First slide

Centre of mass(com)

Question

As shown in fig. when a spherical cavity (centered at O) of radius 1 is cut out of a uniform sphere of radius R (centered at C), the center of mass of remaining (shaded) part of sphere is at G, i.e. on the surface of the cavity. R can be determined by the equation :

Moderate

A

$(R^{2} + R - 1) (2 - R) = 1$
B

$(R^{2} + R + 1) (2 - R) = 1$
C

$(R^{2} - R + 1) (2 - R) = 1$
D

$(R^{2} - R - 1) (2 - R) = 1$

Solution

Moment of masses about COM is zero.
$\begin{array}{l} ∴ M_{r e m a i n i n g} \times G C = M_{c a v i t y} \times O C \\ \Rightarrow (M_{t o t a l} - M_{c a v i t y}) \times (G P - C P) = M_{c a v i t y} \times (C P - O P) \\ S i n c e, M a s s \propto {(r a d i u s)}^{3} \\ \Rightarrow (R^{3} - 1^{3}) (2 - R) = 1^{3} [R - 1] \\ \Rightarrow (R^{2} + R + 1) (2 - R) = 1 \end{array}$

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