Locate the centre of mass of a uniform hemi spherical shell of radius R with reference to point 0 {as shown in figure}

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Solution

The correct option is A

$F i g u r e s h o w s a h o l l o w h e m i s p h e r e o f m a s s M a n d r a d i u s R . N o w w e c o n s i d e r a n e l e m e n t a l c i r c u l a r s t r i p o f a n g u l a r w i d t h d θ a t a n a n g u l a r d i s t a n c e d θ f r o m t h e b a s e o f t h e h e m i s p h e r e . t h i s s t r i p w i l l h a v e a n a r e a d s = 2 π R c o s θ R d θ$

$I t s m a s s d m i s g i v e n a s d m = \frac{M}{2 π R^{2}} 2 π R c o s θ R d θ H e r e y - c o o r d i n a t e o f t h i s s t r i p o f m a s s d m c a n b e t a k e n a s R s i n θ . N o w w e c a n o b t a i n i n t h e c e n t r e o f m a s s o f t h e s y s t e m a s y_{c m} = \frac{1}{M} \int_{0}^{\frac{π}{2}} d m R s i n θ = \frac{1}{M} \int_{0}^{\frac{π}{2}} \frac{M}{2 π R^{2}} 2 π R c o s θ R d θ R s i n θ = R \int_{0}^{\frac{π}{2}} s i n θ c o s θ d θ y_{c m} = \frac{R}{2}$

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