Ch 12: Fluid Mechanics

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 12: Fluid Mechanics

Problem 33c

Chapter 12, Problem 33c

A cubical block of wood, 10.0 cm on a side, floats at the interface between oil and water with its lower surface 1.50 cm below the interface (Fig. E12.33). The density of the oil is 790 kg/m³. (a) What is the gauge pressure at the upper face of the block? (b) What is the gauge pressure at the lower face of the block?

Verified step by step guidance

To find the gauge pressure at the upper face of the block, note that the upper face is at the interface between oil and air. The gauge pressure is the pressure due to the oil above this point. Since the upper face is at the interface, the gauge pressure is zero because it is exposed to the atmosphere.

To find the gauge pressure at the lower face of the block, use the formula for pressure due to a fluid column: \( P = \rho g h \), where \( \rho \) is the density of the fluid (oil in this case), \( g \) is the acceleration due to gravity, and \( h \) is the height of the fluid column above the point. Here, \( \rho = 790 \text{ kg/m}^3 \), \( g = 9.81 \text{ m/s}^2 \), and \( h = 0.085 \text{ m} \) (since the block is 10 cm tall and 1.5 cm is below the interface, 8.5 cm is in the oil).

To find the mass of the block, use the principle of buoyancy which states that the weight of the block is equal to the weight of the displaced fluid. The volume of the block submerged in oil is \( 0.085 \text{ m} \times 0.1 \text{ m} \times 0.1 \text{ m} \). Calculate the mass of the displaced oil using \( \text{mass} = \rho \times \text{volume} \).

To find the density of the block, use the formula \( \text{density} = \frac{\text{mass}}{\text{volume}} \). The volume of the block is \( 0.1 \text{ m} \times 0.1 \text{ m} \times 0.1 \text{ m} \). Use the mass calculated from the previous step.

Ensure all units are consistent when performing calculations, converting cm to m where necessary, and check that the calculated density is reasonable for a wooden block, typically less than 1000 kg/m^3.

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Key Concepts

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

Buoyancy

Buoyancy is the upward force exerted by a fluid on an object submerged in it, counteracting the object's weight. It is governed by Archimedes' principle, which states that the buoyant force is equal to the weight of the fluid displaced by the object. In this scenario, the block floats because the buoyant force from the oil and water equals the gravitational force on the block.

Gauge Pressure

Gauge pressure is the pressure relative to atmospheric pressure, excluding atmospheric pressure itself. It is calculated as the difference between absolute pressure and atmospheric pressure. In this problem, gauge pressure at the block's faces is determined by the depth of the block in the oil and water, considering the density of the oil and the gravitational force.

Density

Density is the mass per unit volume of a substance, expressed in kg/m^3. It is a crucial factor in determining buoyancy, as the density of the block compared to the density of the oil and water affects its floating position. The block's density can be calculated by considering the volume submerged and the densities of the surrounding fluids.

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A hollow plastic sphere is held below the surface of a freshwater lake by a cord anchored to the bottom of the lake. The sphere has a volume of 0.650 m³ and the tension in the cord is 1120 N. The cord breaks and the sphere rises to the surface. When the sphere comes to rest, what fraction of its volume will be submerged?

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Textbook Question

A hollow plastic sphere is held below the surface of a freshwater lake by a cord anchored to the bottom of the lake. The sphere has a volume of 0.650 m³ and the tension in the cord is 1120 N. (a) Calculate the buoyant force exerted by the water on the sphere. (b) What is the mass of the sphere?

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Textbook Question

A cubical block of wood, 10.0 cm on a side, floats at the interface between oil and water with its lower surface 1.50 cm below the interface (Fig. E12.33). The density of the oil is 790 kg/m³. (a) What is the gauge pressure at the upper face of the block? (b) What is the gauge pressure at the lower face of the block?

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