PHYS 1443 Section 002 Lecture #24

Transcription

1 PHYS 443 Section 00 Lecture #4 Monday, Dec. 3, 007 Dr. Jae Yu uoyant Force and Archimedes Principle Flow Rate and Continuity Equation ernoulli s Equation Simple Harmonic Motion Equation of the SHM Simple lock Sprin System Enery of the SHO Today s homework is none!! Monday, Dec. 3, 007 PHYS , Fall 007

2 Announcements Phantom submission problem arose aain over the weekend. Same as before, please send me the list of homework problem numbers that you had this problem Will ive you full credit as lon as you et the answer riht The last quiz is at the beinnin of the class this Wednesday, Dec. 5 Final exam Date and time: am :30 pm, Monday, Dec. 0 Location: SH03 Covers: CH9. what we finish this Wednesday, Dec. 5 (likely to be Ch4.4) Monday, Dec. 3, 007 PHYS , Fall 007

3 Finer Holds Water in Straw p in A m p A A You insert a straw of lenth L into a tall lass of your favorite beverae. You place your finer over the top of the straw so that no air can et in or out, and then lift the straw from the liquid. You find that the straw strains the liquid such that the distance from the bottom of your finer to the top of the liquid is h. Does the air in the space between your finer and the top of the liquid have a pressure P that is (a) reater than, (b) equal to, or (c) less than, the atmospheric pressure P A outside the straw? Less What are the forces in this problem? Gravitational force on the mass of the liquid Force exerted on the top surface of the liquid by inside air pressure F m ρ A( L h) Force exerted on the bottom surface of the liquid by the outside air Since it is at equilibrium Cancel A and solve for p in Fout + F + F 0 in ( ) p p ρ L h in A Fin F out pina p A p A+ ρ ( L h ) A+ p A 0 A So p in is less than P A by ρ(l-h). in A Monday, Dec. 3, 007 PHYS , Fall 007 3

4 uoyant Forces and Archimedes Principle Why is it so hard to put an inflated beach ball under water while a small piece of steel sinks in the water easily? The water exerts force on an object immersed in the water. This force is called the buoyant force. How lare is the buoyant force? The manitude of the buoyant force always equals the weiht of the fluid in the volume displaced by the submered object. This is called Archimedes principle. What does this mean? h M Let s consider a cube whose heiht is h and is filled with fluid and in its equilibrium so that its weiht M is balanced by the buoyant force. M F Therefore, Monday, Dec. 3, 007 PHYS , Fall 007 The pressure at the bottom of the cube is larer than the top by ρh. ΔP / A ρ h ΔPA ρha ρv ρv M F Where M is the weiht of the fluid in the cube. 4

5 More Archimedes Principle Let s consider the buoyant force in two special cases. Case : Totally submered object h M Let s consider an object of mass M, with density ρ 0, is fully immersed in the fluid with density ρ f. The manitude of the buoyant force is The weiht of the object is F Therefore total force in the system is F ρ f M V ρ 0 F V ( ρ ρ )V f 0 What does this tell you? The total force applies to different directions dependin on the difference of the density between the object and the fluid.. If the density of the object is smaller than the density of the fluid, the buoyant force will push the object up to the surface.. If the density of the object is larer than the fluid s, the object will sink to the bottom of the fluid. Monday, Dec. 3, 007 PHYS , Fall 007 5

6 More Archimedes Principle Case : Floatin object h M Let s consider an object of mass M, with density ρ 0, is in static equilibrium floatin on the surface of the fluid with density ρ f, and the volume submered in the fluid is V f. The manitude of the buoyant force is The weiht of the object is Therefore total force of the system is Since the system is in static equilibrium F F ρ f V f ρ f V f M ρ 0 V 0 ρ V F ρ 0 V 0 f V f ρ What does this tell you? V f ρ 0 ρ f V 0 Since the object is floatin, its density is smaller than that of the fluid. The ratio of the densities between the fluid and the object determines the submered volume under the surface. Monday, Dec. 3, 007 PHYS , Fall 007 6

7 Example for Archimedes Principle Archimedes was asked to determine the purity of the old used in the crown. The leend says that he solved this problem by weihin the crown in air and in water. Suppose the scale read 7.84N in air and 6.86N in water. What should he have to tell the kin about the purity of the old in the crown? In the air the tension exerted by the scale on the object is the weiht of the crown In the water the tension exerted T by the scale on the object is water Therefore the buoyant force is Since the buoyant force is The volume of the displaced water by the crown is Therefore the density of the crown is ρ c m V c Vc c Tair m 7.84 N m 6.86N T 0.98N air T water ρ w V w ρ w V c 0.98 N V w m V c c 0.98 N ρ w V c k / m 4 m Since the density of pure old is 9.3x0 3 k/m 3, this crown is not made of pure old. Monday, Dec. 3, 007 PHYS , Fall 007 7

8 Example for uoyant Force What fraction of an iceber is submered in the sea water? Let s assume that the total volume of the iceber is V i. Then the weiht of the iceber F i is F i ρ i V i Let s then assume that the volume of the iceber submered in the sea water is V w. The buoyant force caused by the displaced water becomes ρ w V w Since the whole system is at its static equilibrium, we obtain Therefore the fraction of the volume of the iceber submered under the surface of the sea water is ρ i V i ρ w V w V V w i 97 k 3 ρ i ρ 030 k / m w / m About 90% of the entire iceber is submered in the water!!! Monday, Dec. 3, 007 PHYS , Fall 007 8

9 Flow Rate and the Equation of Continuity Study of fluid in motion: Fluid Dynamics If the fluid is water: Water dynamics?? Hydro-dynamics Two main Streamline or Laminar flow: Each particle of the fluid types of flow follows a smooth path, a streamline Turbulent flow: Erratic, small, whirlpool-like circles called eddy current or eddies which absorbs a lot of enery Flow rate: the mass of fluid that passes the iven point per unit time Δm/ Δ Monday, Dec. 3, 007 PHYS , Fall 007 Δ m ρ Δ V ρ AΔ l Δ t Δ t Δ t since the total flow must be conserved Δ m Δ m ρ Av Δ t Δ t Equation of Continuity ρ Av ρ Av 9 t

10 Example for Equation of Continuity How lare must a heatin duct be if air movin at 3.0m/s throuh it can replenish the air in a room of 300m 3 volume every 5 minutes? Assume the air s density remains constant. A A v v A l v Monday, Dec. 3, 007 PHYS , Fall 007 / t Usin equation of continuity ρ Av ρ Since the air density is constant Av Now let s imaine the room as the lare section of the duct V v t Av Av m 0

11 ernoulli s Principle ernoulli s Principle: Where the velocity of fluid is hih, the pressure is low, and where the velocity is low, the pressure is hih. Monday, Dec. 3, 007 PHYS , Fall 007 Amount of the work done by the force, F, that exerts pressure, P, at point W F Δ l P A Δl Amount of the work done by the force in the other section of the fluid is W Work done by the ravitational force to move the fluid mass, m, from y to y is W m ( y y ) 3 P A Δl

12 ernoulli s Equation cont d The total amount of the work done on the fluid is P AΔl W W + W +W3 From the work-enery principle mv Since the mass m is contained in the volume that flowed in the motion Thus, A Δ l A l and m PAΔ l mv PAΔ l my + my ρ A Δl v P A Δ Δ l ρ A Δl PAΔ l my + my v ρ AΔ l ρ AΔl P AΔl ρ AΔly + ρ AΔly Monday, Dec. 3, 007 PHYS , Fall 007

13 Since We obtain Reoranize ernoulli s Equation cont d ρa Δl v ρaδlv PAΔl P A Δl ρa Δl y + ρaδl y Thus, for any two points in the flow ρv ρv P P ρy + ρy P+ ρv + ρy P + ρv + ρy For static fluid P P+ ρv + ρy P ( ) + ρ y y P For the same heihts ( P ) P+ ρ v v Monday, Dec. 3, 007 PHYS , Fall 007 const. + ρ h ernoulli s Equation Result of Enery conservation! Pascal s Law The pressure at the faster section of the fluid is smaller than slower section. 3

14 Example for ernoulli s Equation Water circulates throuhout a house in a hot-water heatin system. If the water is pumped at the speed of 0.5m/s throuh a 4.0cm diameter pipe in the basement under a pressure of 3.0atm, what will be the flow speed and pressure in a.6cm diameter pipe on the second 5.0m above? Assume the pipes do not divide into branches. Usin the equation of continuity, flow speed on the second floor is Av π r v v m / s A π r 0.03 Usin ernoulli s equation, the pressure in the pipe on the second floor is P P ( + ρ v ) v + ρ ( y ) y N / m Monday, Dec. 3, 007 PHYS , Fall 007 ( ) ( )