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Chapter 12: Problem 12

You are designing a diving bell to withstand the pressure of seawater at a depth of \(250 \mathrm{~m}\). (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 \(30.0 \mathrm{~cm}\) 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.)

Short Answer

Expert verified
The gauge pressure at a depth of \(250 \, m\) can be calculated by using the seawater density, acceleration due to gravity, and the depth. The net force on a circular window, against which the interior pressure of the bell acts to counter the pressure of the seawater, is determined by the area of the window and the atmospheric pressure.

Step by step solution

01

Calculate Gauge Pressure at Depth

Use the equation for pressure in a fluid column to calculate the gauge pressure at this depth which is given by \( P_g = \rho g h \), where \( P_g \) is the gauge pressure, \( \rho \) is the density of the fluid, \( g \) is the acceleration due to gravity and \( h \) is the depth under water. Given that the density of seawater is approximately \( 1029 \, kg/m^3 \), \( g = 9.81 \, m/s^2 \), and the depth under water is \( h = 250 \, m \), substituting these values in will yield the gauge pressure.
02

Calculate the Net Force on the Window

In this step the net force due to the water outside and the air inside the bell should be calculated. The net force is given by \( F = P_a A \) where \( P_a \) is the atmospheric pressure, and \( A \) is the area of the window. Here, air pressure inside is equal to surface air pressure and because it gets applied to the window, it counteracts the force of the seawater. For a circular window, the area can be calculated as \( A = \pi r^2 \), where \( r \) is the radius of the window. Given that the atmospheric pressure \( P_a = 101300 \, Pa \) and the window diameter is \( 30.0 \, cm = 0.3 \, m \), the radius is \( r = 0.15 \, m \). Substituting these values in will provide the net force on the window.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fluid Mechanics
Fluid mechanics is a branch of physics that studies the behavior of fluids (liquids and gases) and how forces affect them. It plays a crucial role in understanding many natural and industrial processes. One of the key principles in fluid mechanics is how pressure changes with depth in a fluid.

In fluids, pressure increases with depth due to the weight of the fluid above. This is described by the equation:
  • Gauge Pressure: \( P_g = \rho g h \)
where \( \rho \) is the fluid density, \( g \) is acceleration due to gravity, and \( h \) is depth. This equation helps calculate the pressure difference with increasing depth.

Understanding fluid mechanics helps design structures like diving bells to withstand the immense pressures experienced underwater. It ensures safety and efficiency in operations involving submersion in fluids.
Buoyancy and Pressure
Buoyancy and pressure are two integral concepts in fluid mechanics that describe the interactions between objects and fluids. When an object is immersed in a fluid, it experiences an upward buoyant force due to the fluid pressure acting on it. This force depends on the pressure differences on various parts of the object's surface.

Pressure in a fluid acts in all directions and increases with depth, influencing the net force on submerged objects.
  • At a given depth, pressure is uniform across a horizontal plane.
  • Buoyancy ensures objects float or sink depending on their density relative to the fluid.
In the context of the diving bell, buoyancy doesn't act on specific components but contributes to the overall design considerations, ensuring any pressure variances don't compromise the structure. Buoyancy helps assess pressure effects on submerged windows, ensuring safe diving bell operations.
Net Force Calculation
Calculating net force on objects submerged in fluids requires understanding pressure interactions across surfaces. For the diving bell's window, it's essential to compute forces resulting from both external water pressure and internal air pressure.

The equation:
  • Net Force: \( F = P_a A \)
helps find the force due to atmospheric pressure. Here, \( P_a \) is the atmospheric pressure, and \( A \) is the window area.

For circular windows, calculate the area:
  • Area: \( A = \pi r^2 \)
with \( r \) as the window radius. This helps determine if structural components can withstand encountered forces. Ensuring accurate net force calculations ensures safety and structural integrity, crucial in diving bell applications. Understanding these calculations aids in maintaining equilibrium and avoiding potential breaches due to excessive forces.

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Most popular questions from this chapter

An ore sample weighs \(17.50 \mathrm{~N}\) in air. When the sample is suspended by a light cord and totally immersed in water, the tension in the cord is \(11.20 \mathrm{~N}\). Find the total volume and the density of the sample.

A golf course sprinkler system discharges water from a horizontal pipe at the rate of \(7200 \mathrm{~cm}^{3} / \mathrm{s}\). At one point in the pipe, where the radius is \(4.00 \mathrm{~cm},\) the water's absolute pressure is \(2.40 \times 10^{5} \mathrm{~Pa}\). At a second point in the pipe, the water passes through a constriction where the radius is \(2.00 \mathrm{~cm} .\) What is the water's absolute pressure as it flows through this constriction?

If the elephant were to snorkel in salt water, which is more dense than freshwater, would the maximum depth at which it could snorkel be different from that in freshwater? (a) Yes - that depth would increase, because the pressure would be lower at a given depth in salt water than in freshwater; (b) yes - that depth would decrease, because the pressure would be higher at a given depth in salt water than in freshwater; (c) no, because pressure differences within the submerged elephant depend on only the density of air, not the density of the water; (d) no, because the buoyant force on the elephant would be the same in both cases.

A gallon of milk in a full plastic jug is sitting on the edge of your kitchen table. Estimate the vertical distance between the top surface of the milk and the bottom of the jug. Also estimate the distance from the tabletop to the floor. You punch a small hole in the side of the jug just above the bottom of the jug, and milk flows out the hole. When the milk first starts to flow out the hole, what horizontal distance does it travel before reaching the floor? Assume the milk is in free fall after it has passed through the hole, and neglect the viscosity of the milk.

A large, cylindrical water tank with diameter \(3.00 \mathrm{~m}\) is on a platform \(2.00 \mathrm{~m}\) above the ground. The vertical tank is open to the air and the depth of the water in the tank is \(2.00 \mathrm{~m}\). There is a hole with diameter \(0.500 \mathrm{~cm}\) in the side of the tank just above the bottom of the tank. The hole is plugged with a cork. You remove the cork and collect in a bucket the water that flows out the hole. (a) When 1.00 gal of water flows out of the tank, what is the change in the height of the water in the tank? (b) How long does it take you to collect 1.00 gal of water in the bucket? Based on your answer in part (a), is it reasonable to ignore the change in the depth of the water in the tank as 1.00 gal of water flows out?
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