Physics 121: Lecture 21 Today,'s Agenda Announcements Homework 8: due Friday Nov. 11@ 6: 00 PM Chap.8:#7,22,28,33,35,44,45,50,54,61,and65 Today' s topics Fluids in motion Bernoullis equation Viscous fluids Simple oscillations Pendulum Physics 121: Lecture 21, Pg
Physics 121: Lecture 21, Pg 1 Physics 121: Lecture 21 Today’s Agenda Announcements Homework 8: due Friday Nov. 11 @ 6:00 PM. Chap. 8: # 7, 22, 28, 33, 35, 44, 45, 50, 54, 61, and 65. Today’s topics Fluids in motion Bernouilli’s equation Viscous fluids Simple oscillations Pendulum
Review: Fluids at rest What parameters do we use to describe fluids? Density Bulk modulus B △p (-△V/V) Pressure F=pAn A For incompressible fiuids(B>>p) p=const p(Ay )=Po+pgAy (y is depth) Pascals Principle Any change in the pressure applied to an enclosed fluid is transmitted to every portion of the fluid and to the walls of the containing vessel Physics 121: Lecture 21, Pg 2
Physics 121: Lecture 21, Pg 2 Review: Fluids at Rest What parameters do we use to describe fluids? Density Bulk Modulus Pressure For incompressible fluids ( ) Pascal’s Principle: Any change in the pressure applied to an enclosed fluid is transmitted to every portion of the fluid and to the walls of the containing vessel. ( V /V) p B − = B p = const. p(y ) = p0 + gy (y is depth) F = pAn ˆ A n
Archimedes' Principle( W,(W2? The buoyant force is equal to the difference in the pressures times the area B=(p2-p1)A=pg(y2-y1)A FB= Pliquidg Vliquid=Liquid.g=Wliq Archimedes The buoyant force is equal to the weight of the liquid displaced y1 The buoyant force determines whether an object will sink or float How does this work? Physics 121: Lecture 21, Pg 3
Physics 121: Lecture 21, Pg 3 The buoyant force is equal to the difference in the pressures times the area. W1 W2? FB = (p2 − p1 ) A = g(y2 - y1 )A FB liquidgVliquid Mliquid g = Wliquid = = Archimedes: The buoyant force is equal to the weight of the liquid displaced. The buoyant force determines whether an object will sink or float. How does this work? y 1 y 2 A p 1 p 2 F 1 F 2 Archimedes’ Principle
Fluids in Motion Up to now we have described fluids in terms of their static properties density p pressure p To describe fluid motion, we need something that can describe flow velocity v There are different kinds of fiuid flow of varying complexity non-steady / steady compressible incompressible rotational irrotational VISCOUS idea Physics 121: Lecture 21, Pg
Physics 121: Lecture 21, Pg 4 Fluids in Motion Up to now we have described fluids in terms of their static properties: density pressure p To describe fluid motion, we need something that can describe flow: velocity v There are different kinds of fluid flow of varying complexity non-steady / steady compressible / incompressible rotational / irrotational viscous / ideal
Ideal fluids Fluid dynamics is very complicated in general (turbulence, vortices, etc.) Consider the simplest case first the Ideal Fluid no" -no flow resistance(no internal friction) incompressible -density constant in space and time Simplest situation: consider streamline A ideal fluid moving with steady flow-velocity at each point in A the flow is constant in time In this case, fluid moves on streamlines Physics 121: Lecture 21, Pg 5
Physics 121: Lecture 21, Pg 5 Simplest situation: consider ideal fluid moving with steady flow - velocity at each point in the flow is constant in time In this case, fluid moves on streamlines A1 A2 v1 v2 streamline Ideal Fluids Fluid dynamics is very complicated in general (turbulence, vortices, etc.) Consider the simplest case first: the Ideal Fluid no “viscosity” - no flow resistance (no internal friction) incompressible - density constant in space and time
Ideal fluids streamlines do not meet or cross velocity vector is tangent to streamline A streamline A volume of fluid follows a tube of flow bounded by streamlines Flow obeys continuity equation volume flow rate Q=Av is constant along flow tube A1V,=A2v2 follows from mass conservation if flow is compressible. Physics 121: Lecture 21, Pg 6
Physics 121: Lecture 21, Pg 6 Flow obeys continuity equation volume flow rate Q = A·v is constant along flow tube. follows from mass conservation if flow is incompressible. A1 A2 v1 v2 streamline A1v1 = A2v2 Ideal Fluids streamlines do not meet or cross velocity vector is tangent to streamline volume of fluid follows a tube of flow bounded by streamlines
Steady Flow of Ideal Fluids (actually laminar flow of real fluid) Physics 121: Lecture 21, Pg 7
Physics 121: Lecture 21, Pg 7 Steady Flow of Ideal Fluids (actually laminar flow of real fluid)
Lecture 21 Act 1 Continuity A housing contractor saves v1/2 some money by reducing the size of a pipe from1” diameter to 1/2 diameter at some point in your house 1)Assuming the water moving in the pipe is an ideal fluid, relative to its speed in the 1 diameter pipe how fast is the water going in the 1/2 pipe? a)2 V1 b)4 V1 c)1/2v1c)1/4v1 Physics 121: Lecture 21, Pg 8
Physics 121: Lecture 21, Pg 8 1) Assuming the water moving in the pipe is an ideal fluid, relative to its speed in the 1” diameter pipe, how fast is the water going in the 1/2” pipe? Lecture 21 Act 1 Continuity A housing contractor saves some money by reducing the size of a pipe from 1” diameter to 1/2” diameter at some point in your house. v1 v1/2 a) 2 v1 b) 4 v1 c) 1/2 v1 c) 1/4 v1
Conservation of Energy for Ideal fluid Recall the standard work-energy relation W=AK Apply the principle to a section of flowing fluid with volume 8V and mass Sm= psV(here W is work done on fluid W=W ravity Pressure y2 gravity=-om g(y2-y1) pavg(y2-y1) SV lese=p1y-P242∞2 (p1-p2)8v W=Ak=1,mvi,=pov(v2-v) Bernoulli Equation p,+pVi+pgy,=p2+2pv2+pgy2 Physics 121: Lecture 21, Pg 9
Physics 121: Lecture 21, Pg 9 Recall the standard work-energy relation Apply the principle to a section of flowing fluid with volume dV and mass dm = dV (here W is work done on fluid) ( p p ) V W p A x p A x 1 2 pressure 1 1 1 2 2 2 d d d = − = − Vg( y y ) W m g( y y ) 2 1 gravity 2 1 = − − = − − d d W K mv mv V(v v ) 2 1 2 2 2 2 1 2 1 2 1 2 2 1 = = d − d = d − 2 2 2 2 1 1 2 2 2 1 1 Bernoulli Equation p1 + v + gy = p + v + gy y 1 y 2 v 1 v 2 p 1 p 2 dV W =Wgravity +Wpressure W = K Conservation of Energy for Ideal Fluid
Lecture 21 Act 2 Bernoulli's Principle A housing contractor saves v1/2 some money by reducing the size of a pipe from1” diameter to 1/2 diameter at some point in your house . )What is the pressure in the 1 /2 pipe relative to the 1”pipe? a smaller b same c larger Physics 121: Lecture 21, Pg 10
Physics 121: Lecture 21, Pg 10 Lecture 21 Act 2 Bernoulli’s Principle A housing contractor saves some money by reducing the size of a pipe from 1” diameter to 1/2” diameter at some point in your house. 2) What is the pressure in the 1/2” pipe relative to the 1” pipe? a) smaller b) same c) larger v1 v1/2
