An expansible balloon filled with air floats on the surface of a lake with `2//3` of its volume submerged. How deep must it be sunk in the water so that it is just in equilibrium neither sinking further nor rising? Is is assumed that the temperature of the water is constant `&` that the height of the water barometer is `9` meters.

Answer

4.5 m

Explanation

Let mass of air in ballon is m and its volume at surface is V.
`mg=B" "=mg=(2)/(3)Vrho_(w)g" "impliesV=(3)/(2)mcm^(3)`
at depth d below let its volume is `V_(2)`
`W=B" "impliesmg=V_(2)rho_(w)g" "impliesimpliesV_(2) = m cm^(3)`
Since `P_(1)V_(1)=P_(2)V_(2)`
`9xx(3)/(2)m=(9+d)xxm" "impliesd=4.5m`