340 Class Notes, Nov 1

Phy 340 class notes for Fall, 1999

Aug 25	Sep 8	Oct 4	Nov 1	Phy 340 page
Aug 30	Sep 13	Oct 11	Nov 8
	Sep 20	Oct 18	Nov 15
	Sep 27	Oct 25	Nov 22
			Nov 29

Phy 340, Week 11

1 Nov Continue to examine the Coriolis term. Set up the equations of motion for a projectile in the rotating system and proceed to solve them by perturbation techniques. I could solve them exactly, but the solution's a mess and it doesn't really show anything. Instead, assume that the solution is close to the solution for a non-rotating Earth, and then find the correction to that term. You could say that I'm trying for a power series solution in powers of w, but only looking at the zeroth and first order terms. After getting the result, start to analyze some special cases in order to get some feeling for the solutions.

3 Nov More analysis of the solutions of the trajectory equations in the presence of the Coriolis force. What happens at the North pole? Figure it out from the standpoint of the inertial observer and see what that says about the rotating observer. Then see what the equations say about the solution. A deflection to the right.

Same thing at the South pole. Then show how this will qualitatively account for the air circulation around a low pressure area in the atmosphere, especially in a hurricane.

The Foucault pendulum. Analyze it at the North pole just by drawing pictures. Say what will have to be done to solve the problem mathematically.

5 Nov Instead of spending much more time on the effects of a rotating coordinate system, go to the problem of planets orbiting the sun.

The force is Newton's gravitational force law, and you can draw some immediate conclusions from it. Remember the definition of angular momentum and where it comes from? Start from F = dp/dt, and take the cross product with r.

r x F = r x dp/dt = r x dp/dt + dr/dt x p - dr/dt x p = d ( r x p ) /dt - 0.

This is torque equals rate of change of angular momentum.

Now apply this to the gravitational force. The torque is zero, so angular momentum is conserved. This turns out to be one of Kepler's laws in disguise (equal areas swept out in equal times). One direct consequence of this is that each planet will have an orbit that stays in a plane, and that in turn make it easier to set up the equations of motion, as they are only two-dimensional.

( - GMm/r² ) e_r = e_r ( d²r/dt² - r( dq/dt )² ) + e_q ( r d²q/dt² + 2 ( dr/dt )( dq/dt ) )

Equate the corresponding components and you get two formidable looking differential equations. They can be handled however, using a few non-obvious and somewhat specialized tricks.

First, notice that one equation can be integrated if only you multiply it by r. Then you get

r² dq/dt = a constant.

The constant is nothing more than L/ m, where L is the angular momentum about the z-azis.

Now you can eliminate q between the two equations by substituting dq/dt = L/mr². The resulting equation is hard, but it will yield to two substitutions:

r = 1/ u,

and use q instead of t as the independent variable.

Now you use the chain rule a lot.

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