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 diagram would show a straight line with a negative slope, representing a linear decrease in air pressure $p$ as the quantity of water $Q$ flowed out increases. The relationship is $p = C - \rho g H$, where $H$ is the depth of the submerged opening from the water surface, and $H$ increases as $Q$ increases.

Answer 2

In a Mariotte's vessel, the pressure at the submerged opening of the tube is constant ($P_{atm} + \rho g h_{tube}$). This pressure equals the pressure inside the vessel at the same level ($p + \rho g H$). Thus, $p = (P_{atm} + \rho g h_{tube}) - \rho g H$. As water flows out, $H$ increases, causing $p$ to decrease linearly with $Q$ (since $Q$ is proportional to the drop in water level, hence the increase in $H$).