A solid sphere of density `rho= rho_(0) (1 - (r^(2))/(R^(2))), 0 lt r le R` just floats in a liquid, then the density of the liquid is (r is the distance from the centre of the sphere)

Options

A. `rho_(0)`
B. `(2)/(5) rho_(0)`
C. `(5)/(2) rho_(0)`
D. `(3)/(5) rho_(0)`

Answer

B

Explanation

Suppose density of liquid is `rho_(L)` and mass of sphere is m
`m= pv [ :. rho = (m)/(v)]`
`:.m= rho xx "Area " xx "Distance"`
`:. m= rho xx 4pi r^(2) dr`
Buoyant force, `F_(B)= rho_(L) xx` volume of liquid equlas to volume of sphere `xx g`
`= rho_(L) xx (4)/(3) pi R^(3) g` where radius of sphere is R
As sphere floats in liquid hence
Buoyant force= weight
`rho_(L) xx (4)/(3) pi R^(3)g = rho xx 4pi r^(2) dr`
`= rho_(0) (1- (r^(2))/(R^(2))) 4pi r^(2) dr`
`= underset(0)overset(R )int rho_(0) (1 - (r^(2))/(R^(2))) 4pi r^(2) dr`
`=4pi rho_(0) underset(0)overset(R )int (r^(2)- (r^(4))/(R^(2))) dr`
`:. rho_(L) xx (4)/(3) pi R^(3) g = 4pi rho_(0)g [(r^(3))/(3)- (r^(5))/(5R^(2))]_(0)^(R )` `=4pi rho_(0)g [(2R^(3))/(15)]`
`:. rho_(L) = (4pi rho_(0) [(2R^(3))/(15)]g)/((4)/(3) pi R^(3)g)`
`:. rho_(L)= (2)/(5) rho_(0)`