Figure shows a large closed cylindrical tank containing water. Initially, the air trapped above the water surface has a height `h_(0)` and pressure `2p_(0)` where `rh_(0)` is the atmospheric pressure. There is a hole in the wall of the tank at a depth `h_(1)` below the top from which water comes out. A long vertical tube is connected as shown.

Find the height `h_(2)` of the water in the long tube above top initially.

Options

A. `(3p_(0))/(rhog)-(h_(0))/(3)`
B. `(2p_(0))/(rhog)-(h_(0))/(2)`
C. `(p_(0))/(rhog)-h_(0)`
D. `(p_(0))/(2rhog)-2h_(0)`

Answer

C

Explanation

`2P_(0)=(h_(2)+h_(0))rhog+p_(0)`
(since liquids at the same level have the same pressure) `P_(0)=h_(2)rhog+h_(0)rhog,h_(2)rhog=P_(0)-h_(0)rhog`
`h_(2)=(P_(0))/(rhog)+(h_(0)rhog)/(rhog)=(P_(0))/(rhog)-h_(0)`
KE to the water `=` pressure energy of the water at that layer `(1)/(2)mV^(2)=mxx(P)/(rho)`
`V^(2)=(2P)/(rho)=(2)/(rho)[P_(0)+rhog(h_(1)+h_(0))]`
`V=[(2)/(rho){P_(0)+rhog(h_(1)-h_(0))}]^(1//2)`
We know `2P_(0)+rhog(h_(1)-h_(0))=P_(0)+rhogX`
`impliesX=(P_(0))/(rhog)+(h_(1)-h_(0))=h_(2)+h_(1)`
i.e., X is `h_(1)` meter below the top or X is `h_(2)` above the top.