As in the figure if a spherical cavity (centered at O) of radius 1 is cut out of a uniform sphere of radius R (centered at C), the centre 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:

Options

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`

Answer

B

Explanation

`M_(1)=4/3piR^(3)rho`
`M_(2)=4/3pi(1)(3)(-rho)`
`M_(2)=4/3pi(1)^(3)(-rho)`
`X_("com")=(M_(1)X_(1)+M_(2)X_(2))/(M_(1)+M_(2))`
`implies([4/3piR^(3)rho]0+[4/3pi(1)^(3)(-rho)][R-1])/(4/3piR^(3)rho+4/3(1)^(3)(-rho))=`
`-(2-R)`
`implies((R-1))/((R^(3)-1))=(2-R)`
`implies((R-1))/((R^(3)-1)=(2-R)(R!=1)`
`((R-1))/((R-1)(R^(2)+R+1))=2-R`
`(R^(2)+R+1)(2-R)=1`
Alternative:
Mremaining `(2-R)="M cavity" (1-R)`
`implies(R^(3)-1^(3))(2-R)=1^(3)[R-1]`
`implies(R^(2)+R+1)(2-R)=1`