19. Fluid Mechanics

When an object of unknown mass and volume is fully immersed in large oil (800 kg/m³ ) container and released from rest, it stays at rest. Calculate the density of this object.

A block floats with 40% of its volume above water. When you place it on an unknown liquid, it floats with 30% of its volume above. What is the density of the unknown liquid?

An 8,000 cm³ block of wood is fully immersed in a deep water tank, then tied to the bottom. When the block is released and reaches equilibrium, you measure the tension on the string to be 12 N. What is the density of the wood?

You want to build a large storage container, with outer walls and an open top, as shown, so that you can load things into it, while it floats on fresh water, without any water getting inside. If the bottom face of the container measures 3.0 m by 8.0 m, how high should the side walls be, such that the combined mass of container and inside load is 100,000 kg?

(II) A spherical balloon has a radius of 7.15 m and is filled with helium. How large a cargo can it lift, assuming that the skin and structure of the balloon have a mass of 890 kg? Neglect the buoyant force on the cargo volume itself.

(II) The specific gravity of ice is 0.917, whereas that of seawater is 1.025. What percent of an iceberg is above the surface of the water?

(II) A crane lifts the 21,000-kg steel hull of a sunken ship out of the water. Determine

(b) the tension when the hull is completely out of the water.

(II) A crane lifts the 21,000-kg steel hull of a sunken ship out of the water. Determine

(a) the tension in the crane’s cable when the hull is fully submerged in the water , and

(III) A 3.75-kg block of wood (SG = 0.50) floats on water. What minimum mass of lead, hung from the wood by a string, will cause the block to sink?

A ship, carrying fresh water to a desert island in the Caribbean, has a horizontal cross-sectional area of 2240 m² at the waterline. When unloaded, the ship rises 8.55 m higher in the sea. How much water (m³) was delivered?

A raft is made of 12 logs lashed together. Each is 45 cm in diameter and has a length of 6.5 m. How many people can the raft hold before they start getting their feet wet, assuming the average person has a mass of 68 kg? Do not neglect the weight of the logs. Assume the specific gravity of wood is 0.60.

(II) A cube of side length 10.0 cm and made of unknown material floats at the surface between water and oil. The oil has a density of 810 kg/m³ . If the cube floats so that it is 72% in the water and 28% in the oil, what is the mass of the cube and what is the buoyant force on the cube?

(II) Archimedes’ principle can be used not only to determine the specific gravity of a solid using a known liquid (Example 13–10). The reverse can be done as well.

(a) As an example, a 3.80-kg aluminum ball has an apparent mass of 2.10 kg when submerged in a particular liquid: calculate the density of the liquid.

(b) Determine a formula for finding the density of a liquid using this procedure.

(b) To ensure that a ship is in stable equilibrium, would it be better if its center of buoyancy was above, below, or at the same point as its center of gravity? Explain.

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A simple model (Fig. 13–62) considers a continent as a block ( density ≈ 2800 kg/m³) floating in the mantle rock around it ( density ≈ 3300 kg/m³) . Assuming the continent is 35 km thick (the average thickness of the Earth’s continental crust), estimate the height of the continent above the surrounding mantle rock.

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A copper (Cu) weight is placed on top of a 0.40-kg block of wood (density = 0.60 x 10³ kg/m³) floating in water, as shown in Fig. 13–60. What is the mass of the copper if the top of the wood block is exactly at the water’s surface?

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(II) A scuba tank, when fully submerged, displaces 15.7 L of seawater. The tank itself has a mass of 14.0 kg and, when “full,” contains 3.00 kg of air. Assuming only its weight and the buoyant force act on the tank, determine the net force (magnitude and direction) on the fully submerged tank at the beginning of a dive (when it is full of air) and at the end of a dive (when it no longer contains any air).

A tub of water rests on a scale as shown in Fig. 13–61. The weight of the tub plus water is 100 N. A 50-N concrete brick is tied by a cord to a fixed arm and lowered into the water but does not touch the bottom of the tub. What does the scale read now? [Hint: Draw two free-body diagrams, one for the brick and a second one for the tub + water + brick.]

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(III) A common effect of surface tension is the ability of a liquid to rise up a narrow tube due to capillary action. Show that for a narrow tube of radius r placed in a liquid of density ρ and surface tension γ , the liquid in the tube will reach a height h = 2γ/ρgr above the level of the liquid outside the tube, where g is the gravitational acceleration. Assume that the liquid “wets” the tube and that the liquid surface is vertical at the contact with the inside of the tube.

"(II) Estimate the diameter of a steel needle that can just barely remain on top of water due to surface tension. (See Figs. 13–38 and 13–39a, and Table 13–1.)

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(II) Calculate the true mass (in vacuum) of an aluminum sphere whose apparent mass is 4.0000 kg when weighed in air.

(II) How many helium-filled balloons would it take to lift a person? Assume the person has a mass of 72 kg and that each helium-filled balloon is spherical with a diameter of 36 cm.

(II) A 0.48-kg piece of wood floats in water but is found to sink in alcohol (SG = 0.79) , in which it has an apparent mass of 0.047 kg. What is the SG of the wood?