Physics 1B03summer-Lecture 12 1 Day of Wrath Tuesday June 16 9:30-11:30 am CNH MC Questions, Cumulative

Physics 1B03summer-Lecture 12 2 Fluid Mechanics Pressure Buoyancy

Physics 1B03summer-Lecture 12 3 Buoyancy h P1P1 P2P2 The fluid exerts an upward force on the object equal to the weight of fluid displaced. This force is result of pressure differences in the fluid; the pressure is greater at the bottom than at the top. This is why objects can ‘float’: the magnitude of the buoyant force equals that of the force of gravity: B = F g so: B = mg = ρ o V o g

Physics 1B03summer-Lecture 12 4 Archimedes’s Principle: the magnitude of the buoyant force always equals the weight of the fluid displaced by the object. Question: A boat filled with bricks floats in a swimming pool. If the bricks are dumped overboard does the water level in the pool rise, or fall? ?

Physics 1B03summer-Lecture 12 5 Totally Submerged Object For an object submerged in a fluid with ρ f the upward force is B=ρ f V o g (=weight of fluid displaced) and so the net force is: B – F g = ρ f V o g - ρ o V o g = (ρ f – ρ o )V o g Floating Object For a floating object, it displaces a volume of fluid equal to its own volume, and so: B = F g ρ f V f g = ρ o V o g

Physics 1B03summer-Lecture 12 6 Example A giant ice cube (   917 kg/m 3 ) floats in a pail of cold water. If the cube is mm on each side, how far is the top surface of the ice above the water? When the ice melts, does the level of water in the pail rise or fall?

Physics 1B03summer-Lecture 12 7 Example A beach ball of mass 0.05 kg and radius of 0.2 m is filled with air (density = 1.29 kg/m 3 ) and placed under water. a) What is the net force on the ball? b) What is the acceleration of the ball?

Physics 1B03summer-Lecture 12 8 Example: How heavy a balloon can 100 litres of helium lift? (ρ He =0.18kg/m 3 )

Physics 1B03summer-Lecture 12 9 Example A 10-kg rock (density 2500 kg/m 3 ) is suspended in a large bucket of water by a cord. What is the tension in the cord? tension = ?

Physics 1B03summer-Lecture min rest

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

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

Physics 1B03summer-Lecture 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

Physics 1B03summer-Lecture Exam Tips + Course Evaluations