Understanding Hydrostatic Pressure: A 12,000 Foot Cylinder

Imagine a scenario where a cylinder is filled with 12,000 feet of water, standing 2 meters wide. Would the hydrostatic pressure at the bottom of this cylinder be the same as if it were filled to the same depth in the ocean? The answer is yes, and this article delves into the underlying principles.

Key Concepts

The pressure at the bottom of the cylinder is determined by the height of the water column, not its width. The pressure P can be calculated using the formula:

P rho g h

P Pressure (in Pascals or Pa) rho (rho) Density of the fluid (in kg/m3) g Acceleration due to gravity (m/s2) h Height of the fluid column (in meters)

In terms of units, rho , m/s^2 cdot m results in Pa, where 100 kPa is approximately equal to 1 bar or 1 atmosphere.

Why Width Doesn't Matter

The width of the cylinder is irrelevant. The pressure at the bottom is solely dependent on the height of the water column, regardless of the cylinder's diameter. Even a cylinder with an effective infinite diameter, such as the ocean, would exhibit the same pressure at a given depth.

Real-World Implications

At depths reaching just 100 meters (328 feet), the hydrostatic pressure becomes so immense that most pipes would burst. This is why high-rise buildings often utilize separate water pipelines connected with pumps at regular intervals, typically every 4-5 floors.

Building a 12,000 feet high cylinder that wouldn't collapse under its own weight presents significant engineering challenges. Even if viable, filling such a cylinder would be nearly impossible with existing technology. If you were to pump water from the bottom, the weight of the water would eventually halt the process. Moreover, at a certain height, the water would freeze, further complicating the process.

Seawater Considerations

For the pressure to be the same, the cylinder would need to contain seawater with the same salinity and temperature profiles as the ocean at the chosen depth. While the width doesn't matter in terms of pressure, ensuring the same seawater properties is crucial for accuracy.

Practical Challenges

Given the immense height, a 2-meter wide cylinder needs to be extraordinarily stable. A thickness of 2 meters in the cylinder walls would be necessary to prevent collapse under the weight of the water.

Conclusion

The pressure at the bottom of a 12,000-foot cylinder filled with water, as with 12,000 feet of ocean water, is determined by the height of the water column, not its width. Despite the theoretical simplicity, the practical challenges of constructing such a structure and filling it with water make the scenario an engineering impossibility with current technology.