# University Physics 226N/231N Old Dominion University. Ch 12: Finish Fluid Mechanics Exam Review

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1 University Physics 226N/231N Old Dominion University Ch 12: Finish Fluid Mechanics Exam Review Dr. Todd Satogata (ODU/Jefferson Lab) Wednesday, November 16, 2016 Reminder: The Third Midterm will be Mon Nov Homework Notebooks are also due when you turn in your exam! Happy Birthday to Maggie Gyllenhaal, Diana Krall, Andy Dalton, Allison Crowe, and Gene Amdahl! Happy Fast Food Day and Have A Party With Your Bear Day! Please set your cell phones to vibrate or silent mode. Thanks! Prof. Satogata / Fall 2016 ODU University Physics 226N/231N 1

2 Midterm #3 Midterm #3: Rotational kinematics and rotational motion Including conservation of energy and angular momentum Rotational equilibrium Stress/Strain/Shear Fluid mechanics I will prepare a handout with all cheat sheat materials Includes tables of material constants, moments of inertia, etc I ll also post something like a sample midterm later today Homework notebook Your homework notebook is due when you turn in your test Hopefully you find it useful while you are taking the test It will be graded on a 3- or 4-point scale and counts for about 5% of your final grade Prof. Satogata / Fall 2016 ODU University Physics 226N/231N 2

3 (Cheat Sheet: Linear and Angular Summary Comparison) x = r v = r! a = r r movie equation equation that doesn t involve time Forces Torques: Newton s 2 nd Law: Momentum: ~ = ~r ~ F ~F net = m~a = d~p/dt ~ net = I~ = d L/dt ~ ~p = m~v L ~ = I~! Kinetic Energy: (inertial) Mass: KE = 1 2 mv2 KE rot = 1 2 I!2 I = X mir 2 m Moment of Inertia: Parallel Axis Theorem: I = I cm + Md 2 i Prof. Satogata / Fall 2016 ODU University Physics 226N/231N 3

4 Tensile stress: Tensile strain: Review: Elastic Moduli Definitions Young s modulus: Pressure or volume stress: Volume strain: Bulk modulus: F? /A l/l 0 Y F?/A l/l 0 V/V 0 p = F? /A B p/( V/V 0 ) Shear stress: Shear strain: Shear modulus: F k /A x/h S F k/a ( x/h) All moduli are material properties All moduli assume elastic deformation All moduli have units of pressure [Pa] x h Prof. Satogata / Fall 2016 ODU University Physics 226N/231N 4

5 Recall force exerted by a spring: F = So force to stretch a spring in elastic limit is F = kx kx Review: The Limits of Hooke s Law Real spring Hooke s Law: F=kx (because of Newton s Third law) Tension Beyond the limits of Hooke s law, extra force must be applied to deform the spring in a non-elastic way. The spring, once deformed by being pulled past its elastic limit, will not return to its initial rest position. Prof. Satogata / Fall 2016 ODU University Physics 226N/231N 5

6 Review: Compression and Tension In many situations, objects can experience both tensile and compressive stresses at the same time. For example, a horizontal beam supported at each end sags under its own weight. Which engineering materials are stronger under compression? Which are stronger under tension? Prof. Satogata / Fall 2016 ODU University Physics 226N/231N 6

7 Chapter 12: Fluid Mechanics (Statics and Dynamics) We consider fluids that are nearly incompressible Fluids are materials that cannot sustain a shear force They basically have zero shear modulus S We assume here that fluid volume doesn t change under pressure They basically have infinite bulk modulus B We saw on Monday this is a reasonable approximation for water Gases are fluids that cannot maintain a well-defined surface Liquids are fluids that can maintain a well-defined surface This is equivalent to assuming that the fluid has nearly constant density m V Density is a material property for solids and incompressible liquids For gases and compressible liquids, it can vary by quite a lot Prof. Satogata / Fall 2016 ODU University Physics 226N/231N 7

8 Review: Pressure In A Fluid A fluid exerts a force perpendicular to any surface in contact with it, such as a container wall or a body immersed in the fluid. This is an example of Newton s Third Law Pressure p F? A Pressure in a small volume of fluid is constant over all surfaces of the volume Pressure certainly varies with height in a fluid column though Pressure is a scalar! (Always perpendicular to the area A) Prof. Satogata / Fall 2016 ODU University Physics 226N/231N 8

9 Review Example: Fluid Column with External Pressure h A p 0 In atmosphere, there is additional air pressure p 0 acting at the top of the liquid column Mass of liquid in column is m = ha So the total pressure at a depth h below the surface of the fluid is p = p 0 + gh F = p 0 A + mg = p 0 A + hag p = F/A Prof. Satogata / Fall 2016 ODU University Physics 226N/231N 9

10 Example Typical household water pressure is 60 psi in first floor plumbing. How much pressure is lost at a second floor faucet that is 14 feet higher than the first floor faucet? p p 0 = gh = (10 3 kg/m 3 )(9.8 m/s 2 )(14 feet) p p 0 =6.06 psi = Pa You lose about 10% of your water pressure per floor without additional pumps. Prof. Satogata / Fall 2016 ODU University Physics 226N/231N 10

11 Pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and the walls of the containing vessel: p = F 1 A 1 = F 2 A 2 Why hydraulics work Acts like a liquid lever Energy (force times distance) is conserved Review: Pascal s law The beer bottle trick is cavitation shock, NOT Pascal s Law: Prof. Satogata / Fall 2016 ODU University Physics 226N/231N 11

12 Pressure at depth in a fluid Each fluid column has the same height, no matter what its shape. The pressure at any depth h within each column is the same: p = p 0 + gh This is a statics problem, where all forces balance on every small volume of fluid. Prof. Satogata / Fall 2016 ODU University Physics 226N/231N 12

13 Archimedes s Principle Prof. Satogata / Fall 2016 ODU University Physics 226N/231N 13

14 Archimedes s Principle A body immersed in water seems to weigh less than when it is in air. When the body is less dense than the fluid, it floats. The human body usually floats in water, and a helium-filled balloon floats in air. These are examples of buoyancy, a phenomenon described by Archimedes s principle: When a body is completely or partially immersed in a fluid, the fluid exerts an upward force on the body equal to the weight of the fluid displaced by the body. Weight (air) Weight (submerged) = (Fluid density) * (Object volume) Prof. Satogata / Fall 2016 ODU University Physics 226N/231N 14

15 Example A cylindrical m=12 kg wooden log of diameter 20 cm and length 50 cm is tied to a heavy anchor and dropped over the side of a boat into a moderately deep fresh water lake. What is the tension in the rope when the anchor comes to rest? F b Log mg T F net =0=F b mg T T = F b mg F b = gv log water V log = m 3 F b = 154 N T = 154 N 118 N = 36 N Prof. Satogata / Fall 2016 ODU University Physics 226N/231N 15

16 Example ice = 917 kg/m 3 water = 1000 kg/m 3 F b mg Prof. Satogata / Fall 2016 ODU University Physics 226N/231N 16

17 Example ice = 917 kg/m 3 water = 1000 kg/m 3 F b F net = F b mg =0 ) F b = mg V ice =(2.0 m) 2 (1.0 m) = 4.0 m 3 m = V ice ice = 3668 kg F b = gv submerged water = mg ) V submerged = m water =3.7 m 3 mg h submerged = V submerged (2.0 m) 2 =0.92 m h exposed =1.0 m 0.92 m = 0.08 m With the woman, h submerged goes up by m Prof. Satogata / Fall 2016 ODU University Physics 226N/231N 17

18 The figure at the right shows a flow tube with changing crosssectional area. The continuity equation for an incompressible fluid is A 1 v 1 = A 2 v 2 The Continuity Equation The volume flow rate is dv dt = Av Describes how incompressible fluid flows through area A with velocity v This is a mathematical way of saying fluid volume is conserved : what goes in must come out Prof. Satogata / Fall 2016 ODU University Physics 226N/231N 18

19 Example Your kitchen faucet produces a volume flow rate of 2.0 gallons/minute through a faucet nozzle of area 0.10 in 2. What is the velocity of water exiting out the nozzle? dv dt = Av ) v = dv/dt A (2.0 gal/min) = (0.1 in 2 ) =1.96 m/s =4.4 mi/hr What is the water pressure? p = p p 0 = 1 1 V 2 mv2 = 1 2 v2 = 10 3 kg/m 3 p = 1900 Pa above air pressure Prof. Satogata / Fall 2016 ODU University Physics 226N/231N 19

20 The continuity equation helps explain the shape of a stream of honey poured from a spoon. A 1 v 1 = A 2 v 2 As velocity v increases, area A must decrease to keep the product Av constant The Continuity Equation Prof. Satogata / Fall 2016 ODU University Physics 226N/231N 20

21 Bernoulli s equation is: Bernoulli s equation Work/volume Potential energy/volume p 1 + gy v2 1 = p 2 + gy v2 2 Kinetic energy/volume It is due to the fact that the work done on a unit volume of fluid by the surrounding fluid is equal to the sum of the changes in kinetic and potential energies per unit volume that occur during the flow. It is really just a fancy conservation of energy statement for incompressible flow Pearson Prof. Satogata / Fall 2016 ODU University Physics 226N/231N 21

22 The Venturi Meter Bernoulli applied at a constant height gives a relationship between the fluid velocity and pressure at any point: p v2 1 = p v2 2 Higher velocity means lower pressure and vice-versa The pressure is less at point 2 because the fluid velocity is greater due to the constriction. Again, higher velocity means lower pressure. Prof. Satogata / Fall 2016 ODU University Physics 226N/231N 22

23 Lift on an airplane wing Bernoulli s principle helps to explain how airplanes fly Pearson Prof. Satogata / Fall 2016 ODU University Physics 226N/231N 23

24 Prof. Satogata / Fall 2016 ODU University Physics 226N/231N 24

25 Prof. Satogata / Fall 2016 ODU University Physics 226N/231N 25

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