Ch 14: Fluids and Elasticity

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 14: Fluids and Elasticity

Problem 73a

Chapter 14, Problem 73a

The tank shown in FIGURE CP14.73 is completely filled with a liquid of density ρ. The right face is not permanently attached to the tank but, instead, is held against a rubber seal by the tension in a spring. To prevent leakage, the spring must both pull with sufficient strength and prevent a torque from pushing the bottom of the right face out. What minimum spring tension is needed?

Verified step by step guidance

Step 1: Analyze the forces acting on the right face of the tank. The liquid exerts a pressure on the face due to its density (ρ) and the height of the liquid column (H). The pressure at a depth y is given by the hydrostatic pressure formula:

P = ρ g y

, where g is the acceleration due to gravity.

Step 2: Calculate the total force exerted by the liquid on the right face. The force is the integral of pressure over the area of the face. The area of the face is

W \times H

. The force is given by:

F = ρ g y W dy

, integrated from y = 0 to y = H.

Step 3: Determine the torque about the bottom edge of the right face due to the liquid force. The torque is calculated by considering the moment arm of the force, which varies with depth. The torque is given by:

τ = ρ g y W y dy

, integrated from y = 0 to y = H.

Step 4: Relate the spring tension to the force and torque. The spring must exert a force equal to the total force exerted by the liquid to prevent the face from being pushed out. Additionally, the spring must counteract the torque to prevent rotation. The spring tension, T, must satisfy both conditions.

Step 5: Solve for the minimum spring tension. Use the expressions for total force and torque derived in steps 2 and 3. Ensure that the spring tension is sufficient to balance both the force and torque simultaneously. This involves solving the equations for T based on the geometry and physical parameters of the tank.

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Key Concepts

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

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at rest due to the force of gravity. It increases with depth and is calculated using the formula P = ρgh, where P is the pressure, ρ is the fluid density, g is the acceleration due to gravity, and h is the height of the fluid column above the point in question. Understanding hydrostatic pressure is crucial for determining the forces acting on the tank's walls.

Buoyant Force

The buoyant force is the upward force exerted by a fluid on an object submerged in it, described by Archimedes' principle. This force is equal to the weight of the fluid displaced by the object. In the context of the tank, the buoyant force acts on the right face, and calculating it is essential for determining the minimum spring tension required to prevent leakage.

Torque and Equilibrium

Torque is the rotational equivalent of linear force and is calculated as the product of force and the distance from the pivot point. For the tank to remain in equilibrium, the net torque acting on it must be zero. This means that the torque due to the hydrostatic pressure on the right face must be balanced by the torque produced by the spring tension, which is critical for preventing the right face from being pushed out.

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Textbook Question

The tank shown in FIGURE CP14.73 is completely filled with a liquid of density ρ. The right face is not permanently attached to the tank but, instead, is held against a rubber seal by the tension in a spring. To prevent leakage, the spring must both pull with sufficient strength and prevent a torque from pushing the bottom of the right face out. If the spring has the minimum tension, at what height d from the bottom must it be attached?

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Textbook Question

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Textbook Question

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Textbook Question

In FIGURE CP14.74, a cone of density ρ₀ and total height l floats in a liquid of density ρ_f. The height of the cone above the liquid is h. What is the ratio h/l of the exposed height to the total height?

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