Physics 203 – College Physics I Department of Physics – The Citadel Physics 203 College Physics I Fall 2012 S. A. Yost Chapter 10 - Part 3 Chapter 11 – Part 1 Fluid Dynamics Simple Harmonic Motion

Physics 203 – College Physics I Department of Physics – The Citadel Exam 3 Average: 32 High: 61.5

Physics 203 – College Physics I Department of Physics – The Citadel Announcements Problem set 10B is due Thursday. We will start discussing Ch. 11 late today or next time: Read sections 1 – 4 and 7 – 9 for next time. Topics: simple harmonic motion, intro to waves. Next Tuesday: Ch. 11, sec. 11 – 13 and Ch. 12, sec. 1 – 4 & 7. If possible, we may finish Ch. 11 next time, leaving Ch. 12 for next week. The end of Ch. 11 and Ch. 12 are related: waves and sound.

Physics 203 – College Physics I Department of Physics – The Citadel Volume Rate of Flow The volume rate of flow of an incompressible fluid is the same throughout a pipe. Q = Av = constant v A

Physics 203 – College Physics I Department of Physics – The Citadel Bernoulli Principle Bernoulli’s Principle is an expression of energy conservation: PV + ½ m v 2 + mgh = constant. Work + kinetic energy + potential energy = constant. P1P1 P2P2 v1v1 v 2 h

Physics 203 – College Physics I Department of Physics – The Citadel Bernoulli Principle P + ½  v 2 +  gh = constant. This is a consequence of the work- energy theorem, assuming the only work is done by gravity and pressure. This neglects friction in the fluid – viscosity. It also assumes smooth “laminar” flow – no turbulance. P1P1 P2P2 v1v1 v 2 h

Physics 203 – College Physics I Department of Physics – The Citadel Question 1. A fluid flows through the pipe shown. In which section is the flow velocity the greatest? Selections: A B C D The same B A C

Physics 203 – College Physics I Department of Physics – The Citadel Answer The volume rate of flow Q = vA is constant for an incompressible fluid. The fluid moves fastest where the pipe is narrowest, section B. (It moves slowest in section C.) B A C

Physics 203 – College Physics I Department of Physics – The Citadel Question 2. In which section is the pressure of the fluid the greatest? Selections: A B C D The same B A C

Physics 203 – College Physics I Department of Physics – The Citadel Answer This is Bernoulli’s principle: The pressure in a fluid decreases when the flow velocity increases. The fluid moves most slowly at C, so the pressure is highest there (and lowest at B). B A C

Physics 203 – College Physics I Department of Physics – The Citadel Water Tower and Fountain A water tower feeds a fountain, which shoots water straight up in the air. How fast does the water leave the fountain? Assume the top of the water is a height h = 55 m above the fountain. h v

Physics 203 – College Physics I Department of Physics – The Citadel Water Tower and Fountain We’ll assume the tank is big, so the top of the water stays fixed: h 1 = 55 m, v 1 = 0, P 1 = 0 h v gauge pressure 1

Physics 203 – College Physics I Department of Physics – The Citadel Water Tower and Fountain At the fountain, h 2 = 0 v 2 = v What is P 2 ? h v 1 2 unknown

Physics 203 – College Physics I Department of Physics – The Citadel Water Tower and Fountain Careful! This is not hydrostatics. If the fountain were turned off, the pressure would be P 2 =  gh = 1000 kg/m 3 x 9.8 m/s 2 x 55 m = 5.4 x 10 5 N/m 2. h 1 2 no flow!

Physics 203 – College Physics I Department of Physics – The Citadel Water Tower and Fountain When the fountain is flowing, this changes! The pressure just outside the pipe is P 2 = 0, normal atmospheric pressure. h v 1 2 P2P2

Physics 203 – College Physics I Department of Physics – The Citadel Water Tower and Fountain The velocity is given by Bernoulli’s equation with P 1 = P 2 = 0 h 1 = h, h 2 = 0 v 2 = 0, v 1 = v h v 1 2 P2P2

Physics 203 – College Physics I Department of Physics – The Citadel Water Tower and Fountain The only terms remaining are ½  v 2 =  gh The result is the same as if the water had fallen from the top of the tower: v = √ 2gh = 33 m/s. h v 1 2

Physics 203 – College Physics I Department of Physics – The Citadel Water Tower and Fountain What is the volume rate of flow if the pipe has diameter 1 cm? Q = Av A =  (0.5 cm) 2 = cm 2 h v 1 2

Physics 203 – College Physics I Department of Physics – The Citadel Water Tower and Fountain Q = Av A = cm 2 v = 33 m/s = 3300 cm/s Q = 2600 cm 3 /s = 2600 mL /s = 2.6 L/s h v 1 2

Physics 203 – College Physics I Department of Physics – The Citadel Water Tower and Fountain How high does the water rise from the fountain? Bernoulli’s equation between points 1 and 2: P 1 = P 2 = 0, v 1 = v 2 = 0 implies  gh 1 =  gh 2. h v 1 2

Physics 203 – College Physics I Department of Physics – The Citadel Water Tower and Fountain The water rises to the height of the tower. This assumes energy conservation: No friction (viscosity or air resistance) or turbulence is considered. h v 1 2

Physics 203 – College Physics I Department of Physics – The Citadel Hooke’s Law From chapter 6: Hooke’s Law describes a linear restoring force when a spring is displaced from its equilibrium position. Elastic potential energy: U = ½ kx 2 x F = -k x

Physics 203 – College Physics I Department of Physics – The Citadel Simple Harmonic Motion This is called simple harmonic motion. When a mass oscillates under a linear restoring force F =  kx, the acceleration is always opposite the displacement from equilibrium, but proportional to it. a = F/m =  (k/m) x.