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A solid sphere of radius R and density r...

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)

A

(a) The net elongation of the spring is `(4piR^3rhog)/(3k)`

B

(b) The net elongation of the spring is `(8piRrhog)/(3k)`

C

(c) The light sphere is partially submerged

D

(d) The light sphere is completely submerged

Text Solution

Verified by Experts

The correct Answer is:
A, D
Consider the equilibrium of the system of both spheres and the spring

The weight of system
`=4/3piR^3(3rho)g+4/3piR^3rhog=4[4/3piR^3rhog]`
This is to be balanced by the buoyant force. This can be possible only when the light sphere is completely submerged. In this way the buoyant force

`B=[(4/3piR^3)xx2]xx(2rho)xxg=4[4/3piR^3rhog]`
Now considering the equilibrium of the heavy sphere `Fs+B=W`
`:.Fs=W-B`
`:.Kx=4/3piR^3(3rho)g-4/3piR^3(2rho)g`
`:x=4/3pi(R^3rhog)/(K)`
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