14-6 The Equation of Continuity
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1 14-6 The Equation of Continuity 14-6 The Equation of Continuity Motion of rea fuids is compicated and poory understood (e.g., turbuence) We discuss motion of an idea fuid 1. Steady fow: Laminar fow, the veocity of the moving fuid at any fixed point does not change with time 2. Incompressibe fow: The idea fuid density has a constant, uniform vaue 3. Nonviscous fow: Viscosity is, roughy, resistance to fow, fuid anaog of friction. No resistive force here 4. Irrotationa fow: May fow in a circe, but a dust grain suspended in the fuid wi not rotate about com 2014 John Wiey & Sons, Inc. A rights reserved.
2 14-6 The Equation of Continuity 14-6 The Equation of Continuity Visuaize fuid fow by adding a tracer Each bit of tracer (see figure 14-13) foows a streamine A streamine is the path a tiny eement of fuid foows Veocity is tangent to streamines, so they can never intersect (then 1 point woud experience 2 veocities) Figure John Wiey & Sons, Inc. A rights reserved. Figure 14-14
3 14-7 Bernoui's Equation 14-7 Bernoui's Equation Figure represents a tube through which an idea fuid fows Appying the conservation of energy to the equa voumes of input and output fuid: Eq. (14-28) The ½ρv 2 term is caed the fuid's kinetic energy density Figure John Wiey & Sons, Inc. A rights reserved.
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5 14-7 Bernoui's Equation 14-7 Bernoui's Equation Equivaent to Eq , we can write: Eq. (14-29) These are both forms of Bernoui's Equation Appying this for a fuid at rest we find Eq Appying this for fow through a horizonta pipe: Eq. (14-30) 2014 John Wiey & Sons, Inc. A rights reserved.
6 14-7 Bernoui's Equation 14-7 Bernoui's Equation Answer: (a) a the same voume fow rate (b) 1, 2 & 3, 4 (c) 4, 3, 2, John Wiey & Sons, Inc. A rights reserved.
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9 2014 John Wiey & Sons, Inc. A rights reserved.
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11 2014 John Wiey & Sons, Inc. A rights reserved.
12 14 Summary 14 Summary Density Eq. (14-2) Fuid Pressure A substance that can fow Can exert a force perpendicuar to its surface Eq. (14-4) Pressure Variation with Height and Depth Eq. (14-8) Pasca's Principe A change in pressure appied to an encosed fuid is transmitted undiminished to every portion of the fuid and to the was of the containing vesse 2014 John Wiey & Sons, Inc. A rights reserved.
13 14 Summary 14 Summary Archimedes' Principe Fow of Idea Fuids Eq. (14-16) Eq. (14-24) Eq. (14-19) Eq. (14-25) Bernoui's Equation Eq. (14-29) 2014 John Wiey & Sons, Inc. A rights reserved.
14 Chapter 15 Osciations Copyright 2014 John Wiey & Sons, Inc. A rights reserved.
15 15-1 Simpe Harmonic Motion 15-1 Simpe Harmonic Motion The frequency of an osciation is the number of times per second that it competes a fu osciation (cyce) Unit of hertz: 1 Hz = 1 osciation per second The time in seconds for one fu cyce is the period Eq. (15-2) Any motion that repeats reguary is caed periodic Simpe harmonic motion is periodic motion that is a sinusoida function of time Eq. (15-3) 2014 John Wiey & Sons, Inc. A rights reserved.
16 15-1 Simpe Harmonic Motion 15-1 Simpe Harmonic Motion The vaue written x m is how far the partice moves in either direction: the ampitude The argument of the cosine is the phase The constant φ is caed the phase ange or phase constant It adjusts for the initia conditions of motion at t = 0 The anguar frequency is written ω Figure John Wiey & Sons, Inc. A rights reserved.
17 15-1 Simpe Harmonic Motion 15-1 Simpe Harmonic Motion The anguar frequency has the vaue: Eq. (15-5) John Wiey & Sons, Inc. A rights reserved. Figure 15-5
18 2014 John Wiey & Sons, Inc. A rights reserved.
19 15-1 Simpe Harmonic Motion 15-1 Simpe Harmonic Motion We can appy Newton's second aw Eq. (15-9)18 Reating this to Hooke's aw we see the simiarity Linear simpe harmonic osciation (F is proportiona to x to the first power) gives: Eq. (15-12)18 Eq. (15-13) John Wiey & Sons, Inc. A rights reserved.
20 15-2 Energy in Simpe Harmonic Motion 15-2 Energy in Simpe Harmonic Motion Learning Objectives For a spring-bock osciator, cacuate the kinetic energy and eastic potentia energy at any given time Appy the conservation of energy to reate the tota energy of a spring-bock osciator at one instant to the tota energy at another instant Sketch a graph of the kinetic energy, potentia energy, and tota energy of a spring-bock osciator, first as a function of time and then as a function of the osciator's position For a spring-bock osciator, determine the bock's position when the tota energy is entirey kinetic energy and when it is entirey potentia energy John Wiey & Sons, Inc. A rights reserved.
21 2014 John Wiey & Sons, Inc. A rights reserved.
22 2014 John Wiey & Sons, Inc. A rights reserved.
23 2014 John Wiey & Sons, Inc. A rights reserved.
24 15-2 Energy in Simpe Harmonic Motion 15-2 Energy in Simpe Harmonic Motion Write the functions for kinetic and potentia energy: Eq. (15-18)18 Their sum is defined by: Eq. (15-20)18 Eq. (15-21)18 Figure John Wiey & Sons, Inc. A rights reserved.
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