Find the center of mass of uniform semicircular ring of radius R and mass M

Answer:

Find the center of mass of uniform semicircular ring of radius R and mass M

The wire is uniform, the mass per unit length of the wire is

\(\frac{M}{\pi R^{2}}\)

The mass of the element is

\(dm = \left ( \frac{M}{\pi R} \right )\left ( R d\theta \right )\)

 

\(dm = \left ( \frac{M}{\pi } \right )d\theta\)

 

The coordinates of the center of mass are

 

\(X = \frac{1}{M}\int xdm = \frac{1}{M}\int_{0}^{\pi }\left ( R cos\theta \right )\left ( \frac{M}{\pi } \right )d\theta = 0\)

 

\(X = \frac{1}{M}\int ydm = \frac{1}{M}\int_{0}^{\pi }\left ( R sin\theta \right )\left ( \frac{M}{\pi } \right )d\theta = \frac{2R}{\pi }\)

 

Therefore, position of centre of mass i

 

\(\left ( 0, \frac{2R}{\pi } \right )\)

 

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