question_answer

Consider a uniform semi-circular wire of radius r Placed, such that its centre coincides of mass are best located in

A)    

B)  

C)

D)

Correct Answer: D

Solution :

By symmetry, \[{{x}_{CM}}=0\]

\[dm=\frac{m}{\pi }d\theta \]

\[{{y}_{CM}}=\frac{\int{ydm}}{\int{dm}}=\int_{0}^{\pi }{\frac{\left( r\sin \theta  \right)\left( \frac{m}{\pi }d\theta  \right)}{\int_{0}^{\pi }{dm}}}\]

\[=\frac{\frac{rm}{\pi }.\int_{0}^{\pi }{\sin \theta }d\theta }{m}=\frac{2r}{\pi }\]