A dull knife has a larger surface area than a sharp knife. So if you apply the same force with each, the sharp knife exerts more pressure. Pressure increases with depth in a fluid: P = Patm + hrg where is the pressure of the air above the fluid, h is the depth in the fluid, r is the density of the fluid and g is the acceleration of gravity. Imagine an object submerged in a fluid. Because pressure increases with depth, the pressure on the bottom of the object is greater than the pressure on the top. The bigger the object the larger the pressure difference. This is known as Archimedes' principle: any object submerged in a fluid experiences a buoyant force proportional to the weight (in Newtons) of the fluid displaced. If the buoyant force is larger than the weight of the object, the object will float (this explains why battleships float even though they are made of iron which is more dense than water). A hydraulic lift (or hydraulic jack) works by having the same pressure at the pump and in the part which lifts the car. The pump end has a small area so the force applied is small whereas the lift end had a large area so the force there can be large: F1/A1 = P = F2/A2. Note: you don't get something for nothing here- the pump end has to move proportionally further than the lift end so that the work at both ends is the same.
Two scientists (Charles' and Boyle's) found empirically (by experiment) several hundred years ago that pressure and volume are both directly proportional to temperature for simple gasses at low pressures. These simple gasses at low pressures became know as idea gasses and the relationship known as the ideal gas law:
PV = nRT where n is the number of moles of gas (1 mole = 6.02x1023 atoms; the umber of objects in a mole is known as Avogadro's number, NA) and R is the universal gas constant (R = 8.31 J/mol K).
Physicists often write the ideal gas law
a different way using Boltzman's constant k = R/NA (so k = 1.38x10
-23):
PV = NkT where N is the number of
atoms (don't get n, N and NA confused!).
Now from our relationship between pressure
and kinetic energy we have that PV = NkT = (2/3) N <mv2/2>.
This gives us a direct relationship between a macroscopic variable, T and
a microscopic measurement, the average kinetic energy:
(3/2)kT = <mv2/2>.
Although we derived this for an ideal gas it turns out to be universally true (for non-ideal gasses, solids, etc.): temperature is directly proportional to the average kinetic energy of the atoms or molecules of the substance. This turns out to be very useful.
Another way to write this same expression
is to solve for v in terms of temperature. We can't really solve for v
because the square root of the average velocity is not the same thing as
the average of velocity (better think about that one). But we can solve
for the square root of the average velocity (the root mean square or rms
value):
vrms = squareroot (3kT/m)
which
is also useful, as we'll see later.
Note that tour above expression defines a temperature scale where no kinetic energy is equivalent to zero degrees. This is called the Kelvin scale; it was already invented from entirely different reasoning having to do with the behavior of the volume of ideal gasses as the temperature was lowered. If you plot the volume of an ideal gas as the temperature is lowered you get a straight line (T and V are directly proportional by the ideal gas law). This is not surprising but what was surprising was that all of these straight line graphs (for any ideal gas) converged to the same temperature at zero volume if they were extrapolated downward. So this temperature was thought to be special and now we know why. As we will see next semester, the idea that absolute zero temperature corresponds to zero kinetic energy is not exactly correct. Turns out you can't actually reach zero kinetic energy (zero Kelvin); there is a residual zero point energy which cannot be removed from a collection of molecules. This is a result of quantum mechanics- the modification of Newtonian mechanics for very small objects such as atoms.
Very small objects such as pollen or
dirt particles which are visible in an optical microscope appear to jiggle.
This is called Brownian motion after Robert Brown who first noticed
it in 1827. In 1905 Einstein published a paper which explained that Brownian
motion was due to the microscopic object being constantly bombarded by
even smaller objects called molecules. His mathematical description of
this effect gave an approximate size for molecules and was considered to
be the first real proof of the existence of molecules.
The official definition of an ideal gas
includes the fact that ideal gasses can only have kinetic energy; they
are assumed to be point particles which never interact with each other.
Real gasses, on the other hand, have molecules which rotate, vibrate and
interact with each other (these interactions are quantum mechanical and
electrical in nature and are generally referred to as chemical potential).
The
total energy a system's molecules have is called the internal energy, U,
measured in Joules. The average kinetic energy is still called the temperature
but the total kinetic, vibrational, and chemical energy is called the internal
energy. So for an ideal gas the temperature and the internal energy
are equivalent because ideal gasses do not have any other kind of energy
(they don't interact, rotate or vibrate).
Let's look at some simple models for the
interaction energy (chemical potential) between atoms or molecules in a
solid. Very similar interactions hold for liquids and non-ideal gasses
but it is easier to start off thinking of how atoms interact in a solid.
