We discussed the conservation of momentum and how it arises from Newton's Laws of Motion. An object's (linear) momentum p is the product of its mass m and its velocity v. To consider conservation of momentum, we first establish a "system" of objects. We were free to choose which objects would be part of our system. The system's momentum is just the sum of the momenta of the objects in the system. The law of conservation of momentum says that if the net external force on a system is zero, then the momentum of the system is conserved (is constant, does not change). An external force originates from outside the set of objects and acts on an object or objects within the system, as opposed to an internal force which would be between objects within the system.

Recall that Newton's Second Law, usually written as F=ma, can also be written as F=dp/dt, the net force on an object is equal to its rate of change of its momentum. Newton's Third Law says that any internal force has an equal and opposite reaction force. Thus while two objects within the system influence each other, one's increase in momentum will offset by the other's decrease in momentum resulting in no change in the system's momentum.

We next returned to results we obtained while discussing Galileo, that is, the equations for one-dimensional motion under constant acceleration.

v = v₀ + a t

x = x₀ + v₀ t + a t² / 2

Solving the first equation for time t = (v - v₀) /a and substituting it into the second equation yields

x = x₀ + v₀ (v - v₀) /a + a (v - v₀)² /(2 a²)

x = x₀ - v₀² / (2 a) + v² / (2 a)

v² / 2 - a x= v₀² / 2 - a x₀

We could that this represents a kind of conservation law in that we have a quantity that is the same "before" and "after". Next, multiplying by the mass m resulted in

m v² / 2 - m a x= m v₀² / 2 - m a x₀

where we identified mv²/2 as the kinetic energy. Thus the quantity we have being conserved is energy, which comes in Joules. Finally we noted that we could replace ma with F using Newton's Second Law.

m v² / 2 - F x= m v₀² / 2 - F x₀

This result is generalizable to constant accelerations (involves integration) and dimensions higher than one (involves what is called a "dot product") but we did not pursue this.

We continued talking about the conservation of energy, which says that the amount of energy in an isolated system does not change -- that the energy may change form but that the total amount does not change. Again here we noted the concept of a "system". The F x term we had derived the previous lecture can be associated either with "work" (if the force is external to our system) or "potential energy" (if the force is internal to our system). For instance if we do not consider the Earth as part of our system we would talk about work done by gravity, but if we include the Earth as part of our system we would talk about gravitational potential energy.

We asked if the Earth was an isolated system as far as energy is concerned, and said no, that the Earth receives energy from the sun and radiates energy out to space. Whether the same amount of energy comes in as goes out is one of issues looked at in global warming studies.

We listed a number of forms of energy, which typically fall into two categories, energy associated with motion: kinetic, heat, wind, waves (as in the ocean), solar; and potential energies: gravitational, spring, chemical (gas, oil, food), nuclear.

We listed several units of energy: Joule, erg, calorie, foot-pounds, kilowatt-hours.

We discussed how especially the concept of heat was difficult to disentangle from related concepts. Heat was most strongly associated with combustion, and there was initial confusion between air, oxygen, other gases, and heat. Experiments began to distinguish among the different gases by have the experimenter and later rats or mice breathe the gas and .... As far as heat goes we need to make two important realizations -- that it was a type of enery and that it was distinct from temperature.

Rumford studied the relationship between heat and work -- the more work applied to an object the more heated produced. Joule studied the equivalence of heat and energy -- a raised weight (having potential energy) was lowered and turned paddles in a closed system of liquid (water?) producing heat. The amount of heat produced was directly related to the inital energy of the raised weight.

Scientists (Black?) realized that temperature was distinct from heat by considered melting ice. Heat was being added to it, but during the melting process the temperature was not changing.

We mentioned that another name for the conservation of energy is the First Law of Thermodynamics, and that as the name suggests there is a Second Law to come.

That second law was about "entropy". One interpretation of entropy is as a measure of disorder in a system. Another related topic is whether a system's energy is "useful" -- whether it can be exploited to do work. The Second Law says that in a closed/isolated system, that the entropy will not decrease. Or that the useful energy tends to decrease.

The two laws of thermodynamics have associated with them a type of perpertual motion machine that they rule out. A perpetual motion machine of the first kind violates the first law of thermodynamics and claims to produce more energy than is put into the system. A perperual motion machine of the second kind violates the second law of thermodynamics and claims to produce more useful energy than is put into the system, i.e. to decrease entropy.