b/This doesn’t happen. If
pressure could vary horizontally
in equilibrium, the cube of water
would accelerate horizontally.
This is a contradiction, since
we assumed the fluid was in
equilibrium.c/The pressure is the same
at all the points marked with dots.d/This does happen. The
sum of the forces from the
surrounding parts of the fluid is
upward, canceling the downward
force of gravity.Variation of pressure with depth
The pressure within a fluid in equilibrium can only depend on
depth, due to gravity. If the pressure could vary from side to side,
then a piece of the fluid in between, b, would be subject to unequal
forces from the parts of the fluid on its two sides. Since fluids do not
exhibit shear forces, there would be no other force that could keep
this piece of fluid from accelerating. This contradicts the assumption
that the fluid was in equilibrium.
self-check A
How does this proof fail for solids? .Answer, p. 1056
To find the variation with depth, we consider the vertical forces
acting on a tiny, imaginary cube of the fluid having infinitesimal
height dy and areas dAon the top and bottom. Using positive
numbers for upward forces, we have
PbottomdA−PtopdA−Fg= 0.The weight of the fluid isFg=mg=ρV g=ρdAdy g, whereρis
the density of the fluid, so the difference in pressure is
dP=−ρgdy. [variation in pressure with depth for
a fluid of densityρin equilibrium;
positiveyis up.]
A more elegant way of writing this is in terms of a dot product,
dP=ρg·dy, which automatically takes care of the plus or minus
sign, depending on the relative directions of thegand dyvectors,
and avoids any requirements about the coordinate system.
The factor ofρexplains why we notice the difference in pressure
when diving 3 m down in a pool, but not when going down 3 m of
stairs. The equation only tells us the difference in pressure, not the
absolute pressure. The pressure at the surface of a swimming pool
equals the atmospheric pressure, not zero, even though the depth is
zero at the surface. The blood in your body does not even have an
upper surface.
In cases wheregandρare independent of depth, we can inte-
grate both sides of the equation to get everything in terms of finite
differences rather than differentials: ∆P=−ρg∆y.
self-check B
In which of the following situations is the equation∆P=−ρg∆yvalid?
Why? (1) difference in pressure between a tabletop and the feet (i.e.,
predicting the pressure of the feet on the floor) (2) difference in air pres-
sure between the top and bottom of a tall building (3) difference in air
pressure between the top and bottom of Mt. Everest (4) difference in
pressure between the top of the earth’s mantle and the center of the
earth (5) difference in pressure between the top and bottom of an air-
plane’s wing .Answer, p.
1057
Section 5.1 Pressure, temperature, and heat 311