Find the y-coordinate of com of a uniform plate having semi-circular inner and outer boundaries of radii R1 and R2 with respect to 0 (as shown below)

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Solution

The correct option is A

$L e t ρ b e t h e m a s s p e r u n i t a r e a o f t h e o b j e c t . T o f i n d i t s c e n t r e o f m a s s w e c o n s i d e r a h a l f r i n g o f m a s s d m a s s h o w n i n f i g u r e o f r a d i u s x a n d w i d t h d x a n d t h e r e w e h a v e d m = ρ . π x d x C e n t r e o f m a s s o f t h i s h a l f r i n g w i l l b e a t a h e i g h t \frac{2 x}{π} f r o m O . T h u s w e h a v e f o r c e n t r e o f m a s s o f o b j e c t y_{c m} = \frac{1}{M} \int_{R_{1}}^{R_{2}} (ρ . π x d x) . \frac{2 x}{π} y_{c m} = \frac{2 ρ}{ρ \frac{π}{2} (R_{2}^{2} - R_{1}^{2})} \int_{R_{1}}^{R_{2}} x^{2} d x = \frac{4 (R_{2}^{3} - R_{1}^{3})}{3 π (R_{2}^{2} - R_{1}^{2})}$

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