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Archimedes’ Principle

Archimedes’ Principle. Physics 202 Professor Lee Carkner Lecture 2. PAL #1 Fluids. Column of water to produce 1 atm of pressure P = r gh P = r = 1000 kg/m 3 g = 9.8 m/s 2 h = P/ r g = Double diameter, pressure does not change On Mars pressure would decrease

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Archimedes’ Principle

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  1. Archimedes’ Principle Physics 202 Professor Lee Carkner Lecture 2

  2. PAL #1 Fluids • Column of water to produce 1 atm of pressure • P = rgh • P = • r = 1000 kg/m3 • g = 9.8 m/s2 • h = P/rg = • Double diameter, pressure does not change • On Mars pressure would decrease • Mars has smaller value of g

  3. Archimedes’ Principle • What happens if you put an object in a fluid? • Called the buoyant force • If you measure the buoyant force and the weight of the displaced fluid, you find: • An object in a fluid is supported by a buoyant force equal to the weight of fluid it displaces • Applies to objects both floating and submerged

  4. Will it Float? • Density • An object less dense than the fluid will float • A floating object displaces fluid equal to its weight • A sinking object displaces fluid equal to its volume

  5. Floating • How will an object float? • The volume of fluid displaced is proportional to the ratio of the densities • Example: ice floating in water, riVig=rwVwg Vw=Vi (ri/rw) rw = 1024 kg/m3 and ri = 917 kg/m3

  6. Ideal Fluids • Steady -- • Incompressible -- density is constant • Nonviscous -- • Irrotational -- constant velocity through a cross section • The ideal fluid approximation is usually not very good

  7. Moving Fluids • What happens if the pipe narrows? Avr = constant • If the density is constant then, Av= constant = R = volume flow rate • Constricting a flow increases its velocity • Because the amount of fluid going in must equal the amount of fluid going out • Or, a big slow flow moves as much mass as a small fast flow

  8. Continuity • R=Av=constant is called the equation of continuity • You can use it to determine the flow rates of a system of pipes • Can’t lose or gain any material

  9. The Prancing Fluids • How can we keep track of it all? • The laws of physics must be obeyed • Neither energy nor matter can be created or destroyed

  10. Bernoulli’s Equation • Consider a pipe that bends up and gets wider at the far end with fluid being forced through it Wg = -Dmg(y2-y1) = -rgDV(y2-y1) • The work of the system due to pressure is, Wp=Fd=pAd=DpDV=-(p2-p1)DV D(1/2mv2)=1/2rDV(v22-v12) p1+(1/2)rv12+rgy1=p2+(1/2)rv22+rgy2

  11. Consequences of Bernoulli’s • Fast moving fluids exert less pressure than slow moving fluids • This is known as Bernoulli’s principle • Energy that goes into velocity cannot go into pressure • Note that Bernoulli only holds for moving fluids

  12. Bernoulli in Action • Blowing between two pieces of paper • Convertible top bulging out • Airplanes taking off into the wind

  13. Lift • If the velocity of the flow is less on the bottom than on top there is a net pressure on the bottom and thus a net force pushing up • If you can somehow get air to flow over an object to produce lift, what happens?

  14. Deriving Lift • Use Bernoulli’s equation: pt+1/2rvt2=pb+1/2rvb2 • The difference in pressure is: pb-pt=1/2rvt2-1/2rvb2 (Fb/A)-(Ft/A)=1/2r(vt2-vb2) L= (½)rA(vt2-vb2)

  15. Next Time • Read: 15.1-15.3 • Homework: Ch 14, P: 37, 42, 47, Ch 15, P: 6, 7

  16. Which of the following would decrease the pressure you exert on the floor the most? • Doubling your mass • Doubling the mass of the earth • Doubling your height • Doubling the size of your shoes • Doubling air pressure

  17. Which of the following would increase the pressure of a column of fluid of fixed mass the most? • Doubling the width of the column • Halving the density of the fluid • Halving the mass of the Earth • Halving the speed of the Earth’s rotation • Doubling the height of the column

  18. Summary: Fluid Basics • Density =r=m/V • Pressure=p=F/A • On Earth the atmosphere exerts a pressure and gravity causes columns of fluid to exert pressure • Pressure of column of fluid: p=p0+rgh • For fluid of uniform density, pressure only depends on height

  19. Summary: Pascal and Archimedes • Pascal -- pressure on one part of fluid is transmitted to every other part • Hydraulic lever -- A small force applied for a large distance can be transformed into a large force over a short distance Fo=Fi(Ao/Ai) and do=di(Ai/Ao) • Archimedes -- An object is buoyed up by a force equal to the weight of the fluid it displaces • Must be less dense than fluid to float

  20. Summary: Moving Fluids • Continuity -- the volume flow rate (R=Av) is a constant • fluid moving into a narrower pipe speeds up • Bernoulli p1+1/2rv12+rgy1=p2+1/2rv22+rgy2 • Slow moving fluids exert more pressure than fast moving fluids

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