Physics 1501: Lecture 32, Pg 1 Physics 1501: Lecture 32 Today’s Agenda l Homework #11 (due Friday Dec. 2) l Midterm 2: graded by Dec. 2 l Topics: çFluid dynamics çBernouilli’s equation çExample of applications l Midterm 2 solutions

Physics 1501: Lecture 32, Pg 2 Pascal and Archimedes’ Principles l 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. l Archimedes’ principle The buoyant force is equal to the weight of the liquid displaced. çObject is in equilibrium

Physics 1501: Lecture 32, Pg 3 ACT 1-A Fun With Buoyancy l Two cups are filled to the same level with water. One of the two cups has plastic balls floating in it. çWhich cup weighs more? Cup I Cup II (A) Cup I (B) Cup II (C) the same (D) can’t tell

Physics 1501: Lecture 32, Pg 4 ACT 1-B Even More Fun With Buoyancy A plastic ball floats in a cup of water with half of its volume submerged. Next some oil (  oil <  ball <  water ) is slowly added to the container until it just covers the ball. çRelative to the water level, the ball will: water oil (A) move up (B) move down (C) stay in same place

Physics 1501: Lecture 32, Pg 5 Fluids in Motion l Up to now we have described fluids in terms of their static properties:  density  çpressure p l To describe fluid motion, we need something that can describe flow: çvelocity v l There are different kinds of fluid flow of varying complexity  non-steady / steady ç compressible / incompressible ç rotational / irrotational ç viscous / ideal

Physics 1501: Lecture 32, Pg 6 l Simplest situation: consider ideal fluid moving with steady flow - velocity at each point in the flow is constant in time l In this case, fluid moves on streamlines streamline Ideal Fluids l Fluid dynamics is very complicated in general (turbulence, vortices, etc.) l Consider the simplest case first: the Ideal Fluid çno “viscosity” - no flow resistance (no internal friction) çincompressible - density constant in space and time

Physics 1501: Lecture 32, Pg 7 l Flow obeys continuity equation ç volume flow rate Q = A·v is constant along flow tube. ç follows from mass conservation if flow is incompressible. streamline A 1 v 1 = A 2 v 2 Ideal Fluids l streamlines do not meet or cross l velocity vector is tangent to streamline l volume of fluid follows a tube of flow bounded by streamlines

Physics 1501: Lecture 32, Pg 8 Steady Flow of Ideal Fluids (actually laminar flow of real fluid)

Physics 1501: Lecture 32, Pg 9 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 32 Act 2 Continuity l 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. v1v1 v 1/2 a) 2 v 1 b) 4 v 1 c) 1/2 v 1 c) 1/4 v 1

Physics 1501: Lecture 32, Pg 10 l Recall the standard work-energy relation  Apply the principle to a section of flowing fluid with volume  V and mass  m =  V (here W is work done on fluid) VV Conservation of Energy for Ideal Fluid Bernoulli Equation

Physics 1501: Lecture 32, Pg 11 Lecture 32 Act 3 Bernoulli’s Principle l 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) smallerb) samec) larger v1v1 v 1/2

Physics 1501: Lecture 32, Pg 12 Some applications l Lift for airplane wing l Enhance sport performance l More complex phenomena: ex. turbulence

Physics 1501: Lecture 32, Pg 13 More applications l Vortices: ex. Hurricanes l And much more …