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)

A solid sphere of density rho= rho_(0) (1 - (r^(2))/(R^(2))), 0 lt r l

1.	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)
Answer» `rho_(0)` `(2)/(5) rho_(0)` `(5)/(2) rho_(0)` `(3)/(5) rho_(0)` Solution :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)`