| Week of | Aug 22 | Sep 5 | Oct 1 | Nov 5
| Phy 205 page |
| Aug 27 | Sep 10 | Oct 8 | Nov 12 | ||
| Sep 17 | Oct 15 | Nov 19 | |||
| Sep 24 | Oct 22 | Nov 26 | |||
| Oct 29 | Notes for Fall, 2000 |
22 Oct Discuss one of the homework problems. Restate the work-energy theorem, using integral notation. Recall the definition of the integral and the basic theorem that allows you to evaluate it.
Potential energy definition
The reason for the minus sign here is so that you don't get a minus sign somewhere else, where it would be very awkward. Evaluate U again for the uniform gravitational field.
Demo: the looping toy roller-coaster. How high up the track must you start the cart so that it will make it all the way through the loop without falling off? Here I used both conservation of energy and F=ma. Can I really assume that the sum of the kinetic and the potential energy is constant? Probably not, the friction is pretty big in this system, so a lot of energy is lost there. I made the assumption anyway, just to get a result that I could compare with experiment. Part of the problem was in figuring out how fast the cart had to be moving in order to make it all the way over the top without falling off. That's where F=ma came in; it was essentially the same problem as whirling a bucket of water overhead and asking not to get wet.
24 Oct Potential energy. The basic definition of potential energy is not mgh or mgy or mgx or any variation on it. The defining equation for potential energy is as a relation to force. In one dimension,
[or derivative with respect to y if that is your coordinate.]
As an example, a spring exerts a force that is not a constant. If you hang a mass from the end of a spring, the greater the mass, the greater is the distance that the spring stretches. It's a decent approximation to say that the amount of stretch is proportional to the force applied, and that's the approximation that I'll use here. I hung a spring vertically and placed a mass on its end. Choose the y-axis to be vertically downward, and put the origin where the spring was before I hung the mass on it. In this system then
The constant, k, depends on the strength of the spring. In this case there is a second force, gravity, with Fy = +m g. This then has a corresponding potential energy
The total potential energy is
I spent some time on the graphical interpretation of this function, showing that you can read a lot of information about what will happen just by looking closely at the graph of U versus y. This sort of analysis is especially important in a subject such as chemistry, where the inter-atomic forces are very complicated. By looking at the potential energies as functions of positions, you can tell a lot about the properties of molecules.
Start a discussion of elastic collisions. This involves the (sometimes very good) approximation that kinetic energy is conserved.
26 Oct Demonstrate one of the homework problems (5-83). Set up and solve the problem of elastic collisions in one dimension. Here momentum is conserved and so is kinetic energy. This last conservation is not general, but is a good approximation in some cases. There's a slightly tricky solution to the equations shown in the text, but I just did it as a straight-forward solution of two simultaneous equations.
To justify the approximation that kinetic energy is conserved in at least some collisions, I used a pair of carts on tracks and with low friction bearings in the wheels. In the case that the carts have the same mass, the originally moving one pretty nearly stops, just as predicted by the equations. If the moving mass is much heavier than the stationery one, then after the collision the lighter one is moving at about twice the original speed, and that seemed to work too.
