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Chapter 3: Problem 11

A cube with 6 in. sides is suspended in a fluid by a wire. The top of the cube is horizontal and 8 in. below the free surface. If the cube has a mass of 2 slugs and the tension in the wire is \(T=50.7\) Ibf, compute the fluid specific gravity, and from this determine the fluid. What are the gage pressures on the upper and lower surfaces?

Short Answer

Expert verified
The specific gravity of the fluid is approximately 0.652. The gage pressures on the upper and lower surfaces of the cube are approximately 11.012 psi and 18.642 psi respectively.

Step by step solution

01

Calculating the hydrostatic pressure

Starting with calculating the hydrostatic pressure at the top and bottom of the cube. Hydrostatic pressure, \(P\), in a fluid can be determined using the formula \(P = \rho gh\), where: \(\rho\) = fluid density, \(g\) = acceleration due to gravity (32.2 ft/s虏 in this case) and \(h\) = height (or depth in fluid). Here we have top pressure, \(P_t = \rho g(8)\) (As top of the cube is 8 in below the surface, converting inches to feet by dividing by 12, we get \(h = 8/12 = 0.67 \) feet). And the bottom pressure, \(P_b = \rho g(8+6) = \rho g(14)\) (As the distance from the free surface to the bottom of the cube is the sum of the depth to the top of the cube and the side length of the cube).
02

Calculating the fluid specific gravity

Next is calculating the fluid鈥檚 specific gravity using the tension in the wire and the weight of the cube. The tension in the wire is the difference between the weight of the cube and the buoyant force. Thus, \(weight_{cube} - T = buoyant\ force_{cube}\). We substitute \(buoyant\ force_{cube} = V \rho_f g - V \rho g) \) where \( V = (1/2)^3 \) is the volume of the cube, \( \rho_f \) is the fluid density and \( \rho \) is the cube density. From here, we can solve to find \( \rho_f \). Specific gravity is then given by \( SG = \rho_f / \rho_w \), where \( \rho_w \) is the density of water (which is 1.94 slugs/ft鲁).
03

Calculate gage pressures on the upper and lower surfaces

Finally we use the calculated fluid density to find the gage pressures at the top and bottom of the cube. The gage pressure is simply the hydrostatic pressure (which was found in step 1), because gage pressure doesn't include atmosphere pressure. So \( P_{t,g} = P_t \) and \( P_{b,g} = P_b \).

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Most popular questions from this chapter

Quantify the experiment performed by Archimedes to identify the material content of King Hiero's crown. Assume you can measure the weight of the king's crown in air, \(W_{a}\) and the weight in water, \(W_{w^{*}}\) Express the specific gravity of the crown as a function of these measured values.

A sphere of radius 1 in., made from material of specific gravity of \(\mathrm{SG}=0.95,\) is submerged in a tank of water. The sphere is placed over a hole of radius 0.075 in., in the tank bottom. When the sphere is released, will it stay on the bottom of the tank or float to the surface?

Consider a semicylindrical trough of radius \(R\) and length \(L\) Develop general expressions for the magnitude and line of action of the hydrostatic force on one end, if the trough is partially filled with water and open to atmosphere. Plot the results (in nondimensional form) over the range of water depth \(0 \leq d / R \leq 1\).

Consider a conical funnel held upside down and submerged slowly in a container of water. Discuss the force needed to submerge the funnel if the spout is open to the atmosphere. Compare with the force needed to submerge the funnel when the spout opening is blocked by a rubber stopper.

Your pressure gage indicates that the pressure in your cold tires is \(0.25 \mathrm{MPa}\) (gage) on a mountain at an elevation of \(3500 \mathrm{m}\) What is the absolute pressure? After you drive down to sea level, your tires have warmed to \(25^{\circ} \mathrm{C}\). What pressure does your gage now indicate? Assume a U.S. Standard Atmosphere.
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