91Ó°ÊÓ

We are given the following variables: \n 1. \(y\), the water height inside the container \n 2. \(H\), the container height \n 3. \(h\), the depth of submersion

Given that the air compression is isothermal, we can use Charles' Law which states that the volume of air is inversely proportional to its pressure. Thus, the initial volume of the air \(V1\) (at sea level) is the total volume of the cylinder, given by: \(V1 = H\). The final volume of the air \(V2\) (after submersion) is the volume of the cylinder not filled with water, given by: \(V2 = H - y\).Under isothermal conditions, the pressure inside the cylinder is always equal to the hydrostatic pressure of the external water, and can be expressed as \(P = P0 + \rho g h\), where \(P0\) is atmospheric pressure, \(\rho\) is the water density and \(g\) is the acceleration due to gravity. As the pressure before submersion was the atmospheric pressure \(P0\), we can set up the relationship \(P0/P = V1/V2\), which simplifies to: \(\frac{P0}{P0 + \rho g h} = \frac{H}{H - y}\).

After rearranging the equation from Step 2 to solve for \(y\), we obtain: \(y = H - \frac{Hâ‹…P0}{P0 + \rho g h}\). This equation gives the height of water inside the container in terms of the container height and depth of submersion.

For this, we first need to express \(y / H\) by dividing the equation from Step 3 by \(H\): \(y / H = 1 - \frac{P0}{P0 + \rho g h}\). The plot of \(y / H\) vs \(h / H\) will show how the height of the water inside the container relative to the container height changes as the relative depth of submersion \(h / H\) changes.