PHYSICS 107. Lecture 8 Conservation Laws. For every action there is an equal and opposite reaction.

Transcription

1 PHYSICS 107 Lecture 8 Conservation Laws Newton s Third Law This is usually stated as: For every action there is an equal and opposite reaction. However in this form it's a little vague. I prefer the form: The exerted by object 1 on object 2 is equal in magnitude and opposite in direction to the force exerted by object 2 on object 1. Perhaps Newton noticed this was true for his gravitation law and then started to apply it other situations. Remember, the attractions of a gravitational force are equal in magnitude and opposite in direction. The Third Law is certainly true of contact forces. Think about jumping. I exert a force downward on the floor, pushing it away from me. What actually happens is that I go UP, which is the opposite direction from the force I exerted. This can only happen if the floor pushes back. The result of any force is that the momentum p = mv changes. In everyday language momentum is a measure of how hard it is to stop something. That s true in physics too. If there are multiple components of the velocity, then there are multiple components of the momentum: p = (p x,p y,p z ) = (mv x, mv y, mv z ) is a vector.

2 Conservation of Momentum Now let us say that a force acts for a certain time Δt between objects 1 and 2. Now we are going to apply both the Second Law F = ma and the Third Law F 1 = - F 2. F 1 = m 1 a 1 = m 1 (Δv 1 /Δt) = - F 2 = - m 2 a 2 = - m 2 (Δv 2 /Δt). Now we see that we can extract two similar looking terms from this big set of equal quantities: m 1 (Δv 1 /Δt) = - m 2 (Δv 2 /Δt). Canceling the Δt we have m 1 Δv 1 = - m 2 Δv 2. Or since the m s don t change we have that Δp 1 = - Δp 2. Since the total momentum is p total = p 1 + p 2 we also have that Δp total = Δp 1 + Δp 2 = Δp 1 Δp 1 = 0. The change in the total momentum is zero. To put it another way, the total momentum is constant independent of time. To put it yet another way, the total momentum is conserved. This is easy to see with our cars on the air track, because we are able to get fid of friction. It s not so easy to see when I jump. What happens then is that I transfer momentum to earth that is exactly equal and opposite to mine, but the mass of the earth is so huge that Δv is unmeasurably small. Energy When two objects act on one another the concept of force tells us the nature of that interaction. The concept of momentum and its conservation gives a constraint on the possible resulting motions. But this cannot be the whole story, somehow. It does not tell us how motion is initiated, at least in many cases that we might think about. Let's go back to the distinction that Aristotle established between living and nonliving things. Living things are those that could initiate motion,

3 while nonliving things were only acted upon. Nothing that we have talked about so far really helps us to understand that distinction. A good illustrative example used for the conservation of momentum is a bullet fired from a gun. A bullet has a small mass but high velocity, so its momentum is quite considerable. When the gun is fired there is a kickback the bullet and the gun have equal and opposite momenta after the firing. Altogether the total momentum is conserved: it starts at zero and ends at zero. But that doesn't at all explain how the bullet gets going in the first place. Clearly it is the explosion of gunpowder that is the source of the motion and this must certainly be included in the explanation. What's missing is the concept of energy. The source of the motion is the chemical energy in the gunpowder. This chemical energy is converted into the energy of motion of the bullet. Energy of motion of an object is called kinetic energy. It is given by the formula KE = mv 2 /2. Kinetic energy is always positive, unlike the momentum. This formula looks a bit similar to that for momentum, but momentum is a vector, but the kinetic energy is a just a single number. The kinetic energy of the bullet in the kinetic energy of the gun must be supplied by the chemical energy stored in the gunpowder. We are not going to stay a physical law that goes far beyond Newton's laws of motion. This is the principle of conservation of energy. We need hardly even write down the statement of principles since it's already contained in the name. But it is anyway the total energy is conserved. Once more, in these contexts conserved means that the total energy is constant, then it does not change in time. So now we have three ways of

4 saying the same thing, "conserved", "constant", and "does not change in time". Let's see how this conservation law applies to the firing of a gun. (Chemical energy) before + (kinetic energy) before = (Chemical energy) after + (kinetic energy) after. Notice that we need to total everything up. That's very important. Then we can equate the total energy at one time (any time before the firing) to the total energy at any other time (anytime after the firing). The chemical energy decreases. The burned gunpowder stores less energy than the fresh gunpowder. The kinetic energy increases. The kinetic energy is zero beforehand but after the firing stuff is moving around, so the kinetic energy is positive. (We actually have to include both the kinetic energy of the bullet and the kinetic energy of the gun.) A possible reason Newton never came up with this concept is that energy comes in very many forms and very often it is really difficult to measure. For example, heat is a form of energy. Heat energy is very difficult to keep track of because it moves around pretty freely. Chemical energy is also tricky. It doesn't manifest itself unless something undergoes a chemical reaction, like burning. So the concept of energy developed relatively slowly over time. The formulation of the concept of the conservation of energy really did not take hold until the late 19th century, 200 years after Newton. The energy concept solves a problem that has been nagging us since the very first lecture. What accounts for the distinction between living things and nonliving things? The answer is now simple. Living things can initiate motion because they contain stored energy. When we eat, we take in fuel that contains chemical energy. When we breathe we take in

5 oxygen. The oxygen combines with the fuel, producing heat energy that raises our body to a higher temperature than its surroundings and that supplies the kinetic energy that we have when we move about. To us of course the distinction between living things and nonliving things was not as clear is that was to old Aristotle, because we see all kinds of things moving about with nothing pushing on them: cars, planes, the little pictures that go across our computer screens. But all of those motions come from objects that can store energy, whether it's gasoline, jet fuel or electrical energy stored in batteries. So that's the final answer of how some things can move on their own and others can't. The former contain an internal source of energy. The latter don't. The conservation of energy also tells us why energy is such an important quantity in technology. Because it is conserved, you cannot create it out of nothing. You have to find substances such as oil which contains chemical energy, or uranium which contains nuclear energy, and figure out how to convert that energy to useful forms. We're lucky that we have the sun because light also contains energy, and it comes down on us all day long. It's that energy that is converted to chemical energy by plants and then is reconverted by animals into heat energy and kinetic energy. But here you might say is a counterexample to this principle: dropping a ball. It seems as if kinetic energy comes from nowhere. But actually that is not true. An object high up in the gravitational field has what is called potential energy. The gravitational potential energy of an object is E = mgh, where h is the height. This potential energy is converted to kinetic energy. This gives us a simple formula for the velocity of an object that has fallen through a distant speech. mv 2 /2 = mgh leads to v = (2gh). A particularly nice example is a swinging pendulum, because potential energy is converted into kinetic energy and then back and this cycle can

6 go on for a long time. This is unlike the falling of a ball in which the energy just ends up as the energy of deformation of the floor and the ball. The same law of conservation of energy applies to the planets in their movement around the sun. When they are closer to the sun in the elliptical orbit they speed up and when they're farther away they slow down. This is also a conversion of kinetic potential energy back again that takes place in. The gravitational potential energy is PE = - GmM/R. This is somewhat similar to the gravitational force law. The main difference is the 1/R as opposed to the 1/R 2. The minus sign, together with the fact that 1/R gets BIGGER as you get closer and closer together, means that the potential energy is greater if the two objects are further apart and less if the two objects are closer together. Energy is a synthetic concept. The whole direction of our course so far has been to start with common sense ideas of motion and slowly refine them, making them simpler but more abstract. We have used the technique of stripping away detail and focusing on a single aspect of a phenomenon. This thought process produces analytic concepts. The idea of energy came about in a different way. It was recognized as an aspect of many seemingly different phenomena, one that they all shared. As such energy a concept that relates many different area of science, a kind of overarching theme.