Ch 12: Fluid Mechanics

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 12: Fluid Mechanics

Problem 14a

Chapter 12, Problem 14a

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.)

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To find the gauge pressure at a depth of 250 m, use the formula for gauge pressure: \( P_{\text{gauge}} = \rho g h \), where \( \rho \) is the density of seawater (approximately 1025 kg/m³), \( g \) is the acceleration due to gravity (9.81 m/s²), and \( h \) is the depth (250 m).

Substitute the known values into the formula: \( P_{\text{gauge}} = 1025 \times 9.81 \times 250 \). This will give you the gauge pressure in pascals (Pa).

To find the net force on the window, first calculate the area of the circular window. The diameter is 30.0 cm, so the radius \( r \) is 15.0 cm or 0.15 m. Use the formula for the area of a circle: \( A = \pi r^2 \).

Calculate the area: \( A = \pi \times (0.15)^2 \). This will give you the area in square meters.

The net force on the window is the difference in pressure between the outside and inside of the bell, multiplied by the area of the window. Since the pressure inside the bell equals the pressure at the surface, the net force is \( F = P_{\text{gauge}} \times A \). Substitute the values for \( P_{\text{gauge}} \) and \( A \) to find the net force.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Gauge Pressure

Gauge pressure is the pressure relative to atmospheric pressure. It is calculated by subtracting atmospheric pressure from the absolute pressure. In the context of a diving bell, gauge pressure at a certain depth is determined by the weight of the water column above that depth, which can be calculated using the formula: P = ρgh, where ρ is the density of the water, g is the acceleration due to gravity, and h is the depth.

Buoyancy and Net Force

The net force on an object submerged in a fluid is influenced by buoyancy, which is the upward force exerted by the fluid. For a diving bell, the net force on a window is the difference between the force due to the water pressure outside and the force due to the air pressure inside. This can be calculated using the formula: F_net = (P_outside - P_inside) × A, where A is the area of the window.

Pressure and Force Relationship

Pressure is defined as force per unit area. The relationship between pressure and force is crucial for understanding how forces act on surfaces submerged in a fluid. For a circular window, the force exerted by the fluid can be calculated by multiplying the pressure difference by the area of the window, which is given by A = πr², where r is the radius of the window.

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Oceans on Mars. Scientists have found evidence that Mars may once have had an ocean 0.500 km deep. The acceleration due to gravity on Mars is 3.71 m/s². (a) What would be the gauge pressure at the bottom of such an ocean, assuming it was freshwater? (b) To what depth would you need to go in the earth's ocean to experience the same gauge pressure?

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Textbook Question

In intravenous feeding, a needle is inserted in a vein in the patient's arm and a tube leads from the needle to a reservoir of fluid (density 1050 kg/m³) located at height h above the arm. The top of the reservoir is open to the air. If the gauge pressure inside the vein is 5980 Pa, what is the minimum value of h that allows fluid to enter the vein? Assume the needle diameter is large enough that you can ignore the viscosity of the liquid.

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Textbook Question

The liquid in the open-tube manometer in Fig. 12.8a is mercury, y₁=3.00 cm,and y₂=7.00 cm. Atmospheric pressure is 980 millibars. What is (a) the absolute pressure at the bottom of the U-shaped tube; (b) the absolute pressure in the open tube at a depth of 4.00 cm below the free surface; (c) the absolute pressure of the gas in the container; (d) the gauge pressure of the gas in pascals?

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Textbook Question

You are designing a diving bell to withstand the pressure of seawater at a depth of 250 m. What is the gauge pressure at this depth? (You can ignore changes in the density of the water with depth.) 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.)

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Textbook Question

Ear Damage from Diving. If the force on the tympanic membrane (eardrum) increases by about 1.5 N above the force from atmospheric pressure, the membrane can be damaged. When you go scuba diving in the ocean, below what depth could damage to your eardrum start to occur? The eardrum is typically 8.2 mm in diameter. (Consult Table 12.1.)

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Textbook Question

A barrel contains a 0.120-m layer of oil floating on water that is 0.250 m deep. The density of the oil is 600 kg/m³. (a) What is the gauge pressure at the oil–water interface? (b) What is the gauge pressure at the bottom of the barrel?