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Lecture 21

Lecture 21. Goals:. Chapter 15 U nderstand pressure in liquids and gases Use Archimedes’ principle to understand buoyancy Understand the equation of continuity Use an ideal-fluid model to study fluid flow. I nvestigate the elastic deformation of solids and liquids. Assignment

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Lecture 21

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  1. Lecture 21 Goals: • Chapter 15 • Understand pressure in liquids and gases • Use Archimedes’ principle to understand buoyancy • Understand the equation of continuity • Use an ideal-fluid model to study fluid flow. • Investigate the elastic deformation of solids and liquids • Assignment • HW9, Due Wednesday, Apr. 8th • Thursday: Read all of Chapter 16

  2. Pressure vs. DepthIncompressible Fluids (liquids) • In many fluids the bulk modulus is such that we can treat the density as a constant independent of pressure: An incompressible fluid • For an incompressible fluid, the density is the same everywhere, but the pressure is NOT! • p(y) = p0 - y g r = p0 + d g r • Gauge pressure (subtract p0, pressure 1 atm, i.e. car tires) F2 = F1+ m g = F1+ rVg F2 /A = F1/A + rVg/A p2 = p1 - rg y

  3. Pressure vs. Depth • For a uniform fluid in an open container pressure same at a given depthindependentof the container • Fluid level is the same everywhere in a connected container, assuming no surface forces

  4. Pressure Measurements: Barometer • Invented by Torricelli • A long closed tube is filled with mercury and inverted in a dish of mercury • The closed end is nearly a vacuum • Measures atmospheric pressure as 1 atm = 0.760 m (of Hg)

  5. (A)r1 < r2 (B)r1 = r2 (C)r1 > r2 Exercise Pressure • What happens with two fluids?? • Consider a U tube containing liquids of density r1 and r2 as shown: • At the red arrow the pressure must be the same on either side. r1 x = r2 (d1+ y) • Compare the densities of the liquids: dI r2 y r1

  6. W2? W1 W1 > W2 W1 < W2 W1 = W2 Archimedes’ Principle: A Eureka Moment • Suppose we weigh an object in air (1) and in water (2). How do these weights compare? • Buoyant force is equal to the weight of the fluid displaced

  7. W2? W1 W1 > W2 W1 < W2 W1 = W2 Archimedes’ Principle • Suppose we weigh an object in air (1) and in water (2). • How do these weights compare? • Why? Since the pressure at the bottom of the object is greater than that at the top of the object, the water exerts a net upward force, the buoyant force, on the object.

  8. y F mg B Sink or Float? • The buoyant force is equalto the weight of the liquid that is displaced. • If the buoyant force is larger than the weight of the object, it will float; otherwise it will sink. • We can calculate how much of a floating object will be submerged in the liquid: • Object is in equilibrium

  9. piece of rock on top of ice Bar Trick What happens to the water level when the ice melts? Expt. 1 Expt. 2 B. It stays the same C. It drops A. It rises

  10. Exercise V1 = V2 = V3 = V4 = V5 m1 < m2 < m3 < m4 < m5 What is the final position of each block?

  11. Exercise V1 = V2 = V3 = V4 = V5 m1 < m2 < m3 < m4 < m5 What is the final position of each block? But this Not this

  12. Pascal’s Principle • So far we have discovered (using Newton’s Laws): • Pressure depends on depth: Dp = r g Dy • Pascal’s Principle addresses how a change in pressure is transmitted through a fluid. 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.

  13. Pascal’s Principle in action: Hydraulics, a force amplifier • Consider the system shown: • A downward force F1 is applied to the piston of area A1. • This force is transmitted through the liquid to create an upward force F2. • Pascal’s Principle says that increased pressure from F1 (F1/A1) is transmitted throughout the liquid. • F2 > F1 with conservation of energy

  14. M dA A1 A10 M dB A2 A10 Exercise • Consider the systems shown on right. • In each case, a block of mass M is placed on the piston of the large cylinder, resulting in a difference di in the liquid levels. • If A2 = 2A1, how do dA and dB compare? V10 = V1= V2 dA A1= dB A2 dA A1= dB 2A1 dA = dB 2

  15. Fluids in Motion • To describe fluid motion, we need something that describes flow: • Velocity v • There are different kinds of fluid flow of varying complexity • non-steady/ steady • compressible / incompressible • rotational / irrotational • viscous / ideal

  16. Types of Fluid Flow • Laminar flow • Each particle of the fluid follows a smooth path • The paths of the different particles never cross each other • The path taken by the particles is called a streamline • Turbulent flow • An irregular flow characterized by small whirlpool like regions • Turbulent flow occurs when the particles go above some critical speed

  17. Types of Fluid Flow • Laminar flow • Each particle of the fluid follows a smooth path • The paths of the different particles never cross each other • The path taken by the particles is called a streamline • Turbulent flow • An irregular flow characterized by small whirlpool like regions • Turbulent flow occurs when the particles go above some critical speed

  18. A 2 A 1 v 1 v 2 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 • Streamline density is proportional to velocity • Flow obeys continuity equation Volume flow rate (m3/s) Q = A·vis constant along flow tube. Follows from mass conservation if flow is incompressible. Mass flow rate is just r Q (kg/s) A1v1 = A2v2

  19. (A) 2 v1 (B) 4 v1 (C) 1/2 v1 (D) 1/4 v1 Exercise 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 • 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?

  20. v1 v1/2 Exercise Continuity (A) 2 v1 (B) 4 v1 (C) 1/2 v1 (D) 1/4 v1 • For equal volumes in equal times then ½ the diameter implies ¼ the area so the water has to flow four times as fast. • But if the water is moving 4 times as fast then it has 16times as much kinetic energy. • Something must be doing work on the water (the pressure drops at the neck) and we recast the work as P DV = (F/A) (A Dx) = F Dx

  21. P1 P2 Conservation of Energy for Ideal Fluid W = (P1– P2 ) DV and W = ½ Dm v22 – ½ Dm v12 = ½ (r DV) v22 – ½ (r DV)v12 (P1– P2 ) = ½ r v22 – ½ r v12 P1+ ½ r v12= P2+ ½ r v22= const. and with height variations: Bernoulli Equation P1+ ½ r v12 + r g y1 = constant

  22. Lecture 21 • Question to ponder: Does heavy water (D2O) ice sink or float? • Assignment • HW9, due Wednesday, Apr. 8th • Thursday: Read all of Chapter 16

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