Locate the COM of a uniform hemisherical solid with respect to 0 as shown in figure. (Radius = r)

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Solution

The correct option is B

$F i g u r e s h o w s a h e m i s p h e r e o f m a s s M a n d r a d i u s R . T o f i n d i t s c e n t r e o f m a s s (o n l y y - c o o r d i a n t e), w e c o n s i d e r a n e l e m e n t a l d i s c o f w i d t h d x, m a s s d m a t a d i s t a n c e x f r o m t h e c e n t r e o f t h e h e m i s p h e r e . T h i s r a d i u s o f t h i s e l e m e n t a l d i s c w i l l b e g i v e n a s r = \sqrt{R^{2} - x^{2}}$

$T h e m a s s d m o f t h i s d i s c c a n b e g i v e n a s d m = \frac{3 M}{2 π R^{3}} \times π r^{2} d x = \frac{3 M}{2 R^{3}} (R^{2} - x^{2}) d x y_{c m} o f t h e h e m i s p h e r e i s g i v e n a s y_{c m} = \frac{1}{M} \int_{0}^{R} d m x = \frac{1}{M} \int_{0}^{R} \frac{3 M}{2 π R^{3}} (R^{2} - x^{2}) d x x = \frac{3}{2 π R^{3}} \int_{0}^{R} (R^{2} - x^{2}) x d x y_{c m} = \frac{3 R}{8}$

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