!! Assignment 121.! The king ordered a crown of solid pure gold, but what he got didn’t feel right, so he asked Archimedes to verify that it was really solid and all gold. Archimedes realized he needed a way to measure the density of the crown, but he couldn’t damage it. There are various stories about what he did. The most likely is this.! He balanced the crown with an equal weight of pure gold as shown. Then he lowered the whole system slowly into water and watched to see if the items were still balanced.! How did this allow him to make a determination of whether the king had been cheated? [He had been.]2.! Questions about hydrostatics.a.! The teapot effect. It is well known that if you pour a liquid too slowly from a spout it will double back under the spout and may miss the container you are pouring into. In the drawing fluid comes slowly out of the opening and runs back underneath the pipe. Adhesion makes it turn the corner, but what holds it against the bottom of the pipe for such a distance? [Consider the pressure at the top and bottom of the stream at point A.]b.! Four u-tubes are partially filled with two different fluids as shown. The tops of the columns of fluid are open to the air. One case shown is impossible. Identify it, and for the others state whether the red fluid is more dense, less dense, or equally dense. Give your reasons.c.! The giraffe’s head is 2 m above its heart, which is 2 m above the ground. Discuss and explain two things about this animal: (a) The skin around its lower legs is very tight and thick; (b) When it drinks from a source at ground level it slowly spreads it legs before lowering its head. crown gold a a b c dPHY 53! Summer 2010! 1!!3.! The Magdeburg hemispheres. In the 1650’s the mayor of the German town of Magdeburg, an amateur scientist, invented the first vacuum pump. To demonstrate the power of the vacuum he could create, he pumped air out of a device consisting of two metal hemispheres with a good seal between them and a valve to pump out the air. When he had reduced the pressure inside the device as much as he could, he closed the valve and attached each side to a team of horses and had them try to pull the two pieces apart, which they couldn’t do. a.! Take the difference in pressure between outside and inside as ΔP. Show that the force necessary to pull one hemisphere away from the other is F =πR2ΔP, where R is the radius of the sphere made of the two parts. ! [This is an exercise in integral calculus. Consider the force on an infinitesimal circular strip of the surface, shown by the dark band, which has area dA = 2πR sinθ⋅ Rdθ. (Prove this.) Then argue that you need consider only the leftward component of the inward force, which is dFx= ΔP ⋅ dA ⋅ cosθ. Do the integral over θ; the substitution α= sinθ helps.] b.! Let R = 30 cm and ΔP = 0.9 atm. Find F.! [See http://upload.wikimedia.org/wikipedia/commons/5/5c/Magdeburg.jpg ]4.! A hydraulic jack consists of a cylinder (1) with a small cross section area a, connected by a pipe to a cylinder (2) with larger cross section A. Each cylinder is fitted with a piston that can move vertically, and the system is filled with an incompressible fluid. A heavy load to be raised rests on the piston in 2; the total mass of the load and the piston is M. A force F1 is exerted downward on the piston in 1, resulting in an upward force F2 on the piston in 2. Comment on the validity of the following statements. a.! The pressure in both cylinders is the same. b.! As the piston in 1 moves down distance d the piston in 2 moves up distance d ⋅ (a/A). c.! To lift the piston in 2 and the load at constant speed, we must have F1= Mg ⋅ (A/ a). d.! The work done by F1 is equal to the work done by F2.θ R 1 2 PHY 53! Summer 2010! 2!!5.! A “Cartesian diver” is a small glass cylinder open at the bottom which has a pocket of air trapped inside. It is in a sealed flexible plastic container nearly filled with water, and initially the diver is floating at the surface as shown. When one squeezes the sides of the container, reducing the volume of the air above the water, the diver sinks toward the bottom. When the squeezing force is removed the diver rises again and floats. Explain this behavior in detail. [What happens to the average density of the diver, and why?]6.! A block of mass m is made of a material with density four times that of water. It is lowered by a string until it is held at rest submerged in a container of water as shown.a.! What is the tension in the string?b.! If the container plus the water have mass M, what is the normal force from the table on the system?c.! If the block is lowered to rest on the bottom, with the string slack, what is the normal force from the table?7.! A tank filled with water has a small hole as shown where water emerges in a stream. At height h above the hole there is a piston of the same cross-section area A as the tank, resting on the water. It cannot sink because the fit is too tight for water to flow around it, but it can slide freely down the sides of the tank as the water level drops.a.! If the water emerges at speed v, what is the pressure P in the water just below the piston, in terms of v, h, the density of water ρ, air pressure P0 and g? Ans: P = P0−ρgh +12ρv2.b.! What must P − P0 be to support the weight of the piston, if its density is 2ρ and its thickness is d? Ans: P − P0= 2ρdg .c.! Find the speed v in terms of h, d and g. Ans: v2= g(2h + 4d).hdPHY 53! Summer 2010! 3!!8.! An aspirator is a device that mixes small amounts of a fluid (e.g., liquid fertilizer) with water so it can be sprayed over a large area. As shown, water from a hose passes through a constriction of cross-section a, where a vertical tube introduces the fluid (moving upward at negligible speed) into the rapidly moving water. The mixture then flows out at speed v0 through an opening of larger cross-section A. The height through which the fluid must rise is h. Assume the fluid and water have the same density ρ, and find the minimum ratio A / a of the two cross-sections. [Find the water pressure in the constriction.] Ans: (A/ a)2= 1 + 2gh/v02.9.! Questions about the Bernoulli effect in air.a.! You can lift a card off a table by blowing horizontally over the top of the card. Explain.b.! In a hurricane a flat roof is …