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 = • Double diameter, pressure does not change • On Mars pressure would decrease • Mars has smaller value of g

3. Archimedes’ Principle • The fluid exerts a force on the object • 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? • What determines if a object will sink or float? • 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 Vw=

6. Continuity • For a moving fluid • Energy must be conserved • Mass must be conserved so, Avr = constant Av= constant = R = volume flow rate • called the equation of continuity • Flow rates in and out must always balance out

7. Moving Fluids • Constricting a flow increases its velocity • Because the amount of fluid going in must equal the amount of fluid going out • Fluids also must obey energy conservation • Pressure work • Kinetic energy

8. 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) Wp=Fd=pAd=DpDV=-(p2-p1)DV D(1/2mv2)=1/2rDV(v22-v12) • Equating work and DKE yields, p1+(1/2)rv12+rgy1=p2+(1/2)rv22+rgy2

9. 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

10. Bernoulli in Action • Getting sucked under a train • Airplanes taking off into the wind

11. Next Time • Read: 15.1-15.3 • Homework: Ch 14, P: 37, 42, 47, Ch 15, P: 6, 7 • (This is just for reference, homework is only done on Webassign)

12. 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

13. 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

14. 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

15. 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

16. 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