Pressure and Temperature

Why Pressure is Important:

Today we will try to learn, from the point of view of pressure,

• Why a star shines
• Why a black hole must exist
• Why an airplane flies

  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?

• The total energy of the system is mgz₁ .

• The potential energy of the water at height z₂ is mgz₂

• The kinetic energy of the moving water is 1/2 mu₂²

By conservation of energy we must have:

mgz₁ = mgz₂ + 1/2 mu₂²

and we have solved the problem, namely

u₂² = 2g(z₁ - z₂)

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.

Now let's rework this idea of conservation of energy into a slightly different form.

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:

E_t= mgh + 1/2 mv² + E₁ where E_t = total energy of the system and E₁ is the energy loss term. Now we divide this equation by V=volume on both sides and remember that mass/volume is density. Density is denoted by the greek letter r.

We then have:

P_t = rgh + 1/2rv² + P₁

This is now an interesting equation in that we see how pressures can be related to velocities. This helps make airplanes fly.

Rearranging terms we see we can express a pressure difference as.

(P_t - P₁) = rgh + 1/2rv² This is known as Bernoulli's equation and its cheif feature is to show that pressure differences increase as the fluid velocity increases.

Bernoulli's equation has some restrictions in its applicability:

• The fluid flow must be steady
• The density must be constant
• Frictional losses are negligible.
• The equation relates the states at two points along a single streamline, (not conditions on two different streamlines).

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.