A hollow cylinder 90cm long in diameter, closed one end is immersed with axis vertical closed end uppermost, this end being held 7.50m below the water surface. The cylinder is at first full of water, but compressed air is admitted from beneath in the immersed position until it has displaced two thirds of the water. Calculate the total absolute pressure from above on tops of the cylinder in kN.

To calculate the total absolute pressure on the top of the cylinder in kilonewtons (kN), we need to follow these steps:

### Step 1: Understanding the Problem
- **Length of the cylinder**: \( L = 90 \, \text{cm} = 0.9 \, \text{m} \)
- **Depth of the top of the cylinder below water surface**: \( h = 7.5 \, \text{m} \)
- The cylinder is initially full of water, but air is pumped in to displace two-thirds of the water.

### Step 2: Pressure due to Water Column
The pressure due to the water column at the top of the cylinder is given by the hydrostatic pressure formula:

\[
P_{\text{water}} = \rho g h
\]

where:
- \( \rho \) is the density of water (\( 1000 \, \text{kg/m}^3 \)),
- \( g \) is the acceleration due to gravity (\( 9.81 \, \text{m/s}^2 \)),
- \( h \) is the depth of the top of the cylinder below the water surface (\( 7.5 \, \text{m} \)).

So,

\[
P_{\text{water}} = 1000 \times 9.81 \times 7.5 \, \text{Pa} = 73575 \, \text{Pa}
\]

### Step 3: Atmospheric Pressure Contribution
The total absolute pressure includes both the atmospheric pressure and the pressure due to the water column. The atmospheric pressure at sea level is approximately \( 101325 \, \text{Pa} \).

So, the total pressure at the top of the cylinder is:

\[
P_{\text{total}} = P_{\text{water}} + P_{\text{atm}} 
\]

\[
P_{\text{total}} = 73575 \, \text{Pa} + 101325 \, \text{Pa} = 174900 \, \text{Pa}
\]

### Step 4: Converting Pressure to Kilonewtons
1 Pascal is equivalent to 1 Newton per square meter. To convert the pressure to kilonewtons per square meter (kN/m²):

\[
P_{\text{total}} = \frac{174900 \, \text{Pa}}{1000} = 174.9 \, \text{kN/m}^2
\]

Thus, the total absolute pressure on the top of the cylinder is **174.9 kN/m²**.

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Would you like any more details or have any further questions?

Here are some related questions you might be interested in:
1. How does the depth of submersion affect the pressure on the cylinder?
2. What would happen to the pressure if the cylinder were placed at a different depth?
3. How does the displacement of water affect the overall pressure?
4. What would the pressure be if the cylinder were filled completely with air instead?
5. How does atmospheric pressure influence measurements underwater?

**Tip:** Always account for both atmospheric pressure and hydrostatic pressure when calculating total pressure under a fluid.

Learn how to calculate the total absolute pressure on a hollow cylinder immersed in water with one end closed. Includes detailed steps and formulas for hydrostatic pressure. Suitable for high school students studying hydrostatics and pressure calculations.

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