A solid sphere of radius R and density `rho` is attached to one end of a mass-less spring of force constant k. The other end of the spring is connected to another solid sphere of radius R and density `3rho`. The complete arrangement is placed in a liquid of density `2rho` and is allowed to reach equilibrium. The correct statements(s) is (are)

Options

A. The net elongation of the spring is ` (4pi R^3 rho g )/(3 k )`
B. The net elongation of the spring is ` (8pi R ^ 3 rho g )/( 3 k ) `
C. The light sphere is partially submerged
D. the light sphere is completely submerged

Answer

A, D

Explanation

At equilibrium, for small sphere
`(4)/(3)piR^(3)2rhog=(4)/(3)piR^(3)rhog+F_(s),"Here" F_(s)=kx` = spring force (in Mag.)
`rArrF_(s)=(4)/(3)piR^(3)rhog" "therefore" "x=(4)/(3k)piR^(3)rhog`
For equilibrium of the complete system, net force of buoyancy must be equal to the total weight of the sphere which holds true in the given problem. So, both the spheres are completely submerged.