Monday:
We discussed the upcoming test. The material for the test will include Chapters 1 to 5 of Pigliucci
and Chapter 1 of Rothman (I accidentally said Chapter 2 (atoms) of Rothman in class when I mean Chapter 3
(energy) so I will not hold you responsible for that.) You are responsible for any physics theory, history,
and philosphy discussed in class. The test will include material from six labs (Levers, Buoyancy, Incline,
Projectile, Netwon's second law, and Momentum Conservation.)
We continued our discussion of energy. We recalled our other conservation laws such as momentum and that
a conservation law often involves the identification of a "system", and that under some isolation
conditions (no external forces in the case of momentum) that the quantity is conserved within the system --
that it does not change in time -- that it has the same value "before" and "after".
Types of energy. When we derived our first concept of energy (from the equations for the velocity and
position when the acceleration was constant) we saw that there were two basic categories of energy:
kinetic and potential. Kinetic Energy is the "energy of motion" while Potential Energy is
an energy associated with a particular arrangement or relative position of object that can be converted
to motion -- for instance an object raised to some height that can be released and start to move.
We considered other groupings of energy
- Chemical energy (e.g. food or gasoline)
- Nuclear energy (e.g. fission -- breaking bigger atoms into smaller -- and fusion -- combining
smaller atoms into bigger). With nuclear we said that we really need to combine what we originally
thought to be two separate conservation laws, matter and energy, into one conservation law
matter/energy -- with the two quantities related by Einstein's famous equation E=mc2.
- Mechanical -- including gravitational and spring potential energy.
- Wind (a type of kinetic energy)
- Heat (a disorganized version of kinetic energy)
With heat we considered that we will eventually need to discuss useful/organized energy versus
non-useful/disorganized energy. We will eventually introduce the concept of entropy to make this
distinction.
If an object is thrown up from the surface of the Earth, it has three possible fates:
1) it may return to the surface of the earth; 2) it may start orbiting the earth; and
3) it may "escape" the earth's gravitational hold. We will consider this last scenario from an energy point of
view. The object is moving and thus has kinetic energy. The acceleration due to gravity is opposite to the
direction of the velocity and thus the object is slowing down -- or in energy language the
kinetic energy is being converted to gravitational potential energy. But the question is
whether we always reach a point where the object turns around -- where all of its kinetic
energy is converted to potential energy.
Since we are asking the question about whether we can get far from the surface of
the Earth, we cannot use the simplified gravitation potential energy m g h (based on Galileo's
terrestrial version of gravity) for that applies only near the surface. Instead we must the more general
form Ugrav = - G M m / R (based on
Newton's universal law of gravitation). Note this formula looks similiar to Newton's form for the
gravitational force F = - G M m / R2, but has only one R in the denominator
while the force has R2. (They are of course similar looking because the energy is related to
the integral of the force.)
Etotal = m v2 / 2 - G M m / R
where m is the mass of the object, M is the mass of the Earth and R is the distance from the object to the center of the
Earth. We say that we can "escape" the Earth if R goes to infinity. If we take R to infinity and v to zero, we have the
situation of just being able to escape, and if we plug these values in above we get Etotal = 0. Because energy
is conserved, if the energy is zero or higher at the surface on the Earth, it remains so -- unless something from the outside
the system (not the Earth) does work on the object.
m vescape2 / 2 - G M m / R = 0
Solving for v yields the so-called escape velocity.
vescape = ( 2 G M / R )1/2.
If one has this velocity or more when one is at the Earth's surface then one can escape the Earth's gravitional field.
We went on to say that for objects other than the Earth that the combination ( 2 G M / R )1/2 might
exceed the speed of light c. In such an instance then an object cannot escape since it cannot have a velocity
greater than the speed of light (a limit that comes from Einstein's theory of special relativity). Considerations
like these give rise to the concept of a Black Hole.
Another thing we mentioned is that the concept of ionization in chemistry -- an electron escaping the hold of its
positive nucleus is completely analogous. The mathematics is pretty much the same since as we will eventually
discuss the (Coulomb) force between charges has an "inverse square law" just like gravitation. That the concept of
escaping is an energy consideration is perhaps more apparent in the chemistry situation since one talks of
ionization energies.
