Archimedes’ Principle Physics 202 Professor Lee Carkner Lecture 2

PAL #1 Fluids  Column of water to produce 1 atm of pressure  P =  gh  P =   = 1000 kg/m 3  g = 9.8 m/s 2  h = P/  g =  Double diameter, pressure does not change   On Mars pressure would decrease  Mars has smaller value of g

Archimedes’ Principle  What happens if you put an object in a fluid?   Called the buoyant force   If you measure the buoyant force and the weight of the displaced fluid, you find:  An object in a fluid is supported by a buoyant force equal to the weight of fluid it displaces   Applies to objects both floating and submerged

Will it Float?   Density  An object less dense than the fluid will float  A floating object displaces fluid equal to its weight   A sinking object displaces fluid equal to its volume

Floating  How will an object float?   The volume of fluid displaced is proportional to the ratio of the densities  Example: ice floating in water,  i V i g=  w V w g V w =V i (  i /  w )  w = 1024 kg/m 3 and  i = 917 kg/m 3

Ideal Fluids  Steady --  Incompressible -- density is constant  Nonviscous --  Irrotational -- constant velocity through a cross section   The ideal fluid approximation is usually not very good

Moving Fluids   What happens if the pipe narrows?  Av  = constant  If the density is constant then, Av= constant = R = volume flow rate   Constricting a flow increases its velocity  Because the amount of fluid going in must equal the amount of fluid going out  Or, a big slow flow moves as much mass as a small fast flow

Continuity  R=Av=constant is called the equation of continuity   You can use it to determine the flow rates of a system of pipes   Can’t lose or gain any material

The Prancing Fluids   How can we keep track of it all?  The laws of physics must be obeyed   Neither energy nor matter can be created or destroyed

Bernoulli’s Equation  Consider a pipe that bends up and gets wider at the far end with fluid being forced through it  W g = -  mg(y 2 -y 1 ) = -  g  V(y 2 -y 1 )  The work of the system due to pressure is, W p =Fd=pAd=  p  V=-(p 2 -p 1 )  V   (1/2mv 2 )=1/2  V(v 2 2 -v 1 2 )  p 1 +(1/2)  v  gy 1 =p 2 +(1/2)  v  gy 2

Consequences of Bernoulli’s   Fast moving fluids exert less pressure than slow moving fluids  This is known as Bernoulli’s principle   Energy that goes into velocity cannot go into pressure  Note that Bernoulli only holds for moving fluids

Bernoulli in Action  Blowing between two pieces of paper   Convertible top bulging out  Airplanes taking off into the wind 

Lift   If the velocity of the flow is less on the bottom than on top there is a net pressure on the bottom and thus a net force pushing up   If you can somehow get air to flow over an object to produce lift, what happens?

Deriving Lift   Use Bernoulli’s equation: p t +1/2  v t 2 =p b +1/2  v b 2  The difference in pressure is: p b -p t =1/2  v t 2 -1/2  v b 2  (F b /A)-(F t /A)=1/2  (v t 2 -v b 2 )  L= (½)  A(v t 2 -v b 2 ) 

Next Time  Read:  Homework: Ch 14, P: 37, 42, 47, Ch 15, P: 6, 7

Which of the following would decrease the pressure you exert on the floor the most? a)Doubling your mass b)Doubling the mass of the earth c)Doubling your height d)Doubling the size of your shoes e)Doubling air pressure

Which of the following would increase the pressure of a column of fluid of fixed mass the most? a)Doubling the width of the column b)Halving the density of the fluid c)Halving the mass of the Earth d)Halving the speed of the Earth’s rotation e)Doubling the height of the column

Summary: Fluid Basics  Density =  =m/V  Pressure=p=F/A  On Earth the atmosphere exerts a pressure and gravity causes columns of fluid to exert pressure  Pressure of column of fluid: p=p 0 +  gh  For fluid of uniform density, pressure only depends on height

Summary: Pascal and Archimedes  Pascal -- pressure on one part of fluid is transmitted to every other part  Hydraulic lever -- A small force applied for a large distance can be transformed into a large force over a short distance F o =F i (A o /A i ) and d o =d i (A i /A o )  Archimedes -- An object is buoyed up by a force equal to the weight of the fluid it displaces  Must be less dense than fluid to float

Summary: Moving Fluids  Continuity -- the volume flow rate (R=Av) is a constant  fluid moving into a narrower pipe speeds up  Bernoulli p 1 +1/2  v  gy 1 =p 2 +1/2  v  gy 2  Slow moving fluids exert more pressure than fast moving fluids