Question

Find the centre of mass of uniform circular arc of radius $R$ as shown below:-

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

Answer 1

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

$x_{com}=\frac{\underset{\frac{-\pi }{6}}{\overset{\frac{\pi }{6}}{\int }}R\cos \theta \cdot \left(\lambda Rd\theta \right)}{\int dm}$

$=\frac{\lambda R^2\underset{\frac{-\pi }{6}}{\overset{\frac{\pi }{6}}{\int }}\cos \theta d\theta }{\lambda \cdot \left(R\frac{\pi }{3}\right)}$

$=\frac{\lambda R^2\times \pi }{\lambda R\times \pi }=\frac{3R}{\pi }$

Hence,

option $\left(C\right)$ is correct answer.