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Unformatted text preview: lecture 4 Topics: Where are we? Conservation laws Work and Energy and the second Law Energy in the harmonic oscillator Work and Energy in three dimensions Examples of potentials in 3-dimensions A particle on a frictionless track Forced oscillation and resonance Harmonic driving forces Energy in the driven oscillator Breaking the wine glass Where are we? We have now seen a number of examples of the use of F = ma . You should all be familiar at this point with the techniques of determining the trajectory of a classical system that is picked out by a given set of initial conditions. We also discussed the beautiful and surprisingly general behavior of the harmonic oscillator, and the particularly simple description of its motion that obtains when we allow our trajectories to involve i = √- 1 . Now, we will start the process of going beyond F = ma to more general and powerful ap- proaches. This will occupy us for the next two weeks. This week, we will discuss two of the great conservation laws of classical mechanics — energy and momentum. I also have some organizational remarks. First, a bit of good news — we have now seen all the ways of solving differential equations that we are going to use. There are really only two of them. One is integrating, perhaps after moving things around a little bit to separate variables on two sides of the equation before integrating. The other is hoping that the solution is an exponential, plugging in and checking to see whether it works! Nothing more complicated is going to happen! We will, however, do a little more multivariable calculus today. Please please please stop me and ask questions if you see something you don’t understand. Next, you should soon (I hope) find a Q&A question on the web page to be filled out by next Monday. If you cannot access this, then either the system is not working or you are not on my list, and you need to send me an email right away and check with the registrar. Now to work (literally and figuratively)! Conservation laws It is somewhat unfortunate that “energy conservation” has come to mean two very different things. When we read in a newspaper about energy conservation, the article is usually about using energy carefully and not wasting it. What we mean in physics when we say energy conservation is some- thing very different. We mean that there is a quantity called “energy” that is unchanged with time. I suppose that there is some connection between these two meanings. If energy were not conserved 1 in the physics sense, if we could simply make new energy whenever we need it, then perhaps we would not have to be so worried about using it sparingly and efficiently. It is the fact that energy is conserved in the physics sense that makes it such an important quantity....
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- Spring '09