must be opposed by the pressure in the uid, which exerts an equal
and opposite upwards force: F = Fp = P0A (731)
where A is the cross sectional area of the piston and where weve put
the cylinder face at z = 0, which we are obviously free to do. For all
pr

We will ignore these in this course. Pressure is the force per unit
area exerted by a uid on its surroundings: P = F/A (687)
Its SI units are pascals where 1 pascal = 1 newton/meter squared.
Pressure is also measured in atmospheres (the pressure of air at

which implies that the pressure at the left and right conning walls is
the same:
Pleft = Pright = P (707)
, and that this pressure describes the force exerted by the uid on the
walls and vice versa. Again, the exact same thing is true for the other
four s

properties of and laws pertaining to static uids, uids that are in
static equilibrium.
Static Fluids
8.1.6: Pressure and Connement of Static Fluids
Fleft Fright
V
A
Fluid (density )
Confining box
Figure 102: A uid in static equilibrium conned to a sealed

displays features that are the results of moderate interaction,
depending on the pressure and temperature. Water131 is, as noted, a
remarkable liquid. H2O is a polar molecules with a permanent dipole
moment, so water molecules are very strongly interactin

In b):
Tw + Fb mg = 0 Tw = mgFb = crownV gwaterV g = (crown
water)V g (749)
We know (we measured) the values of Ta and Tw, but we dont know V
or crown. We have two equations and two unknowns, and we would
like most of all to solve for crown. To do so, we

Heres the general idea. If we consider a tiny (eventually di erentially
small) chunk of uid in force equilibrium, gravity will pull it down
and the only thing that can push it up is a pressure dierence between
the top and the bottom of the chunk. By requi

omit the modier dynamical from the term viscosity in this course,
although there is actually another, equivalent measure of viscosity
called the kinematic viscosity, = /. The primary di erence is the
units has the SI units of pascal-seconds where has unit

the large end can be neglected in Bernoullis Equation. In that case
the exit speed is the same as the speed of a mass dropped the same
distance: v =p2gH (695)where H is the depth of the hole relative to the
top surface of the uid in the tank.
8.1: General

opposite in their y-direction) and similarly for the force on the front
and back faces in the x-direction, which will always be true if the
pressure does not vary horizontally with variations in x or y. In the zdirection, however, force equilibrium requir

Newton meter2 (698) A Pascal is a tiny unit of pressure a Newton
isnt very big, recall (one kilogram weighs roughly ten Newtons or 2.2
pounds) so a Pascal is the weight of a quarter pound spread out over a
square meter. Writing out pascal is a bit cumbers

Example 8.1.2: Variation of Oceanic Pressure with Depth
The pressure on the surface of the ocean is, approximately, by
denition, one atmosphere. Water is a highly incompressible uid with
w = 1000 kilograms per cubic meter142. g 10 meters/second2.
Thus: P(

used the pythagorean theorem and/or inspection of the gure to
determine r for each of the two forces. No torque due to N is
present, so Fm in this case is indeed the minimum force F at the
marginal point where rotation just starts to happen:
Fm =
mgpR2 (R

In the gure above, a force
~ F = 2 x + 1 y
Newtons is applied to a disk at the point
~r = 2 x2 y as shown. (That is, Fx = 2 N, Fy = 1 N, x = 2 m, y = 2
m). Find the total torque about a pivot at the origin. Dont forget that
torque is a vector, so specify

12r
Figure 99: A Force Couple is a pair of equal and opposite forces that
may or may not act along the line between the points where they are
applied to a rigid object. Force couples exert a torque that is
indenpedent of the pivot on an object and (of cou

At one standard atmosphere, we can easily determine what a mercury
barometer at room temperature will read (the height H of its column
of mercury above the level of mercury in the reservoir):
P0 = 13534
kg m3 9.80665
m sec2 H = 101325Pa (719)
Note well, w

between the two hinges, nd the total force (magnitude and direction)
exerted by each hinge.
Neglect the mass of the doorknob and assume that the center of mass
of the door is at W/2,H/2. The force directions drawn for you are NOT
likely to be correct or e

original denition of the standard atmosphere. For better or worse,
Torricellis original observation dened one standard atmosphere to be
exactly 760 millimeters of mercury (which is a lot to write or say) or
as we would now say, 760 torr139.
Mercury barome

s
Figure 98: A uniform rectangular block with dimensions W by H
(which has its center of mass at W/2, H/2) is pushed at a height h by a
force F. The block sits on a horizontal smooth table with coe cient of
static friction s.
A uniform block of mass M bei

where we assume the upper surface is at depth z (this wont matter, as
well see in a moment). Since P(z +h) = P(z)+gh, we can nd the net
upward buoyant force exerted on this little cross-section
360 Week 8: Fluids
h
A
A
A
F = g h A = g V b (up)
F = P(z +

with 10% silver would have decreased its density by around 5%. If the
mass of the crown was (say) a kilogram, the goldsmith would have
stolen 100 grams almost four ounces of pure gold at the cost of 100
grams of silver. Even if he stole twice that, the 9%

A
V
Figure 111: Fluid in uniform ow is transported down a pipe with a
constant cross-section at a constant speed v. From this we can easily
compute the ow, the volume per unit time that passes (through a
surface that cuts the pipe at) a point on the pipe.

P (726)
where M is the molar mass148 , the number of kilograms of the gas
per mole. Note well that this result is idealized thats why they call it
the Ideal Gas Law! and that no real gases are ideal for all
pressures and temperatures because sooner or lat

massless) sheet of Plexiglas placed on top. What is the vertical force
exerted by the Plexiglas on each leg when the mass is in the position
shown?
Week 7: Statics 331
Problem 7.
T
m
M
L
P
This problem will help you learn required concepts such as:
Force

(in a barometer or otherwise) is that you can easily spill it, and once
spilled it is fairly likely to sooner or later make its way into either
mercury vapor or methyl mercury, both of which are biologically
active and highly toxic. Liquid mercury itself

When they do, their momentum perpendicular to those walls is
reversed. Since many, many of these collisions occur each second,
there is a nearly continuous momentum transfer between the walls
and the gas and the gas and the walls. This transfer, per unit

where P0 is the constant of integration for both integrals, and
practically speaking is the pressure in the uid at zero depth (wherever
that might be in the coordinate system chosen).
Example 8.1.1: Barometers
Mercury barometers were originally invented b

This principle is enormously important and ubiquitous. Buoyancy is
why boats oat, but rocks dont. It is why childrens helium-lled
balloons do odd things in accelerating cars. It exerts a subtle force on
everything submerged in the air, in water, in beer,

y
x
z
P(z + ) z)
z + z
P(z)
P(0) = P 0
mg
Figure 107: A solid chunk of stu of mass m and the dimensions
shown is immersed in a uid of density at a depth z. The vertical
pressure dierence in the uid (that arises as the uid itself becomes
static static) exe

breathe with order of Avogadros Number (61023) molecules per liter
hundreds of billions of billions per cubic centimeter.
At this point we cannot possibly track the motion and interactions of
all of the individual molecules, so we coarse grain and averag

http:/www.wikipedia.org/wiki/Eects of high altitude on humans.
This is the height where air pressure drops to where humans are at
extreme risk of dying if they climb without supplemental oxygen
support beyond this height hypoxia reduces ones ability to ma

To summarize, uids have the following properties that you should
conceptually and intuitively understand and be able to use in working
uid problems:
They usually assume the shape of any vessel they are placed in
(exceptions are associated with surface ee

M = V (705)
133This state will also entail thermodynamic equilibrium with the box
(which must be at a uniform temperature) and hence the uid in this
particular non-accelerating box has a uniform density.
Week 8: Fluids 347
where V is the volume of the box

the integral will be too complicated to cope with. Pascals Principle is
more commonly given in English words as:
Any change in the pressure exerted at a given point on a conned uid
is transmitted, undiminished, throughout the uid.
Pascals principle is the

R
h
F
pivot
Figure 100:
One classic example of static equilibrium and force couples is that of a
ball or cylinder being rolled up over a step. The way the problem is
typically phrased is:
a) Find the minimum force F that must be applied (as shown in gure