AP Physics 2 Assignment – Fluids

 

Reading   Sections:  11.1 – 7; 12.1 – 3, Open Stax College Physics
                Sections:  10.1 – 9, Giancoli

 

Objectives/HW

 

 

The student will be able to:

HW:

1

Define and apply concepts of density, specific gravity, and pressure, and solve related problems.

1 – 6

2

Define, distinguish, and apply concepts of absolute, atmospheric, and gauge pressure, and solve related problems including application of Pascal’s principle and relationship with depth, density, and gravitational field.

7 – 14

3

Define and apply Archimedes principle and the concept of buoyancy and solve related problems.

15 – 19

4

Define and apply the concept of continuity of flow and conservation of matter and solve related problems.

20 – 21

5

State and apply Bernoulli’s principle and equation and the conservation of energy and solve related problems.

22 – 29

 

Homework Problems

 

1.      The core of the Sun is an incredible place – plasma at temperature 15 million kelvin and density 150 000 kg/m3 (and it is the site of fusion that transforms hydrogen to helium).  (a) If you could put this material in a 1.00 liter water bottle what would be the mass?  (b) In spite of being so dense the material is not solid or liquid – how is this possible?  Think of it as a compressed gas – what does this mean about the empty space in “ordinary gas”?

2.      Generally speaking materials made of elements that have a greater atomic number have greater density.  Explain this by incorporating the following.  With elements of greater and greater atomic number how is the nucleus changing in terms of mass and density and volume?  With elements of greater and greater atomic number how is the electron “cloud” of orbitals changing in terms of mass and density and volume?  Therefore overall density…

3.      The mean density of the human body is fairly close to that of water.  (a) Use this fact to determine the approximate volume of a person that weighs 170 pounds, which is equivalent to 756 N.  (b) The volume calculated is equivalent to a cube with sides of what length?

4.      The specific gravity of alcohol is 0.789.  Determine the resulting density of the following mixtures of water and alcohol:  (a) 100 mL of water and 100 mL of alcohol, and (b) 789 g of water and 789 g of alcohol.

5.      A particular hot tub is a cylindrical cedar barrel of mass 370 kg with height 1.2 m and diameter 2.1 m.  A homeowner wishes to put it on her wooden deck.  (a) What amount of force is required to move the empty tub up the stairs to the deck?  (b) What amount of normal force must the deck withstand when it is full of water?  (c) What pressure is this on the deck?

6.      Your diaphragm moves downward and you suck in a breath of air with volume 500 mL.  (a) Use the density of air to determine the mass of air inhaled.  (b) Explain how the diaphragm’s downward motion causes air to enter your lungs – is it being “pulled” in or “pushed” in?

7.      An Olympic size pool is 50.0 m long, 25.0 wide, and 2.00 m deep.  (a) Determine the mass and weight of water in the pool.  (b) Determine the force of the air pushing downward on the surface of the pool.  (c) Determine the force of the water pushing downward on the bottom of the pool.  (d) Determine the absolute pressure at the bottom of the pool.

8.      How do suction cups work?  Suppose a suction cup of radius R is stuck on a glass window pane.  When pressed onto the window, air is forced out of the space in between.  Friction between the suction cup and the glass can be modeled by the static coefficient of friction μ.  (a) Explain how atmospheric pressure P0 and friction act to prevent the suction cup from falling.  (b)  Derive an expression for the maximum mass m that can be supported by the suction cup in terms of R, g, P0, and μ.  (c) Calculate this amount for a cup of radius 2.0 cm and μ = 0.90.  (d) Explain why this will not be achievable if there is any amount of air left between the suction cup and the window pane.

9.      The cabin door on a certain airplane is 1.8 m by 0.80 m.  While in flight the cabin is pressurized to 0.80 atm for the sake of the passengers, but the atmospheric pressure at cruising altitude can be as low as 0.25 atm.  (a) Find the force on the exterior of the door while at altitude.  (b) Find the force on the interior of the door.  (c) If the door latch were to fail how much force would be required to hold the door closed by its handle, which is on the opposite side of the hinge?

10.  Hydraulic brake systems have pipes or “lines” that are filled with brake fluid and connect a wheel (or “slave”) cylinder with a master cylinder.  In one such system when force is applied to the piston of the master cylinder it causes a force 1.5 times greater to act on the piston in the wheel cylinder, (which is effectively the force pushing the brake pads against the drum or disc).  Suppose the wheel cylinder has diameter 18 mm.  (a) Determine the diameter of the master cylinder.  (b) If the master piston moves 2.0 mm, how much does the “slave” piston move?  (c) Explain or show mathematically how this problem illustrates conservation of matter (considering fluid amounts in the cylinders) and energy (considering work done).

11.  The Earth’s atmosphere extends into space getting thinner and thinner, with less and less density.  It is thought that about 99% of the atmosphere is within 50 km of the surface.  (a) Suppose hypothetically the density of the atmosphere were uniform 1.3 kg/m3 – determine how thick it would have to be and what mass it would have to be to cause the pressure at the surface to be 101.3 kPa.  (b) How is the real atmosphere different than the hypothetical?

12.  Your ears may “pop” when you go up on a mountain and the atmospheric pressure changes.  As you travel up the mountain the pressure in the fluids behind your eardrum may not change the same as the atmospheric pressure does.  (a) Assuming a relatively constant density of air, determine the change in pressure that occurs while ascending from Farragut to Newfound Gap – an elevation change of 1270 m (4170 ft).  (b) Find the force on the eardrum caused by the change in pressure, given its radius is 3.5 mm (and assuming the pressure in the fluid does not change). 

13.  The tires on a bicycle are inflated to a gauge pressure of 448 kPa (65 psi).  The total mass of bicycle and rider is 95.0 kg.  (a) Determine the area of each tire that is actually in contact with the road.  (b) What forces are acting on this “patch of rubber” that is touching the road?  Are any assumptions necessary to work this problem?  (c) If the width of the tire is 4.0 cm, what is the approximate length of the contact area (assumed rectangular).  (d) Describe the change in the “footprint” of each tire if the pressure is half as great – how does this affect the ride of the bicycle?

14.  Scuba divers ascending from a dive must follow careful procedures to prevent the “bends” (decompression sickness).  This occurs when nitrogen bubbles form in the bloodstream.  There are many variables involved, but ascent rates would typically not exceed 10 meters per minute through seawater (ρ = 1020 kg/m3).  (a) Determine the change in pressure on a diver that ascends this amount.  (b) The same thing can happen when ascending in the Earth’s atmosphere – how far approximately?  (Just describe this amount – can’t be calculated easily, why not?)  (c) Use Pascal’s principle and compressibility vs. incompressibility to explain why nitrogen dissolved in blood may become gas and form bubbles.

15.  An aluminum “jon boat” is a simple rectangular aluminum boat with a flat bottom.  One such boat has dimensions 3.0 m × 0.81 m × 0.17 m and mass 150 kg.  (a) How deep in the water does the boat sit when empty?  (b) If filled to the maximum suggested load of 180 kg how deep will it sit?  (c) Suppose the boat sinks (and no air is trapped in it).  Given that the density of aluminum is 2.70 g/cm3, find the force of buoyancy on the submerged boat.

16.  An object with a density ρ1 is submerged in a fluid of density ρ2 and then released.  (a) Derive an expression for its initial upward acceleration in terms of these variables and relevant constants.  Note:  if the object sinks, the expression should yield a negative result because of this.  (b) Explain why this expression is technically only valid at an instant when the speed is zero (but should yield an approximate result for low speeds).  (c) With a choice of fluids water, mercury, or alcohol and objects of gold or aluminum, determine the combination that gives the greatest acceleration and find the value and direction.  (Look up the densities.)

17.  A submerged 30.0 kg rock that was anchoring a boat is pulled up by a rope and it requires tension 185 N to hold it at rest or 192 N to raise it steadily at 0.500 m/s.  (a) Determine the force of buoyancy.  (b) Find the volume of the rock and its approximate diameter (assumed spherical).  (c) Suppose the rope breaks, estimate the terminal speed of the rock falling through the water assuming drag is proportional to speed squared.

18.  A certain helium balloon has radius 15 cm and density 0.19 kg/m3.  A mass of 10.0 grams is attached to the balloon and it is released.  Use density of air 1.3 kg/m3.  (a) Determine the initial acceleration.  (b) What is the greatest mass that could be held aloft by the balloon?

19.  Determine the force of buoyancy from the atmosphere (not water) on a kid of mass 60.0 kg by assuming the density of a human is about the same as water.  Explain why we usually can ignore this force when solving physics problems – especially when using only 3 or less significant digits.

20.  A garden hose has an inner diameter of 16 mm and water exits the end at speed 1.8 m/s.  (a) Determine the mass flow rate.  (b) Determine the time to fill a bucket of volume 20.0 L using this hose.

21.  A “wide” straw of diameter 1.0 cm is used to drink a cola of density 1.1 g/cm3 from a tumbler with radius 5.0 cm.  While sucking on the straw the flow rate is 3.0 mL/s.  (a) Determine the speed of cola moving through the straw.  (b) Determine the speed at which the level of the drink is dropping in the tumbler.  (c) If the top of the straw is 0.10 m above the level of the drink what is the pressure at the top relative to atmospheric?

22.  Gasoline of density 0.75 g/cm3 flows in a horizontal pipe of diameter 6.00 cm at a flow rate 1.4 L per minute.   Continuing on, the pipe narrows to a diameter half its original value.  (a) Determine the change in speed of the oil.  (b) Determine the change in pressure caused by the reduction in pipe size.  (c) Find the power associated with this flow.

23.  A Venturi tube arrangement allows the determination of flow rates through a pipe by simply measuring pressure difference as the fluid moves into a pipe of smaller diameter and then back into a pipe of the original diameter.  Suppose a pipe of diameter 2.5 cm carries water with gauge pressure 225.5 kPa. The water then flows into a pipe of diameter 1.0 cm, where the gauge pressure is 223.0 kPa.  (a) Find the speed of the water in the larger pipe.  (b) Find the volume flow rate in the pipe.  (c) Explain why the volume flow rate should be the same in either diameter pipe.

24.  One way to generate pressure for use in the plumbing of a building is to simply store water in an elevated tank – the height of the water is sometimes called the “pressure head”.  Suppose the water level in the tank is 8.00 m above a particular fixture in the building with a faucet of opening diameter 12 mm.  (a) Determine the gauge pressure at the faucet when it is closed.  (b) Determine the volume flow rate when it is open.  (c) Discuss how it makes sense to refer to the elevation as the pressure head.  e.g. What type of relation is there between height and pressure?  How does it help facilitate the distribution of water in a system of pipes?

25.  Water enters at the base of a tall building in a pipe of diameter 5.00 cm.  The water flows through various pipes and fittings and emerges from a faucet of diameter 1.50 cm at a sink on the sixth floor, 25.0 m up.  It takes 4.50 s to fill a 1.80 L jug at the faucet.  Assume no other taps are open in the building.  (a) Determine the speed of water exiting the faucet.  (b) Find the gauge pressure in the pipe at the base of the building.  (c) How would things change if other fixtures in the building are being used to obtain water at the same time?  Would this solution still be valid?   

26.  When a smooth “coherent” stream of fluid falls straight downward through air (like water from a kitchen faucet), the diameter of the fluid gets more and more narrow.  Suppose the stream has initial speed v0, initial diameter d0 and then falls a distance h.  (a) Derive an expression for the final diameter d of the stream in terms of these variables and relevant constants.  (b) Does the density of the stream matter?  How is this shown in your derivation?  (c) Discuss how this solution requires assumptions of laminar flow and zero viscosity – how might an actual stream differ from this ideal one?       (Just for fun:  how could you take your equation and use a graphing calculator to “draw” a picture of the stream?  Try it!)

27.  A drinking fountain has a pump that takes water with essentially zero speed and moves it to the nozzle of diameter 0.60 cm, where it emerges and forms an arc of water 0.10 m high and 0.20 m wide.  (a) Based on the nozzle size and arc of water what is the flow rate in mL/s?  (b) What gauge pressure must the pump produce at the base of the fountain, which is 1.2 m below the nozzle?  (c) Determine the power of the pump.  (d) Because the viscosity of water was ignored in your solution the actual power pump required would compare how to your answer?  Explain in terms of conservation of energy.

28.  Wind with speed 35 m/s blows over the top of the flat roof of a one story rectangular building with dimensions (h, w, l) =  4.0 m, 20.0 m, 25.0 m.   Estimate the force (and direction) on the building’s roof according to Bernoulli’s principle.  Use a density of air 1.2 kg/m3.

29.  The wings of a particular type of airplane generates lift of an amount 11 kN when flying at airspeed 63 m/s.  The effective area of the wings is 16 m2.  (a) Determine the pressure difference of the top of the wing versus the bottom of the wing.  (b) Estimate the airspeed of the air passing over the top of the wing assuming the bottom of the wing is horizontal.

 

 

 

Selected Answers

 

1.3 mm

15 mm

2.6 cm

6.2 cm
14 cm
0.43 m (or 16 in)

8.0 km, 5.3 × 1018 kg

approx. 50 km

10.4 cm2
0.0111 m3; 0.277 m
0.077 m3 (or 77 L)

0.65 g

16 g

12 kg

150 kg

2.50 × 106 kg, 24.5 MN
0.895 g/mL (895 kg/m3)
0.882 g/mL (882 kg/m3)

55 s

0.038 cm/s
2.5 cm/s
3.8 cm/s
0.36 m/s
2.26 m/s
2.57 m/s

72 m/s

4.4 m/s2 up

4g

44 mL/s
0.18 L/s

1.4 L/s

0.36 kg/s

0.62 N outward

0.764 N upward

109 N
540 N, up

3600 N
36 kN
40 kN (about 9000 pounds of force!)

44 kN
120 kN
370 kN upward

127 MN
151 MN
–16 kPa
– 1100 Pa

–690 Pa
–0.38 Pa
12.8 kPa

13 kPa
78.4 kPa
1­00 kPa
121 kPa

248 kPa

8.9 μW

0.57 W