Physics 123 Unit #1 Review

Transcription

1 Physics 123 Unit #1 Review I. Definitions & Facts Density Specific gravity (= material / water) Pressure Atmosphere, bar, Pascal Barometer Streamline, laminar flow Turbulence Gauge pressure II. Mathematics & Tools Fluids obey the equations of motion. III. Basic Concepts An ideal fluid: incompressible, laminar, irrotational, no viscosity. Pascal s principle: A pressure applied at any point in the fluid at rest is spread throughout the entire fluid. Archimedes principle: The buoyant force equals the weight of the displaced liquid. Fluid in the air is always at atmospheric pressure. Bernoulli s equation is a result of conservation of energy. IV. Equations to Memorize Continuity equation: A vflow rate, A! v 1 A 2 v 2 Bernoulli s equation: P gh½ v 2 constant

2 Physics 123 Unit 1 Reading and Homework Assignments 15.1 Understand density and pressure. The device in Figure 15.2 only works when fluid is not flowing. To measure pressure in a flow, we have to be careful to separate the effects of the stream s momentum from the pressure. A better device would have a cap on it with little holes to allow the fluid to enter. The differential form Eq is not important to use, but you should understand why it s mentioned. Many units of pressure are in use. Note that 1 atmosphere = Pa, 1 bar = Pa. In vacuum work, we use 1 torr = 1 mm Hg = 1/760 atm Especially important are Eq. 15.4, Pascal s Principle, and the idea of hydraulics. Eq only works for static fluids that can t be compressed, so it does not work for the air. Work Example 15.2 (without looking at the solution). If you like a challenge, try Know gauge pressure. We ll see more of the barometer later. What is the pressure at the surface level of the Hg in the air to the side of the barometer tube? In the air just under the barometer tube? 15.4 Know Archimedes principle well (top of p. 466). Understand why it is true. Do Quick Quizzes 15.6, 15.7 (same), and 15.8 (falls). Understand Examples 15.5, and Know what an ideal fluid is. Memorize the four items Note that what we say hereafter only applies to ideal fluids and tubes of flow. (15.7) is very important. Recognize that Av is the volume flow per unit time Bernoulli s equation is a consequence of energy conservation. Understand its derivation and know it (15.9) well! It is important to remember that P + ½ v 2 + gy is NOT the total pressure (P is that). It is related to total energy/work per unit volume and has the name total head in engineering. Understand all three Examples. Try doing the trick in These are a few of many applications. You should understand all of these.

3 Homework Questions: Questions 15.5 (think of no water under the fish), 15.6, 15.7, Essential Problems: Problems 15.2, 15.6, 15.8, Application Problems: Problems 15.19, Computational Problems: A. The area to the right of the line x=a in the circle to the right to the right is A 1 2 R 2 R 2 sin 1 a R a R 2 a 2 A cylindrical log of radius 10 cm is placed in water. If the density of the wood is 0.5 g/ cm 3, how much of the log is above the surface of the water? Make a graph of this height as a function of the density of the wood. Contextual Problem: B. A drink cup is placed on the seat of a car. As the car undergoes a uniform acceleration, a, in the forward direction, what angle does the surface of the liquid make to the horizontal? Find an expression for the pressure as a function of the distance, d, measured perpendicularly from the surface of the liquid. (Remember that no experiment can differentiate between a stationary elevator in a gravitational field and an elevator accelerating upward with acceleration g. This is called the Principle of Equivalence and is the basis of Einstein s General Theory of Relativity.) Homework Questions: Question 15.18, A. Why does the water level in a toilet go up and down on a windy day? (A genuine question from one of my neighbors...) Essential Problems: Problems B. A tank 6 m high is drained by a hose extending horizontally from the bottom of the tank. What is the velocity of the water coming out the hose? What is the pressure at the bottom of the tank? What is the pressure of the water in the hose? Application Problems: Problems 15.35, 15.38, (the vapor pressure is nearly zero) Computational Problems: C. A pipe runs horizontally along the ground. The pipe gradually tapers from a radius of 5 cm to a radius of 2 cm. Water comes out of the pipe at a speed of 1.25 m/s. Plot the speed of the water within the nozzle as a function of the nozzle radius. Plot the pressure of the water also as a function of the radius of the pipe.

4 Contextual Problems: D. How long does it take to drain a half-gallon milk carton (filled with water) if we punch a hole in the bottom with a pencil? You have to model the flow with a differential equation. To do this, you will need to determine the flow rate when the water is at an arbitrary level, y, in the container. E. A hose comes horizontally outward from the bottom of a tank of water. When real water flows through the hose, energy is lost in friction between the water and the hose. Assume that the energy lost by a small volume of water, V, is proportional to the surface area of the water in contact with the hose and is also proportional to the length of the hose. (a) Find the surface area, A, of the water in terms of V. (b) By appropriately modifying Bernoulli s Equation, show that the velocity of the water coming out the hose is: v 2gh r where h is the height of the water, is a constant, is the length of the hose, and r is the radius of the hose.

5 Physics 123 Section 2 Sample Exam #1 Assume all fluids are ideal. Possibly useful information: The density of water is 1000 kg / m 3. 1 bar = Pa. Use g = 9.80 m / s 2. Terminology 1. (5 points) Density 2. (5 points) Barometer 3. (5 points) What is the SI unit of pressure? 4. (5 points) What is streamline flow?

6 Conceptual Applications 5. (5 points) Bernoulli s Equation, as we derived it, holds only for an ideal fluid. An ideal fluid must have each of the following characteristics except: A. it must have streamline flow. B. it must be incompressible. C. it must have no viscosity. D. it must not have rotational motion. E. All of the above are characteristics of an ideal fluid. 6. (5 points) An open-topped tank has water flowing out through a hole in the bottom, as illustrated. The height of the water level above the ground is h + H = 2.00 m. In order to increase the flow rate through the hole, you could: A. Increase H, leaving H + h constant. B. Increase H, leaving h constant. C. Increase the size of hole to allow more water to flow through the hole. D. Decrease the size of the hole in order to increase the velocity of the water out of the hole. E. None of the above. 7. (5 points) A hydraulic lift is best understood by applying A. Pascal s Principle. B. Euler s Angles. C. Bernoulli s Equation. D. The continuity equation. E. None of the above.

7 Equations and Tools For each of the following equations: A) Tell what each symbol means B) Write a short problem which can be solved with this equation The problem must be a word problem, not just a=5, b=4, what is c? 8. (5 points) P gh½ v 2 P 0 9. (5 points) P 1 gy 1 P 2 gy (5 points) Av

8 Problems We can usually evaluate how well you understand the physics of a problem by the way you attempt to solve the problem. Occasionally, however, you may be unable to work a problem or a part of a problem even if you do understand the physics. If you cannot work a problem, please do the following so that you may receive partial credit: 1. Discuss the basic physics concepts involved. Include enough detail that the grader will know if you understand the physics. 2. Write down the equation(s) you need to solve the problem. Explain the meaning of each symbol in the equations. 3. Explain as much of the problem as you understand. 4. Explain why you feel you have difficulties with the problem. 11. (25 points) A block of iron of dimensions 50.0 cm 50.0 cm 10.0 cm is placed in a large tank of mercury. The specific gravity of iron is 7.86 and the specific gravity of mercury is (a) How much of the iron will float above the mercury? (b) What is the maximum buoyant force this block would experience in mercury?

9 12. (25 points) A tank 1.50 m in height is filled with water to a level of 1.32 m. At the bottom of the tank a hose is attached. The hose goes up at an angle of 55.0with respect to the ground. The end of the hose at a height of 58.0 cm above the bottom of the tank. Atmospheric pressure is 1.00 bar m 58.0 cm (a) Find the velocity of the water coming out of the hose. (b) Find the velocity of the water inside the hose at the bottom end of the hose (next to the tank). (c) Find the pressure of the water inside the hose at the bottom end of the hose (next to the tank). (d) Is the pressure you found in part (c) the same as the pressure at the bottom of the tank (not inside the hose)? Briefly explain.

10 13. (25 points) A person is drinking water from a cylindrical tumbler by using a straw. The dimensions of the system are shown in the diagram to the right. The person maintains a gauge pressure of 3000 Pa in her mouth. The diameter of the straw is 5.00 mm. (a) What is the flow rate of the water into the person s mouth? (Assume the straw is narrow enough that you may ignore the velocity of the water in the tumbler.) (18 points) (b) If the person keeps the pressure in her mouth constant while drinking, what will happen to the flow rate as the level of water in the tumbler drops? (Give a descriptive answer, you do not need to calculate the flow rate as a function of water level.) (7 points)

11 14. (25 points) A horizontal pipe consists of three sections as shown in the diagram to the right. The right section, C, is open to the air where water flows out at a speed of m/s. (The speed, of course, is the same just inside and just outside the pipe.) Section A is not open to the air. The diameter of sections A and C is 2.00 cm. The diameter of section B is 4.00 cm. (a) What is the velocity of the water in section B? (8 points) (b) What is the pressure of the water in section B? (9 points) (c) What is the pressure of the water in section A? (8 points)

12 Answers. Terminology 1. Mass per unit volume of a material. 2. A device used to measure air pressure. 3. Pascal ( or newton / m 2. ) 4. Flow that is not turbulent (or chaotic). Flow where fluid at a given point will always follow the same path. Conceptual Applications 5. E. 6. C. 7.A. Equations and Tools 8. P pressure density of the fluid g gravitational acceleration h height v velocity of the flow P atmospheric pressure If the level of water in a closed tank is 2.00 m above the ground and water leaving a hose at the ground level exits with a speed of 4.0 m/s, what is the air pressure in the top of the tank? 9. (5 points) P pressure density of the fluid g gravitational acceleration y height, vertical position Water flows through a hose down a slope. The bottom end of the hose is open to air. If the top end of the hose is 1.24 m above the bottom end, what is the pressure at the top of the hose? 10. (5 points) A v flow rate cross sectional area of a pipe velocity of the flow Water flows from a 6" pipe at 4.0 m/s. How many liters of water flow through the tank in one minute?

13 Problems 11. (a) We know that if the iron floats, the buoyant force must be equal in magnitude to the weight of the iron. Thus: F B W Hg V displaced g Fe V Fe g Hg Ah Fe AH where A is the crosssectional area h is the height of the block below the surface H is the total height of the block water is 4.19 cm. h Fe Hg h5.81 cm Thus the amount out of the (b) The maximum buoyant force occurs when the block is totally submerged. 12. (25 points) F B,max Hg V Fe g kg/m m m m 9.80 m/s N (a) We apply Bernoulli s Equation to the water surface in the tank and the end of the hose: Total head at the surface: Total head at the hose opening: v open 2g (h top h open ) P 0 gh top P 0 gh open ½ v 2 open m/s 3.81 m/s (b) By the continuity equation, the velocity of the water in the hose must be the same as in part (a), 3.81 m/s. ( c) Total head at the bottom of the hose: P bot ½ v 2 open P bot ½ v 2 open P 0 ½ v 2 open gh open P bot P 0 gh open Pa Pa 106 kpa (d) No, the pressure of the hose is less than the pressure in the bottom of the tank. Since the height of the water in the bottom of the tank is the same as in the bottom of the hose, and the velocity of the water increases as it enters the hose, the pressure must drop to keep the total head constant.

14 13. (a) First we find the velocity of the water by using Bernoulli s Equation: Total head at the surface: P 0 gh top Total head at the straw opening: P Pa gh s ½ v 2 ½ v Pa g( h s h top ) v open ( ) m/s m/s sectional area of he straw, the flow rate is: Letting A denote the cross A 2.06 m/s m m/s m 3 /s (b) The difference in height between the mouth and the surface gets larger as so the flow rate will decrease. (It is harder to pull the water up a larger distance.) 14. (a) We apply the continuity equation: A C v c A B v B v B A C v A C 1 B 4 v C 3.00 m/s (b) Note that the velocity of the water and the height of the water are the same at C and just after the water leaves the hose. Therefore, the pressure at point A must be P 0. By Bernoulli s Equation: P B ½ v B P 0 ½ v 2 C P B P 0 ½ ( v 2 C v 2 B ) Pa½ 1000 (1449.0) Pa 168 kpa velocity of the water in A must be the same as in C. Hence: ( c) By the continuity equation, the Total head at A: Total head at C: P A ½ v 2 C P 0 ½ v 2 C P A P 0

15 Physics 123 Unit Summary Unit #1 ID Number: Unit Score: (sum all boxes below): 1. Homework problems completed on time 10 = Homework problems completed late 8 = Sum of previous lines 18 problems =, average score per problem Homework score: Average score per problem 2.5 = ( Maximum = 25 ) 2. Work Item Hours Item Hours Total Hours (expected = 7, maximum = 12): Work Score: Total Hours 5.83 = ( Maximum = 70 ) 3. Reading Checks Reading score: 10 # correct = ( Maximum = 10 ) 3 4. Quizzes Quiz score: 10 # taken = ( Maximum = 10 ) 3 5. Walk-in Labs Lab score: = 15 #completed = ( Maximum = 15 )