Question

A narrow tube open at both ends is passed through the cork of a vessel filled with water. The tube does not reach the bottom of the vessel (Mariotte's vessel shown in figure). Draw a diagram showing how the pressure p of the air in the vessel depends on the quantity of water Q that has flowed out.

Answer 1

The relationship between the pressure $p$ of the air in the vessel and the quantity of water $Q$ that has flowed out is represented by a straight line graph with a positive slope. The graph starts at $Q=0$ with an initial air pressure $p_{initial}$ (where $p_{initial} < P_{atm}$) and ends at the maximum quantity of water flowed out, $Q_{max}$, where the air pressure reaches the atmospheric pressure, $P_{atm}$.

Answer 2

The pressure of the air inside the vessel ($p$) is related to the atmospheric pressure ($P_{atm}$) and the height of the water column above the vent tube opening ($h$) by the equation $p + \rho g h = P_{atm}$. As water flows out of the vessel, the quantity of water $Q$ increases, and the water level drops. This causes the height $h$ to decrease. To maintain the pressure balance, the air pressure $p$ must increase as $h$ decreases. This relationship is linear, so $p$ increases linearly with $Q$ until the water level reaches the vent tube opening ($h=0$), at which point the air pressure becomes equal to the atmospheric pressure ($p=P_{atm}$).