therefore x_{c m} \, =\frac{1}{m}\displaystyle \int x d m$
$=\frac{1}{m}\displaystyle \int _{0}^{\pi / 2} R cos \theta \frac{2 m}{\pi } d \theta $
Similarly $y_{c m}=\frac{1}{m}\displaystyle \int _{0}^{\frac{\pi }{2}} R \, sin . \theta \frac{2 m}{\pi } d \theta $