Archimedes’ Principle Physics 202 Professor Lee Carkner Lecture 2 “Got to write a book, see, to prove you’re a philosopher. Then you get your … free official philosopher’s loofah.” --Terry Pratchett, Small Gods

PAL #1 Fluids  Column of water to produce 1 atm of pressure     = 1000 kg/m 3   h = P/  g = 10.3 m  Double diameter, pressure does not change   On Mars pressure would decrease 

Archimedes’ Principle   The fluid exerts a force on the object    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

Buoyancy

Will it Float?  What determines if a object will sink or 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 denser the object, the lower it will float, or:   Example: ice floating in water, W=  Vg  V i /V w =  w /  i  w = 1024 kg/m 3 and  i = 917 kg/m 3

Iceberg

Ideal Fluids  Steady --  Incompressible --  Nonviscous --  Irrotational --  Real fluids are much more complicated  The ideal fluid approximation is usually not very good

Moving Fluids  Consider a pipe of cross sectional area A with a fluid moving through it with velocity v   Mass must be conserved so,  If the density is constant then, Av= constant = R = volume flow rate    Because the amount of fluid going in must equal the amount of fluid going out 

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

Continuity

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  The work of the system due to lifting the fluid is,  The work of the system due to pressure is, W p =Fd=pAd=  p  V=-(p 2 -p 1 )  V  The change in kinetic energy is,   Equating work and  KE yields, p 1 +(1/2)  v  gy 1 =p 2 +(1/2)  v  gy 2

Fluid Flow

Consequences of Bernoulli’s Equation   Fast moving fluids exert less pressure than slow moving fluids  This is known as Bernoulli’s principle  Based on conservation of energy   Note that Bernoulli only holds for moving fluids

Constricted Flow

Bernoulli in Action  Blowing between two pieces of paper   Convertible top bulging out   Shower curtains getting sucked into the shower

Shower Physics

Lift  Consider a thin surface with air flowing above and below it   This force is called lift  If you can somehow get air to flow over an object to produce lift, what happens?

December 17, 1903

Deriving Lift  Consider a wing of area A, in air of density   Use Bernoulli’s equation:  The difference in pressure is: p b -p t =1/2  v t 2 -1/2  v b 2  Pressure is F/A so:  L=F b -F t and so:  If the lift is greater than the weight of the plane, you fly

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