# Fluids, Archimedes' Principle and Equation of Continuity

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Conceptual Questions:

A person could not balance her entire weight on the pointed end of a single nail, because it would penetrate her skin. However she can lie safely on a "bed of nails" consists of many nails driven through a sheet of wood so that the pointed ends form a flat array. What is the "nail of beds" trick safe?

As you climb a mountain, you ears "pop" because of the changes in atmospheric pressure. In which direction does your eardrum move (a) as you climb up and (b) as you climb down? Give your reasoning.

Put an ice cube in a glass, and fill the glass to the brim with water. When the ice cube melts and the temperature of the water returns to its initial value would the water level drop, remain the same, or rise (causing the water to spill out)? Explain what would you observe in terms of Archimedes principle.

As a person dives toward the bottom of a swimming pool, the pressure increases noticeably. Does the buoyant force also increase? Justify your answer. Neglect any change in water density with depth.

The cross-sectional area of a stream of water becomes smaller as the water falls from the faucet. Account for this phenomenon in terms of equation of continuity. What would you expect to happen to the cross-sectional area when the water is shot upward, as it is in a fountain?

Problems

Mass Density

One end of a wire is attached to a ceiling, and a solid brass ball is tied to the lower end. The tension in the wire is 120 N. What is the radius of the brass ball?

Accomplished silver workers in India can pound silver into a incredibly thin sheets, as thin as 3.00 x 10-7 m (about one hundredth of the thickness of this sheet of paper). Find the area of such a sheet that can be performed from 1.00 kg of silver.

Pressure

An airtight box has a removable lid of area 1.3 x10 -2 m2 and negligible weight. The box is taken up a mountain where the air pressure outside the box is 0.85 x 105 Pa. The inside of the box is completely evacuated. What is the magnitude of the force required to pull the lid off the box?

A glass bottle of soda is sealed with a screw cap. The absolute pressure of the carbon dioxide inside the bottle is 1.80 x 105 Pa. assuming the top and the bottom surfaces of the cap each have an area of 4.10 x 10-4 m2, obtain the magnitude of the force of the screw thread exerts on the cap in order to keep it on the bottle. The air pressure outside the bottle is one atmosphere.

Pressure and Depth in a Static Fluid, Pressure Gauges

The main water line enters a house on the first floor. The line has a gauge pressure of 1.90 x 105 Pa. (a) A faucet on the second floor, 6.50 m above the first floor, is turned off. What is the gauge pressure at this faucet? (b) How high could the faucet be before no water would flow from it, even if the faucet were open?

The Mariana trench is located in the floor of the Pacific Ocean at a depth of about 11000 m below the surface of the water. The density of seawater is 1025 kg/m3. (a) If an underwater vehicle were to explore such a depth, what force would the water exerts on the vehicle observation window (radius = 0.10 m)? (b) For comparison, determine the weight of a jetliner whose mass is 1.2 x 105 kg.

The human lungs can function satisfactorily up to a limit where the pressure difference between the outside and the inside of the lungs is one-twentieth of an atmosphere. Is a diver uses a snorkel for breathing, how far below the water can she swim? Assume the diver is in salt water whose density is 1025 kg/m3.

Archimedes' Principle

A duck is floating on a lake with 25% of its volume beneath the water. What is the average density of the duck?

A 0.10 m x 0.20 m x 0.30 m block is suspended from a wire and is completely under water. What buoyant force acts on the block?

The density of ice is 917 kg/m3, and the density of sea water is 1025 kg/m3. A swimming polar bear climbs onto a piece of floating ice that has a volume of 5.2 m3. What is the weight of the heaviest bear that the ice can support without sinking completely beneath the water?

A paperweight, when weighed in air, has a weight of W = 6.9 N. When completely immersed in water, however, it has a weight of Win water = 4.3 N. Find the volume of the paperweight.

The Equation of Continuity

Oil is flowing with a speed of 1.22m/s through a pipeline with a radius of 0.305 m. How many gallons of oil (1 gal = 3.79 x 10-3 m3 flow in one day?

A patient recovering from a surgery is being given fluid intravenously. The fluid has a density of 1030 kg/m3, and 9.5 x 10-4 m3 of it flows into the patient every six hours. Find the mass flow rate in kg/s.

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Conceptual Questions:

A person could not balance her entire weight on the pointed end of a single nail, because it would penetrate her skin. However she can lie safely on a "bed of nails" consists of many nails driven through a sheet of wood so that the pointed ends form a flat array. What is the "nail of beds" trick safe?

Answer: Pressure = Weight of the person/Area on which the weight rests. In case of a the person balancing on a single nail, entire weight of the person is balanced on the negligibly small area of the nail end, which results in an enormous pressure due to the nail on the person's body at the point of contact. However, in case of the person lying on a bed of nails, consisting of a large number of nails, person's weight is distributed over a large number of nail ends, resulting in drastic reduction in the pressure at point of contact of each nail and the body. Hence, it is much less painful to lie on a bed of nails as compared to balancing on a single nail.

As you climb a mountain, you ears "pop" because of the changes in atmospheric pressure. In which direction does your eardrum move (a) as you climb up and (b) as you climb down? Give your reasoning.

Answer: As one climbs up a mountain, the atmospheric pressure keeps decreasing. Hence, the ear drums will move outwards (inside pressure being greater than the outside pressure).However, soon, the pressure inside the ear adjusts itself to the outside atmospheric pressure and the ear drums return to their normal position. On climbing down the mountain, the atmospheric pressure rises and the ear drums move inwards (outside pressure being greater than the inside).

Put an ice cube in a glass, and fill the glass to the brim with water. When the ice cube melts and the temperature of the water returns to its initial value would the water level drop, remain the same, or rise (causing the water to spill out)? Explain what would you observe in terms of Archimedes principle.

Answer: Density of ice is lower than that of water by approximately 9%. (that is the reason ice floats on the water surface). Hence, taking density of water as 1000 kg/m3, density of ice is 910 kg/m3. Let the volume of ice cube be V m3.

Mass of ice cube = Density of ice x Volume = 910V kg.

As per Archimedes principle, the ice floating on the surface will displace water equal to 910V kg.

Volume of the displaced water = Mass/Density = 910V/1000 = 0.91V m3.

As the cube displaces volume of water equal to the volume of the immersed portion of the cube, from the above calculation we conclude that the volume of the immersed portion of the cube is 0.91V m3.

When whole of the cube melts, volume of the resulting water is equal to 0.91V.

Hence, when cube floats in water, water level rises by 0.91V on account of the water displaced by its immersed portion. When whole of the cube melts, water level rises by 0.91V on account of the water from the molten cube. Hence, there is no change in the water level in the glass.

As a person ...

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