(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.

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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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?