Height of Water Column Inside Submerged Inverted Cylinder

To determine the height $H$ of the water column inside the cylinder, we analyze the pressure and volume of the trapped air inside the cylinder before and after it is submerged.

1. Initial State of the Trapped Air:
Before the cylinder is submerged, the piston is removed and the hole at the top is sealed. The cylinder of total length $L_0$ and cross-sectional area $A$ is completely filled with air at atmospheric pressure $P_0$.

• Initial Pressure: $P_i = P_0$
• Initial Volume: $V_i = A L_0$

2. Final State of the Trapped Air (in Equilibrium):
The cylinder is placed vertically such that its sealed top is flush with the water surface in the tank. Water enters from the open bottom up to a height $H$.

• The length of the trapped air column remaining at the top of the cylinder is:
  $L_{\text{air}} = L_0 - H$
• The volume of the trapped air is:
  $V_f = A(L_0 - H)$
• The depth of the water-air interface below the free surface of the water in the tank is:
  $h = L_0 - H$
• Therefore, the hydrostatic pressure of the water at this interface (and hence the pressure of the trapped air) is:
  $P_f = P_0 + \rho g (L_0 - H)$

3. Applying Boyle's Law:
Assuming the process occurs at constant temperature (isothermal conditions), the ideal gas law gives:
$P_i V_i = P_f V_f$

Substitute the initial and final values:
$P_0 (A L_0) = \left[ P_0 + \rho g (L_0 - H) \right] A (L_0 - H)$

Canceling the common cross-sectional area $A$:
$P_0 L_0 = P_0(L_0 - H) + \rho g(L_0 - H)^2$

Rearranging the terms:
$\rho g(L_0 - H)^2 + P_0(L_0 - H) - L_0 P_0 = 0$

Thus, the correct option is C.