Lunch With George!

September 20, 2001 - Fazoli's

Agenda

• Special Relativity (reading assignment: Six Not-So-Easy Pieces, Chapter One: Vectors.
• If there's any time left...
  • skateboard - DEFERRED
  • tides - DEFERRED
  • nanotech - DEFERRED

Miscellaneous Topics During the Meal

We decided to defer the relativity/vectors discussion until we finished our main entree. George mentioned that he read or heard about a genetic study involving mice. Two groups of mice were fed a high calorie diet. The control group got fatter... but the test group of mice (who all posessed a particular gene) got warmer! Here's the story. George wants to get that gene. I remarked that I would prefer a gene for living longer... which leads to a story combining both issues here. But in many of these cases, you either have a gene or you don't.

This reminded me of the Robert Heinlein novel Methuselah's Children. The focus of this novel is the Howard Foundation, which was created and mandated to spend the very large estate of a man obsessed with long life in any manner which would prolong human life. This happened around the beginning of the 20th century, so a technical solution was not at hand. Instead, the Foundation identified young adults whose grandparents were all living, and offered significant financial incentives to marry someone on the same list. In the story, the decendants of the Foundation members naturally lived longer and longer lives with each generation...

George made a point that research has shown that a diet in which calories are significantly restricted, test subjects (monkeys) live considerably longer. Of course, these individuals tended to be nervous and stressed, and would fight over available food... doesn't sound like much fun.

George pointed out a great photo from his high school physics book which shows a "puck" swinging around in a circle. Here's the image:

From George:

"For a caption, I dunno. The picture itself is just something like "Demonstration of how a body flies tangential to an orbital path when the binding force is removed (as opposed to the radial path one might expect from feeling the outward 'pull' of the body on the string one is holding to maintain it in a circular path)". But it's really the contraption for the demonstration that intrigued me, more than the demonstration itself. The rubber band beautifully illustrates for the camera the force acting on the body at each strobe in the sequence, and the torch is a great frictionless way to cut the string and release the object. And the fact that the object itself looks like a brass float from a toilet tank adds a nice Rube Goldberg touch. Plus the fact that it's really a 'dry ice puck', sort of an air-hockey puck with the gas pressure built into the puck instead of the table, through the simple mechanism of dry ice sublimation, with the brass toilet tank float serving as both pressure vessel and heat sink to keep the insides warming up. I just really liked the use of the combination of simple mechanical devices to elegantly demonstrate the concept. So maybe the caption should be something like 'High school physics text uses cool contraption to demonstrate centripital force'."

Special Relativity

This chapter is NOT about relativity, but rather about vectors. It is foundation material we must master prior to talking about special relativity. Of course, since Calculus & Physics are concepts from our distant youth, we're missing foundation material needed just to understand this chapter!

Here is some stuff one needs to know:

Basic Trigonometry Stuff, like

   sin = opposite / hypotenuse

   cos = adjacent / hypotenuse

   tan = opposite / adjacent

Calculus stuff, like

   dx/dt is the derivative of x with respect to time t.

   the derivative of x² = 2x

   the derivative of 2x = 2

   the derivative of 2 = 0

The Meat of the Chapter

Vectors are used to describe any concept which posesses a value and a direction. They're great for representing velocity and acceleration. In three dimensional space, a vector is specified by three values x, y, and z, where each is a value along each of the three coordinate axes. The line which results between the origin and the point represented by (x,y,z) is the vector. The length of this line is the value, or magnitude, of the vector.

Even though this chapter is about vectors, Feynman's objective is to show that physical laws (specifically Newton's laws of mechanics in this chapter) are symmetrical.

In figure 1.7, Feynman describes a complicated, curved trajectory for a particle. He attempts to show how we can determine the acceleration for this particle using vectors. In the figure, the particle has a velocity v₁ at time t₁, and it has a different velocity v₂ at time t₂. How do we get the acceleration? It is the difference between the two velocities. All we need to do is redraw the two vectors with their "tails" at the same point, and then subtract vector v₁ from vector v₂. Figure 1.8 shows this, but I found it confusing... so I broke it up into three figures, each one building on the previous one:

Figure 1.8a Vectors v1 and v2.	Figure 1.8b adding component acceleration vectors.	Figure 1.8c adding resultant acceleration vector.

Figure 1.8a just shows the two velocity vectors associated with times t₁ and t₂. In figure 1.8b, I've added the two component acceleration vectors: the acceleration Dv_^ which is perpendicular to the second velocity vector v₂, and the acceleration Dv_||, which is parallel (or tangent) to the same velocity vector. Figure 1.8c adds the resulting acceleration, Dv.

After many complicated equations, Feynman finally gets to the equation

a = v²/R          (eq. 1)

An earlier equation of note which led to the equation above:

Dq = v(Dt/R)    (eq. 2)

Experimentation!

In Six Easy Pieces, Feynman says "The test of all knowledge is experiment." Taking him at his word, George and I devised a simple experiment using only pencil and paper (and George's memory of radians and p).

The intent of the experiment is to prove that by modeling the movement of a particle along a circular path, we can validate the equation 2 above.

Besides the pencil, I created another tool: I made a scale by tearing off a strip of the page in my notebook (the lines of the paper provide the graduated scale).

Here's what we did:

1. We picked a point on the paper to act as the center of a circle.
2. Using the scale, we drew a circle with radius R equal to 5 units.
3. We marked a point on the curve to represent the particle's position at time t₁, and another point to represent its position at time t₂ (conveniently, the point for t₂ happens to be located at the intersection of the circle and a radius line which is separated from the t₁ point by 45 degrees) .
4. I then used my scale to measure the distance traveled by the particle along the circle (approx. 3.9 units).
5. Assuming a time delta of 1 second, v=3.9/s.
6. Let's "Plug n' Play:"
   Dq = v(Dt/R)
   Dq = 3.9(1/5)
   Dq = .785 radians
   Dq = .24987 p
7. Yahoo! We deliberately started with a Dq of 45 degrees, which is .25p. Close enough for government work!

George still has an issue with the intermediate equation at the top of page 18:

Dv_^ = vDq

Other Sources

I mentioned that the book refers back to other chapters from the Lectures on Physics that we don't have, and that we could sorely use it. After lunch, I ordered a used copy of the Lectures on Physics, Volume 1!!! I also brought along a layman's book on Relativity by the master himself-- Albert Einstein. I hope to read this after finishing the Feynman material. If you wish to read to read this book online, you can do so here.