| Week of | Aug 23
| Sep 6
| Oct 2
| Nov 6
| Phy 205 page |
| Aug 28 | Sep 11 | Oct 9 | Nov 13 | ||
| Sep 18 | Oct 16 | Nov 20 | Notes for Fall, 1999 | ||
| Sep 25 | Oct 23 | Nov 27 | |||
| Oct 30 |
20 Nov Look at the pendulum again with an eye toward experiment. Get the differential equation that yields the motion of the pendulum (in the small angle approximation) and solve it again. This gives the value for the period as
When I did the experiment, the observed value was very close to the predicted one -- within about a percent or less.
When I derived the equations for a mass on a spring, I simplified the problem so as to ignore gravity -- I had the mass moving horizontally. When I show the motion though I have a mass hanging from a spring, so gravity is important. Use a coordinate system with x pointing down from the point where the spring exerts no force. Then
This says that the second derivative of x (with respect to t) is the sum of x and a constant. After discussing the possible forms of the solution, I tried
When everyone in class had plugged this into the equation to see whether it worked or not, I went through the plugging-in and found that to make it work requires
This effectively re-defines the origin by moving it down to the new equilibrium. The constant term says how much the spring will stretch under a constant load. It's how a common spring scale weighs things.
When I hold a mass on a spring and move my hand up and down, I can get an effect totally out of proportion to the size of the motion of my hand. You get the same effect if you push a child on a swing or tune a radio to your favorite station.
The simplest version of this to picture is to take a mass on a spring, moving horizontally and with an extra oscillating force on the mass. The differential equation of motion (F=ma) is then
Re-arrange this into
You want to find a function x(t) so that a sum of the function and its second time derivative gives a cosine of w't. What will do this is a cosine of w't itself!
Try x( t ) = C cos w't. Do the two derivatives and plug that and the original x into the equation that they're supposed to satisfy. It works provided the constant C is chosen correctly. It requires
The next step is to interpret this. Notice that the denominator can vanish. This usually means that I made a mistake, but not this time. This time it means that there's a real phenomenon that I didn't expect. "Resonance."
When the denominator approaches zero, the response of the oscillator becomes very large. This says that when the denominator reaches zero, the response is infinite, and that's wrong, but it's wrong not because of a mathematical mistake but because I neglected friction. With friction in the system, the denominator will still become small, but it won't actually reach zero.
