2c: Density and Pressure

The density of something is its mass divided by its volume (m/V) and is measured in kilograms per cubic meter, kg/m3 (or sometimes grams per cubic centimeter; g/cm3). So a kilogram of feathers and a kilogram of iron have the same mass and weigh the same but since the kilogram of feathers takes up more space (larger volume) it is less dense. One other version of density we will use is called the linear density which is the mass per length in kg/m. Bass strings on guitars and in pianos have a larger linear density than the strings used for the higher notes. We will see why later on.
Pressure is defined to be a force acting over an area; P = F/A. There are several units of pressure; we will use the Pascal, Pa, which is a Newton per meter squared, N/m2. Other units are the bar; atmospheres; millimeters of mercury, mmHg; inches of water; Torr, etc. A larger force over the same area increases pressure but the same force over a larger area decreases pressure. A dull knife does not apply the same pressure as a sharp knife because the area of contact of the dull blade is larger than the area of contact of the sharp blade. Being stepped on by the heel of a high heeled shoe hurts a lot more than if the same person steps on you with a low heel because the same force (the persons weight) applied with a high heel acts over a smaller area so the pressure is much higher. As we will see, the loudness of a sound wave is related to pressure; high volume sound exerts more pressure on average, and therefore more force on the surface of your eardrum.

For gasses in a closed container, pressure and volume are inversely proportional (pressure increases as volume decreases). Pressure and volume are also each directly proportional to temperature (either pressure or volume or both will increase if the temperature increases). These properties are sometimes summarized as the ideal gas law which can be written as PV = nRT. Here P is pressure, V is volume, T and is temperature in Kelvin. The variable n indicates how much gas there is (in moles where a mole is 6.0 × 1023 atoms or molecules) and R is a constant equal to 3.14 J/mol K.

Pressure in a liquid or gas is the weight of the liquid times the depth (and is measured in the same units as pressure) or P = mgh where h is the depth, m is the mass and g is gravitational acceleration. We sit at the bottom of a sea of air that pushes down on us. This pressure is called atmospheric pressure and it varies a little bit from day today because the air above us is moving and also because of changes in temperature and humidity (and so its density changes). When you use a straw you are decreasing the pressure inside the straw and atmospheric pressure outside the straw pushes the liquid up into the straw. This is why a straw would not work in a vacuum. If you are under water, the water above you pushes down on you in addition to the air above the water which pushes down on the water. Since water is much more dense than air, pressure changes a lot faster as you go deeper under water than it does if you change altitude in the air.

Bernoulli’s principle says that if the speed of a fluid (liquid or gas) increases, the internal pressure in the fluid decreases. Take a strip of paper one inch wide and 12 inches long. Hold the short end up to your lips and blow. You’ll notice that the strip pulls upward to meet the flowing air. This is because the moving air above has a slightly lower pressure than the stationary air below. A similar effect causes baseballs to change direction (curve balls) and airplane wings to have lift. Some wind instruments and the human voice operate in part because of forces due to the Bernoulli effect, as we shall see.

Video/audio examples:

Definition of pressure.

Bernoulli:

Why does the ball not follow Newton’s first law and travel in a straight line?

Why do the balloon’s come together?

Bed of nails: What would happen to the pressure if the number of nails is reduced? What would this do to the person lying on them?

Simulation exercise 2B (turn in answers on a separate sheet of paper):
    1. There are three simulations. In the first, raise the temperature by sliding adding heat. What happens?
    2. Now look at the second simulation. Raise the temperature until the solid turns into a gas. Pump molecules into the simulator. What is the effect of increasing the pressure?
    3. What do you think will happen if the volume is reduced?
    4. Click on the finger at the top to push the container wall, reducing the volume. Was your prediction correct? Explain.
    5. What happens to pressure and temperature in the simulation when you add heat? Remove heat?
    6. On the right you can choose a different kind of molecule. What happens if you change the type of molecules?
Simulation exercise 2C:
Molecular Dynamics Simulation
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