Physics 201, Lecture 26 Today s Topics n Fluid Mechanics (chapter 14) n Review: Pressure n Buoyancy, Archimedes s Principle (14.4) n Fluid Dynamics, Bernoulli s Equation (14.5,14.6) n Applications of Fluid Dynamics
q When and where About Final Thursday Dec 21st 2:45-4:45 pm See my email yesterday. q Format Closed book 1+3 8x11 formula sheets allowed, must be self prepared. (Absolutely no photocopying/printing of sample problems, examples, class lectures, HW etc.) 30 multiple choice questions. (count as 200 points) Bring a calculator (but no computer). Only basic calculation functionality can be used. q Special needs/ conflicts: All special requests should be made by 6pm Thursday December 7th. (except for medical emergency) All alternative test sessions are in our lab room, only for approved requests.
Chapters Covered q The final exam is cumulative. q ~50 % will be on old chapters (Ch 1-12) q ~50 % will be on new chapters (Ch 13,14,15) Chapter 13: Gravitation Chapter 14: Fluid Mechanics Chapter 15: Oscillation Motion. Special Review Lecture: Thursday December 14 th 9:55-10:45am : Chapters since midterm 3 (Note the special date which is after the last class day) Super Friday: December 15 th. 10am-5pm in the lab room.
College of Engineering Policies On Final Exam Rescheduling q Regulation 9 Student responsibility for scheduling: " Each student is responsible for arranging a course list that will permit satisfactory progress towards degree requirements and a class schedule that (a) avoids class and final exam scheduling conflicts, (b) avoids an excessively demanding final exam schedule, and (c) verifies registration in chosen classes. q Regulation 25 Final exam rescheduling: " A student may be permitted to take an examination at other than the regularly scheduled time only with permission of the instructor. Permission will be granted only for illness or other unusual and substantiated cause beyond the student's control. (See also Regulation 9). ( http://www.engr.wisc.edu/current/coe-enrollment-regulations.html )
Review: Pressure q Fluid exerts force on the objects it contact: molecules constantly hit the surface F =Σ(Δmv/Δt) q Force exerted by fluid distributes over contact surface q Pressure: P = Force / Area P F/A Unit: N/m 2 Pascal (Pa) Pressure depends on only the magnitude of F Pressure also definable for solids when the contact surface is regular Be careful with p : momentum/power/pressure? Atmospheric (room air) pressure: 1.01 x10 5 Pa
Review: Pascal s Principle q Pascal s Principle: Pressure is a contained fluid is transmitted to every point, in every direction, regardless of the shape of the container. Pressure is the same at same height/depth. water oil Same level same pressure Regardless of shape, etc. (in the same fluid) q At different level: P = P 0 + ρgh
Quick Quiz q Compare P A, P B? Ø Define a reference level CD : P C =P D Δh A B C D Ø Compare level AB to level CD: P A =P C - ρ Hg gδh < P D ρ H2o gδh =P B
Exercise: Hydraulic Jack q Explain the working principle of the hydraulic press Force ratio F 1 :F 2 =? P 2 = P 1 F 2 A 2 = F 1 A 1 F 2 F 1 = A 2 A 1 >> 1 Displacement ratio Δx 1 :Δx 2 =? oil is not compressible Δx 1 A 1 = Δx 2 A 2 Δx 2 Δx 1 = A 1 A 2 Work ratio W 1 :W 2 =? recall: W = F Δx W 2 W 1 = F 2 Δx 2 F 1 Δx 1 =1:1! One can save force, but not work (energy)!
Buoyancy and Archimedes's Principle q All object immersed in fluid is subject to an buoyant force (B) exerted by the fluid. q Archimedes Principle: The magnitude of buoyant force always equals the weight of the fluid displaced by the object: Archimedes of Syracuse ( 287 212 BC) B=M disp g= ρ fluid V disp g ü Buoyant force is always upwards ü It is independent of shape ü It is independent of the density of the object ü It is caused by pressure difference over depth
Floating or Sinking If ρ obj > ρ fluid F g = m obj g = ρ obj V obj g B = ρ fluid V obj g < F g If ρ obj < ρ fluid F g = m obj g = ρ obj V obj g B = ρ fluid V obj g > F g sinking! (eventually, sunken at the bottom) floating! (eventually, floating at top)
Recall: Density Density = Mass/Volume ρ m/v Definable for solids and fluids Basic Unit: kg/m 3 Independent of shape Solids and liquids: ρ very weakly depends on temperature and pressure Gases: ρ strongly depends on temperature and pressure Solid form is not necessarily heavier than liquid form Substance Density ρ (10 3 kg/m 3 ) Water 1.00 Ice 0.917 Mercury 13.6 Lead 11.3 Copper 8.92 Iron 7.86 Aluminum 2.70 Wood 0.550 Blood 1.06 Oil 0.92-0.98 Alcohol 0.82 Room Air 0.00129
Exercise: Floating Iceberg q What is the portion of a floating iceberg that is under water? ( ρ ice =0.9, ρ water =1.0 ) Ø Floating: F g = B F g = ρ ice V whole_ice g B =ρ water V ice_in_water g à ρ ice V whole_ice = ρ water V ice_in_water à V ice_in_water : V whole_ice = ρ ice : ρ water = 0.9 :1 = 90%
Quiz: Ice on a Cup of Water q Convince yourself that the only 10% of the ice is showing above the water level. q Quiz: when all ice is melted, is the water level higher, lower, or remains the same Solution: Before melting B= m ice g = ρ water V disp g à V disp = m ice /ρ water After melting, m ice becomes V ice_melt ρ water of water. à V ice_melt =m ice /ρ water =V disp Follow up quiz: What about Ice on Oil (ρ oil =0.94, ρ Ice =0.9)?
Exercise: King Hiero II of Syracuse s Crown q The legend says that Archimedes was once asked by the King of Syracuse to tell whether a new crown he had just acquired was made of pure gold. Here is what Archimedes did: Ø Weigh the crown in air, he got: W in_air = 7.84 N = ρ crown V crown g then weigh the crown in water and got: W in_water = 6.84 N = W in_air - B Ø So the buoyancy B= 7.84-6.84 = 1.00N = ρ water V crown g then he can obtain ρ crown = ρ water 7.84/1.00 = 7.84 x10 3 kg/m 3 <<ρ Gold =19.3 x10 3 kg/m 3 So the king was indeed cheated!
Fluid Dynamics q Pascal s Principle and Archimedes principle are for fluids that are in equilibrium. What about fluid that flows? Ø The subject is called Fluid Dynamics (a course by itself!) q For this course, we are considering a simpler (but still very useful ) model: ideal fluid flow. q Ideal Fluid Flow: flow of streamline The flow is of steady streamline (i.e. non-turbulence) The fluid is non-viscous (i.e. ignore all internal friction) The fluid is incompressible. (i.e. density is constant) The fluid is ir-rotational (i.e. zero angular momentum)
Fluid Dynamics: Continuity The amount of (ideal) fluid moving in at point 1 shall equal the amount moving out at point 2 Continuity Equation A 1 v 1 = A 2 v 2 (=constant)
Quiz: Water Flow From Faucet q Quiz: Why the water is getting narrower while going down? q Answer: By gravity, water is accelerated while going down v 2 > v 1 Per continuity: A 1 v 1 =A 2 v 2 A 2 <A 1! v 1 A 1 v 2 A 2
Fluid Dynamics: Bernoulli s Equation The net work done between point 1 and 2 shall equal change energy Bernoulli s Equation P 1 + ½ ρv 1 2 + ρgy 1 =P 2 + ½ ρv 2 2 + ρgy 2 direct result of energy conservation
Demo: Paint Sprayer q Explain why water can be sucked up (and sprayed out) q Also: Ping-Pong balls in air tubes, Bernoulli ball etc.
Demo: The Venturi Tube q Larger A, lower v same height: P+ ½ ρv 2 = constant lower v higher P higher P lower liquid height
Discussion: Airplane Wing to be countered by propeller/jets Air flow faster flying direction Air flow slower