Class Notes, Fall 2001

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

Phy 205, Week 14

19 Nov I briefly rederived the equation of motion for a simple pendulum.

Before going into detail on this equation, what about the first such equation that I got? The one for a mass on the end of a spring?

d2x( t ) d t2 = - k m x( t )

The dependent variable here is x. The independent variable is t. You are looking for a function of t with the property that its second derivative with respect to t is a (negative) multiple of itself (x). The cosine and sine functions do this, so try

Substitute into the equation to be solved and it works provided that the parameter ω₀ is chosen to be k/m.

Try another equation:

d2 Ξ d Λ 2 = - ξ  Ξ

What is the solution to this, Ξ( Λ )?

Finally I went back to the pendulum problem and noted that this is not the same equation. If however you make the approximation that the angle θ is small, then

With this approximation the equation for the pendulum becomes the same as the others, so it has the same sort of solution. I then compared the theory to an experiment to see how well the predicted period agreed. It agreed quite well.

21 Nov It's easier to demonstrate the motion of a mass on a spring if it's hanging vertically. I has said before that the motion in this case is pretty much the same as what I had done before, but I didn't show why.

A mass m is hanging down on the end of a spring of spring constant k. Take the origin to be where the spring is relaxed and measure y downward. The y-component of force then comes from gravity and from the spring:

Rearrange this into the equation

This is similar to the equations that have come before, but it has an extra term in it. Now I need to find a function whose second derivative is a negative multiple of itself, plus a constant. Cosines (or sines) worked before; you can perhaps guess that the solution to this new equation is a cosine plus a constant.

Substitute it into the preceeding equation that I want to solve, and you find that it will work provided that the constants are correct:

All that this extra term does is to shift the point about which the oscillations occur. You could even change to a new origin shifted by this amount.

If there is still another type of force on the mass, one that is itself oscillating, you can get a new phenomenon called resonance. If you push a child on a swing, you know that you have to push in the right rhythm to get it up high. If I have a mass on a spring and I add an oscillating force I get an equation

The frequency ω' is one that I control by how I apply the force. Rearrange this to

This says that I want a function such that if I take two derivatives, and add it to the original function, the result is a cosine (up to constants). The function that will do this is a cosine!

Try it. Choose the guess for the solution

It has to have the ω' in it because that's what has to match the right-hand side. Plug it in and you find that it works provided that the constant C matches:

This has the notable property that the denominator can vanish. Usually that's an indication that I made a mistake, but not this time. Now it's an indication that a new phenomenon is taking place -- 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.