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Why Pressure is Important:
Today we will try to learn, from the point of view of pressure,
In the last lecture, we considered a rather unusual application of pressure when we considered the early universe. The early universe was characterized by a very large ratio of radiation to matter. The radiation exerts a pressure on the matter and tries to push the matter apart, even though gravity is trying to pull matter together.
If the radiation had succeeded in producing a smooth distribution of matter there would be no galaxy formation and no intelligent life in the Universe. Hence, overcoming pressure is a big deal. You can read more about this right here
Towards the Bernoulli Equation.
Flow from a reservoir:
We can use an application of energy conservation to determine the velocity of flow along a pipe from a reservoir. Consider the 'idealised reservoir' in the figure below.
The level of the water in the reservoir is z1. The velocity of the water in this reservoir u1 is zero. The potential energy of the reservoir is mgz1 and this is also the total energy of the system.
If a pipe is attached at the bottom water flows along this pipe out of the tank to a level z2 and a mass of water, m flows from
the top of the reservoir to the nozzle it will have gained some
velocity u2 what is this velocity?
and we have solved the problem, namely
Note that the diameter of the pipe would be
irrelevant to this caculation. The velocity of the fluid is only
determined by the height difference (z1 - z2).
However, if you multiply the velocity times the cross sectional area
of the pipe you get:
m/s * m2 = m3/sec
Which is the volume flow rate. Thus a larger diameter pipe will empty
the reservoir faster than a small diameter but not because the velocities
are different. The velocity of the flow is independent of the diameter
of the pipe.
This is just one example of a problem that can be solved using energy
conservation. Study it carefully, it will be quite relevant for the
final.
We know that Pressure = Force/Area
We also know that Force x distance = Energy (e.g. mgh).
So we can write that Pressure = Force/Area * d/d (d = distance).
Force x distance = energy; area*d = volume and so we can also write pressure as Pressure = Energy/Volume.
So let's return to our equation for energy conservation in a more general case involving kinetic energy, potential energy and energy losses. We then have:
We then have:
Rearranging terms we see we can express a pressure difference as.
Bernoulli's equation has some restrictions in its applicability:
An air foil works because it is a surface that forces the air to flow faster over the top of the surface than the bottom of the surface. This creates a pressure difference which causes the air foil to lift.
In baseball, the Bernoulli principle produces a curve ball as the spinning ball through the air will have a higher air flow on one side of the ball compared to the other. That creates a pressure difference which tends to continuously push the ball.
Stars and Black Holes
Any object with some mass, M, has a gravitational force which is
trying to pull the object together. The force of gravity on your
body is compenstated for by the pressure exerted outwards by your
skeletal structure.
The earth is prevented from collapsing because of the crystal
structure of its interior which is sufficient strong to exert
a pressure.
What about a star? Why doesn't the sun collapse in to a black hole?
What causes it to be stable against gravitational collapse?
This was discussed in class but for now, we will leave these as
questions to think about for the Final exam.