Find the position of centre of mass of the quarter solid sphere from 'C' in which mass per unit volume is given as `rho(r )=rho_(0)(1-(r )/(R ))`, where r is radial distance from centre and R is radius of solid quarter sphere

Options

A. `(3)/(10)R`
B. `(3sqrt(2))/(7)R`
C. `(3sqrt(2))/(10)R`
D. `(3sqrt(2))/(5)R`

Explanation

`y_(cm)=(int_(0)^(R )[rho_(0)(1-(r )/(R ))(2pi r^(2)dr)](r )/(2))/(int_(0)^(R )rho_(0)(1-(r )/(R ))(2pi r^(2))dr)`
`=((1)/(2)((R^(4))/(4)-(R^(5))/(5R)))/(((R^(3))/(3)-(R^(4))/(4R)))=(12)/(2).(R )/(20)=(3)/(10)R`
Position of C.M. from C
`=((3sqrt(2))/(10)R)`