You are designing a diving bell to withstand the pressure of seawater at a depth of 250 m. (a) What is the gauge pressure at this depth? (You can ignore changes in the density of the water with depth.) (b) At this depth, what is the net force due to the water outside and the air inside the bell on a circular glass window 30.0 cm in diameter if the pressure inside the diving bell equals the pressure at the surface of the water? (Ignore the small variation of pressure over the surface of the window.)

Verified step by step guidance

1. Calculate the gauge pressure at a depth of 250 m using the formula: \( P = \rho \cdot g \cdot h \), where \( \rho \) is the density of seawater (approximately 1025 kg/m^3), \( g \) is the acceleration due to gravity (approximately 9.8 m/s^2), and \( h \) is the depth (250 m).
2. Determine the diameter of the circular glass window in meters to use in further calculations (30.0 cm = 0.30 m).
3. Calculate the area of the circular window using the formula: \( A = \pi \cdot (\frac{d}{2})^2 \), where \( d \) is the diameter of the window.
4. Compute the net force on the window by using the formula: \( F = P \cdot A \), where \( P \) is the gauge pressure calculated in step 1 and \( A \) is the area of the window calculated in step 3. Since the internal pressure of the bell equals the atmospheric pressure at the surface, the net force is due to the external water pressure only.
5. Summarize the results and implications for the design of the diving bell, ensuring that the structure can withstand the calculated forces at the specified depth.

Key Concepts

Gauge Pressure

Gauge pressure is the pressure relative to atmospheric pressure. It is calculated by subtracting atmospheric pressure from the absolute pressure at a given depth. In the context of a diving bell, gauge pressure at a depth can be determined using the formula P = ρgh, where ρ is the density of seawater, g is the acceleration due to gravity, and h is the depth. This concept is crucial for understanding the forces acting on the diving bell.

Hydrostatic Pressure

Hydrostatic pressure refers to the pressure exerted by a fluid at equilibrium due to the force of gravity. It increases linearly with depth in a fluid, which means that deeper points experience greater pressure. This principle is essential for calculating the pressure at the specified depth of 250 m, as it directly influences the design and structural integrity of the diving bell.

Net Force on a Surface

The net force on a surface, such as the circular glass window of the diving bell, is determined by the difference in pressure between the inside and outside of the surface multiplied by the area of that surface. If the pressure inside the bell equals the atmospheric pressure, the net force can be calculated by considering the gauge pressure outside due to the water. Understanding this concept is vital for ensuring that the window can withstand the external pressures without failing.