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Chapter

Physics 1029A/B Chapter Notes -Electric Potential, Shear Modulus, Thermal Energy


Department
Physics
Course Code
PHYS 1029A/B
Professor
Martin Zinke- Allmang

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CHAPTER 12: STATIC FLUIDS JUNE 17, 2014
Introduction
Term fluid includes liquids and gases
oAre deformable – can evolve towards a mechanical equilibrium
Intermolecular forces dominate = fluid is found in liquid state
Thermal energy of system dominates = fluid found in gaseous state
12.1 Model System: The Ideal Stationary Fluid
Fluids are systems that yield to any force that attempts to alter their shape, causing
system to flow until it reaches mechanical equilibrium
oFluid conforms to shape of the container
oRefers to liquid and gases (not solids b/c remain unchanged when placed in
containers of different sizes)
Molecules in a liquid maintain a fixed intermolecular distance
Liquid put into container with volume larger than volume of liquid, liquid forms a
surface
Gases have no surface b/c gases adjust intermolecular distance and fill any space
Fluids in mechanical equilibrium are called stationary
Properties essential for an ideal stationary fluid
oThe ideal stationary fluid is incompressible
Volume and density (ρ = m/V) are constant and doesn’t depend on P
Does not apply to gases (use ideal gas model)
oThe ideal stationary fluid is deformable under influence of forces and seeks
mechanical equilibrium
Mechanical equilibrium needs to be established for fluid to be
stationary
Applies to both liquids and gases

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12.2 Pressure in an Ideal Stationary Fluid
Pressure variations in liquids are more profound
12.2.1 Pascal’s Law
A small rectangular prism of fluid is identified in a beaker with three forces acting on
it:
oTwo contact forces due to remaining fluid on top and below the object (Fup
and Fdown) and weight of fluid element (W)
oElement is at rest in an ideal stationary fluid
FNet,y = 0 = Fup – W – Fdown
Rectangular prism has horizontal surface area A and height of Δy = yup – ydown
oThis is a small length b/c pressure in the fluid expected to vary in vertical
direction
Volume is given by V = AΔy
All three forces are rewritten
oMass is rewritten as  m = ρV = ρAΔy
oWeight of fluid element is  W = mg = ρgAΔy
Fluid element stretches from y0 to y0 + Δy
oAt y0 the pressure is labeled p and at y0 + Δy it is labeled p + Δp
oFup = pA
oFdown = (p + Δp)A
Rewrite Newton’s law in the form of
opA – (p + Δp)A – ρgAΔy = 0
oCombine first two terms and then divide by A, we get
oΔp = -ρgΔy
This equation applies whether the density is constant or varies with depth
oRestrict further discussion to cases with constant density b/c

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Ideal stationary fluid is incompressible
General case requires calculus
When ρ = constant, previous equation applies for any depth difference Δy
Pascal’s Law: p2 – p1 = -ρg(y2 – y1)
Used when surface of fluid can’t be identified and can’t be used as reference point
Pascal’s law states that difference b/w pressures at two different positions in a fluid
of constant density is proportional to vertical distance b/w two positions
oProportionality factor is product of the ρ of the fluid and gravitational
acceleration
12.2.2 Pressure in Liquids with a Visible Surface
In systems with an identifiable surface of the liquid, index 1 in above equation is
referred to the surface of the liquid
y1 = 0 and p1 = patm
patm is proper value for pressure of the fluid surface since in mechanical equilibrium
Force pushing surface upwards equals force caused by air pressure pushing
downwards
More useful to define depth below the surface as a positive distance (y2 = d)
op = patm + ρgd
Pascal’s law doesn’t describe pressure variations in the atmosphere
oIllustrated by calculating height of upper end of atmosphere by using y2 = ymax
o(py,max – pground) = -ρg(ymax – yground)
At max height pressure drops to 0 atm b/c ymax is where vacuum of
outer space would begin, yground = 0 m and pground = patm, and ρ = 1.2
kg/m^3
ymax = 8614 m
Therefore, assumption of constant density throughout atmosphere is
inadequate b/c gases are compressible and density depends on
pressure
Pascal’s law doesn’t contain any info about shape of container & regardless of
shape of container, pressure increases below surface and results in fixed value at
given depth
Pressure data are often given in non-standard units  convert
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