General Physics (PHY 2130)

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1 General Physics (PHY 30) Lecture 3 Solids and fluids buoyant force Archimedes principle Fluids in motion

2 Lightning Review Last lecture:. Solids and fluids different states of matter; fluids density, pressure, etc. Review Problem: The surface pressure on the planet Venus is 95 atm. How far below the surface of the ocean on Earth do you need to be to experience the same pressure? The density of seawater is 05 kg/m 3. ( 3)( 05 kg/m 9.8 m/s ) P atm 95 atm atm + ρgd ρgd d d P 94 atm m + ρgd 6 N/m 6 N/m

3 Pascal s Principle A change in pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid and to the walls of the container. The hydraulic press is an important application of Pascal s Principle F F P A A Also used in hydraulic brakes, forklifts, car lifts, etc. Since A >A, then F >F!!!

4 4 ΔP at point ΔP at point F A F F A A A F The work done pressing the smaller piston (#) equals the work done by the larger piston (#). F d F d Using an hydraulic lift reduces the amount of force needed to lift a load, but the work done is the same.

5 Example: Assume that a force of 500 N (about 0 lbs) is applied to the smaller piston in the previous figure. For each case, compute the force on the larger piston if the ratio of the piston areas (A /A ) are, 0, and Using Pascal s Principle: A A F 500 N N 00 50,000 N

6 Example: In the previous example, for the case A /A 0, it was found that F /F 0. If the larger piston needs to rise by m, how far must the smaller piston be depressed? 6 Since the work done by both pistons is the same, d F d F 0 m

7 7 Example: Depressing the brake pedal in a car pushes on a piston with crosssectional area 3.0 cm. The piston applies pressure to the brake fluid, which is connected to two pistons, each with area.0 cm. Each of these pistons presses a brake pad against one side of a rotor attached to one of the rotating wheels. See the figure for this problem. (a) When the force applied by the brake pedal to the small piston is 7.5 N, what is the normal force applied to each side of the rotor? The pressure in the fluid P F b A b. Also, Pr essure P Normal Force N Area of the brake pad piston A the normal force applied to each side of the rotor A.0 cm N PA Fb (7.5 N) 30 N A 3.0 cm b

8 Measuring Pressure The spring is calibrated by a known force The force the fluid exerts on the piston is then measured One end of the U-shaped tube is open to the atmosphere The other end is connected to the pressure to be measured Pressure at B is P o +ρgh A long closed tube is filled with mercury and inverted in a dish of mercury Measures atmospheric pressure as ρgh

9 9 Manometer A manometer is a U-shaped tube that is partially filled with liquid. Both ends of the tube are open to the atmosphere.

10 If there is a pressure difference between the gas and the atmosphere, a force will be exerted on the fluid in the U-tube. This changes the equilibrium position of the fluid in the tube. From the figure: At point C Also P c P atm P B P B' The pressure at point B is the pressure of the gas. 0 P P P B B P P gauge B' C P P B ρgd C + ρgd P atm ρgd Gauge Pressure P gauge P abs - P atm

11 Barometer The atmosphere pushes on the container of mercury which forces mercury up the closed, inverted tube. The distance d is called the barometric pressure. P P A B P atm P ρgd A Atmospheric pressure is equivalent to a column of mercury 76.0 cm tall.

12 How would you measure blood pressure? Has to be: (a) accurate (b) non-invasive (c) simple sphygmomanometer

13 Example: An IV is connected to a patient s vein. The blood pressure in the vein has a gauge pressure of mm Hg. At least how far above the vein must the IV bag be hung in order for fluid to f low into the vein? Assume the fluid in IV has the same density as blood. At a minimum, the gauge pressures must be equal. When h is large enough, fluid will flow from high pressure to low pressure. 3 P gauge h ρ ρ ρ Hg Hg gh blood gh g ρhg h ρblood 54 mm ρ blood gh 3,600 kg/m kg/m 3 ( mm)

14 Question Suppose that you placed an extended object in the water. How does the pressure at the top of this object relate to the pressure at the bottom?. It s the same.. The pressure is greater at the top. 3. The pressure is greater at the bottom. 4. Whatever Since pressure is related to force: there is a force present!

15 Buoyant Force This force is called the buoyant force. What is the magnitude of that force? P A ΔF B P B ( P P ) P + ρ gh, so : A, but : ( P + ρ gh P ) A gha gv! ρ fluid ρ fluid mg P A

16 Buoyant Force The magnitude of the buoyant force always equals the weight of the displaced fluid B ρ Vg fluid w fluid The buoyant force is the same for a totally submerged object of any size, shape, or density The buoyant force is exerted by the fluid Whether an object sinks or floats depends on the relationship between the buoyant force and the weight

17 Archimedes' Principle Any object completely or partially submerged in a fluid is buoyed up by a force whose magnitude is equal to the weight of the fluid displaced by the object. This force is buoyant force. Physical cause: pressure difference between the top and the bottom of the object

18 Example: Archimedes Principle for Totally Submerged Object The upward buoyant force is Bρ fluid gv obj The downward gravitational force is wmgρ obj gv obj The net force is B-w(ρ fluid -ρ obj )gv obj Depending on the direction of the net force, the object will either float up or sink!

19 The net force is B-w(ρ fluid -ρ obj )gv obj The object is less dense than the fluid ρ fluid <ρ obj The object experiences a net upward force The object is more dense than the fluid ρ fluid >ρ obj The net force is downward, so the object accelerates downward

20 Archimedes Principle: Floating Object The object is in static equilibrium The upward buoyant force is balanced by the downward force of gravity Volume of the fluid displaced corresponds to the volume of the object beneath the fluid level If B mg : ρ gv fluid fluid ρ object gv object, or ρ ρ obj fluid V V fluid obj

21 Example: floating frog A frog in a hemispherical pod finds that he just floats without sinking in a fluid of density.35 g/cm3. If the pod has a radius of 6.00 cm and negligible mass, what is the mass of the frog?

22 Question Suppose that you have a steel bar. Will it float on water? Why?

23 Question Suppose that you have a steel bar. Will it float on water? Why? How come that ships (which are made of steel) can float?

24 Question 3 Suppose that your friend gave you a necklace (crown, piece of yellow metal, ). He claims that this object is made of pure gold. How can you check his statement (without going through his credit history)?

25 Question 3 Suppose that your friend gave you a necklace (crown, piece of yellow metal, ). He claims that this object is made of pure gold. How can you check his statement (without going through his credit history)? Idea: determine density! Let s weight the object in and outside the water container: Out: In: mg ρ gv or " weight" mg B ρ mg gv V mg ρ " weight" mg ρ g fluid fluid gv If ρ is not the same as ρ gold, your friend is lying

26 ConcepTest Two identical glasses are filled to the same level with water. One of the two glasses has ice cubes floating in it.which weighs more?. The glass without ice cubes.. The glass with ice cubes. 3. The two weigh the same.

27 ConcepTest Two identical glasses are filled to the same level with water. One of the two glasses has ice cubes floating in it.which weighs more?. The glass without ice cubes.. The glass with ice cubes. 3. The two weigh the same. Ice cubes displace exactly their own weight in water.

28 ConcepTest 3 Two identical glasses are filled to the same level with water. One of the two glasses has ice cubes floating in it.when the ice cubes melt, in which glass is the level of the water higher?. The glass without ice cubes.. The glass with ice cubes. 3. It is the same in both.

29 ConcepTest 3 Two identical glasses are filled to the same level with water. One of the two glasses has ice cubes floating in it.when the ice cubes melt, in which glass is the level of the water higher?. The glass without ice cubes.. The glass with ice cubes. 3. It is the same in both. Previous problem: weight was the same. Since density is also the same, the volumes are going to be the same as well.

30 30 Fluid Flow A moving fluid will exert forces parallel to the surface over which it moves, unlike a static fluid. This gives rise to a viscous force that impedes the forward motion of the fluid. A steady flow is one where the velocity at a given point in a fluid is constant. Steady flow is laminar; the fluid flows in layers. An ideal fluid is incompressible, undergoes laminar flow, and has no viscosity. V constant V constant v v

31 Fluids in Motion: Streamline Flow Streamline flow every particle that passes a particular point moves exactly along the smooth path followed by particles that passed the point earlier also called laminar flow Streamline is the path different streamlines cannot cross each other the streamline at any point coincides with the direction of fluid velocity at that point Fluids in Motion: Turbulent Flow The flow becomes irregular exceeds a certain velocity any condition that causes abrupt changes in velocity

32 3 Equation of Continuity The amount of mass that flows though the cross-sectional area A is the same as the mass that flows through cross-sectional area A. ΔV Δt Av is called the volume flow rate (units m 3 /s)

33 33 Δm Δt ρ Av is the mass flow rate (units kg/s) The continuity equation is A v ρ Av ρ If the fluid is incompressible, then ρ ρ.

34 34 Example: A garden hose of inner radius.0 cm carries water at.0 m/s. The nozzle at the end has radius 0.0 cm. How fast does the water move through the constriction? ( ) m/s 50.0 m/s 0.0 cm.0 cm v r r v A A v A v A v π π

35 Spring Break! 35

36 Additional Example: A piece of metal is released under water. The volume of the metal is 50.0 cm 3 and its specific gravity is 5.0. What is its initial acceleration? (Note: when v 0, there is no drag force.) 36 FBD for the metal y F B w x Apply Newton s nd Law to the piece of metal: F FB w F B ρ water Vg ma The magnitude of the buoyant force equals the weight of the fluid displaced by the metal. Solve for a: a F B ρwatervg ρwaterv g g g m ρ objectv object ρobjectv object

37 37 Example continued: Since the object is completely submerged VV object. specific gravity ρ ρ water where ρ water 000 kg/m 3 is the density of water at 4 C. object Given specific gravity 5. 0 ρ ρ water a ρ ρ water g g g object V V object S. G m/s

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