1. Lecture 26 Goals: • Understand pressure in liquids • Use Archimedes’ principle to understand buoyancy • Understand the equation of continuity • Use an ideal-fluid model to study fluid flow.

2. Fluids • At ordinary temperatures, matter exists in one of three states • Solid: Has a shape and forms a surface • Liquid: Has no shape but forms a surface • Gas : No shape and does not form a surface gas liquid solid Increasing temperature

3. Fluids • Fluids: Liquids or Gases • Shape determined by container • Close contact at boundaries. No macroscopic gaps • Forces exerted by fluids are always normal to the surface. • Fluids can flow • Fluids: • They have mass • They have weight • They carry energy • They have velocity when flow • Newton’s Laws apply

4. Fluids • An intrinsic parameter of a fluid • Density units : kg/m3 = 10-3 g/cm3 r(water) = 1.000 x 103 kg/m3 = 1.000 g/cm3 r(ice) = 0.917 x 103 kg/m3 = 0.917 g/cm3 r(air) = 1.29 kg/m3 = 1.29 x 10-3 g/cm3 r(Hg) = 13.6 x103 kg/m3 = 13.6 g/cm3 r(W or Au) = 19.3 x103 kg/m3 = 19.3 g/cm3

5. A Fluids • Another parameter: Pressure • A Fluid exerts force on the objects it contacts • Molecules constantly bounces off the surface with F =Σ(Δmv/Δt) (momentum transfer / time) • Any force exerted by a fluid is perpendicular to a surface of contact, and is proportional to the area of that surface. • Force (avector) in a fluid can be expressed in terms of pressure (a scalar) as: Do not confuse pressure with momentum

6. What is the SI unit of pressure? • Pascal • Atmosphere • Bernoulli • Young • p.s.i. Units : 1 N/m2 = 1 Pa (Pascal) 1 bar = 105 Pa 1 mbar = 102 Pa 1 torr = 133.3 Pa 1 atm = 1.013 x105 Pa = 1013 mbar = 760 Torr = 14.7 lb/ in2 (=PSI)

7. A bed of nails • Distributing the force over a large area ….

8. Ping pong ball bazooka • What is the initial force on the ping pong ball? • Assuming this force is constant, how fast is the ping pong ball travelling when it leaves the tube? F This is a crude model….the speed of sound in air is 330 m/s

9. P0 F1 y1 y2 P1 A P2 mg F2 Pressure vs. DepthIncompressible Fluids (liquids) • When the pressure is small, relative to the bulk modulus of the fluid, we can treat the density as constant independent of pressure: incompressible fluid • For an incompressible fluid, the density is the same everywhere, but the pressure is NOT! • P(y) = P0 - y g r • Gauge pressure (subtract P0) • PGauge = P(y) - P0 F2 = F1+ m g = F1+ rVg F2 /A = F1/A + rVg/A

10. P0 F1 y1 y2 P1 A P2 mg F2 Pressure vs. Depth in water Every 10 meters the pressure increases by 1 atm

11. Pascal’s Principle • 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

12. Exercise • Does PA=PB? B A

13. (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: Compare the densities of the liquids: dI r2 r1

14. 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 • PA=PB • Measures atmospheric pressure as 1 atm = 0.760 m (of Hg)

15. 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. P2 P1 P1 = P2 F1 / A1 = F2 / A2 A2 / A1 = F2 / F1

16. 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 = Weight of the fluid displaced

17. 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

18. 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

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

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

21. For Tuesday • All of chapter 14