Fluid Mechanics Practice Problems for AP Physics B & C (page 2)
Review the following concepts if necessary:
- Fluid Mechanics for AP Physics B
- Buoyancy and Archimedes' Principle for AP Physics B
- Pascal's Principle for AP Physics B
- Flowing Fluids for AP Physics B
- A person sips a drink through a straw. At which of the following three positions is the pressure lowest?
- Inside the person's mouth
- At the surface of the drink
- At the bottom of the drink
- only at position I
- only at position II
- only at position III
- both at positions I and III
- both at positions I and II
- The circulatory system can be modeled as an interconnected network of flexible pipes (the arteries and veins) through which a pump (the heart) causes blood to flow. Which of the following actions, while keeping all other aspects of the system the same, would NOT cause the velocity of the blood to increase inside a vein?
- expanding the vein's diameter
- cutting off blood flow to some other area of the body
- increasing the heart rate
- increasing the total amount of blood in the circulatory system
- increasing the pressure difference between ends of the vein
- A pirate ship hides out in a small inshore lake. It carries twenty ill-gotten treasure chests in its hold. But lo, on the horizon the lookout spies a gunboat. To get away, the pirate captain orders the heavy treasure chests jettisoned. The chests sink to the lake bottom. What happens to the water level of the lake?
- The water level drops.
- The water level rises.
- The water level does not change.
- Brian saves 2-liter soda bottles so that he can construct a raft and float out onto Haverford College's Duck Pond. If Brian has a mass of 80 kg, what minimum number of bottles is necessary to support him? The density of water is 1000 kg/m3, and 1000 L = 1 m3.
- 1600 bottles
- 800 bottles
- 200 bottles
- 40 bottles
- 4 bottles
- The water tower in the drawing above is drained by a pipe that extends to the ground. The amount of water in the top spherical portion of the tank is significantly greater than the amount of water in the supporting column.
- What is the absolute pressure at the position of the valve if the valve is closed, assuming that the top surface of the water at point P is at atmospheric pressure?
- Now the valve is opened; thus, the pressure at the valve is forced to be atmospheric pressure. What is the speed of the water past the valve?
- Assuming that the radius of the circular valve opening is 10 cm, find the volume flow rate out of the valve.
- Considering that virtually all of the water is originally contained in the top spherical portion of the tank, estimate the initial volume of water contained by the water tower. Explain your reasoning thoroughly.
- Estimate how long it would take to drain the tank completely using this single valve.
Information you may need: Density of water = 1000 kg/m3.
- A—The fluid is pushed into the mouth by the atmospheric pressure. Because the surface of a drink is open to the atmosphere, the surface is at atmospheric pressure, and the pressure in the mouth must be lower than atmospheric.
- A—Flow rate (volume of flow per second) is the area of a pipe times the speed of flow. So if we increase the volume of flow (choices B, C, and D), we increase the speed. By Bernoulli's equation, choice D also increases the fluid speed. Expanding the vein increases the area of the pipe, so if flow rate is constant, then the velocity must decrease.
- A—When the treasure is floating in the boat, it displaces an amount of water equal to its weight. When the treasure is on the lake bottom, it displaces much less water, because the lake bottom supports most of the weight that the buoyant force was previously supporting. Thus, the lake level drops.
- D—Since Brian will be floating in equilibrium, his weight must be equal to the buoyant force on him. The buoyant force is ρwaterVsubmergedg, and must equal Brian's weight of 800 N. Solving for Vsubmerged, we find he needs to displace 8/100 of a cubic meter. Converting to liters, he needs to displace 80 L of water, or 40 bottles. (We would suggest that he use, say, twice this many bottles—then the raft would float only half submerged, and he would stay drier.)
- If the valve is closed, we have a static column of water, and the problem reduces to one just like the example in the chapter. P = P0 + ρgh = 105 N/m2 + (1000 kg/m3)(10 N/kg)(15 m) = 250,000 N/m2, (atmospheric pressure is given on the constant sheet).
- Use Bernoulli's equation for a flowing fluid. Choose point P and the valve as our two positions. The pressure at both positions is now atmospheric, so the pressure terms go away. Choose the height of the valve to be zero. The speed of the water at the top is just about zero, too. So, four of the six terms in Bernoulli's equation cancel! The equation becomes ρgytop = 1/2 ρvbottom2. Solving, vbottom = 17 m/s.
- Flow rate is defined as Av. The cross-sectional area of the pipe is π (0.1 m)2 = 0.031 m2. (Don't forget to use meters, not centimeters!) So the volume flow rate is 0.53 m3/s.
- The volume of the spherical portion of the tank can be estimated as (4/3) πr3 (this equation is on the equation sheet), where the radius of the tank looks to be somewhere around 2 or 3 meters. Depending what actual radius you choose, this gives a volume of about 100 m3. The tank looks to be something like 3/4 full… So call it 70 m3. (Any correct reasoning that leads to a volume between, say, 30–300 m3 should be accepted.)
- The flow rate is 0.53 m3/s; we need to drain 70 m3. So, this will take 70/0.53 = 140 seconds, or about two minutes. (Again, your answer should be counted as correct if the reasoning is correct and the answer is consistent with part (d).)