Physics 1B03summer-Lecture 13 Final Exam April 18 2 hours long – 30 MC questions Covers all material with approximately equal weight, up to and including today’s final lecture on fluids.

Physics 1B03summer-Lecture 13 Fluid Dynamics Equation of Continuity Bernoulli’s equation and examples

Physics 1B03summer-Lecture 13 Fluid Dynamics Approximations: 1)no viscosity (frictionless flow) 2)steady, “laminar” flow. If the flow is turbulent, mechanical energy is lost (converted to thermal energy). 3)“incompressible” fluid. Sufficiently accurate for gases if pressure differences are small.

Physics 1B03summer-Lecture 13 Streamlines -the paths followed by particles in steady flow -velocity is parallel to the streamline - particles never cross streamlines; the streamlines mark out imaginary “tubes of flow” area A 1 speed v 1 area A 2 speed v2

Physics 1B03summer-Lecture 13 Equation of Continuity “Volume flow rate” (volume per unit time) = (cross-sectional area)  (linear velocity) “Mass flow rate” (mass per unit time) = (density)  (volume flow rate) So, if mass in = mass out, then  1 A 1 v 1 =  2 A 2 v 2  Av = mass flow rate = constant or for steady flow. “Incompressible” fluids (density remains uniform): cancel out density to get Volume flow rate = constant or A 1 v 1 = A 2 v 2

Physics 1B03summer-Lecture 13 radius r 1 = 10mm radius r 2 = 5mm A fluid if flowing through a pipe of 10mm radius at a velocity of 10m/s. How fast will it be flowing if the pipe narrows to 5mm in radius ?

Physics 1B03summer-Lecture 13 Bernoulli’s Equation: work and energy in fluids Conditions: steady flow, incompressible fluid. Look at energy balance along a streamline: Change in (kinetic energy/volume) + change in (potential energy/volume) = (net work by pressure)/volume then, or, Note: the above equation looks similar what we have seen before if we replace ρ by m.

Physics 1B03summer-Lecture 13 Example a) What is the velocity of the water leaving the little hole b) How far (horizontally) from the hole does the water hit the ground? d h x

Physics 1B03summer-Lecture 13 Example What is the speed of the water leaving the hole in the tank? gauge pressure P 0 h v

Physics 1B03summer-Lecture 13 Example Water moving at 10m/s through a 1m radius pipe at a pressure of 50kPa. It then falls 50m and goes into a 0.3m radius pipe. What is the water pressure at the bottom ? h