### Compton's Water Based Foucault Pendulum

A further review of the article[1] I mentioned yesterday, (Dr. Tatum on the Coriolis effect as a navigational aid for birds), turned up a very interesting reference to Arthur Compton of Compton scattering fame (picture 1)!

It turns out that Compton built an apparatus in 1915 at Wooster University, (now Wooster College) that accurately measured the rotation speed of the Earth allowing Compton to determine the length of a day, his latitude and longitude all without any astronomical references.  It was a sort of Foucault pendulum constructed with a tube of water.

To read all about Compton's apparatus and method in Scientific American, go to [5]:

or in the blog version of this post, just scroll to the bottom of the post for an embedded version of the article.

Picture 2, below, shows a simplified version of Compton's apparatus sitting directly on the axis of rotation of the Earth.  Here's the basic idea.  The circular tube is filled with water and the tube is rotating along with the Earth.  After some time, the rotational speed of the water will match the rotational speed of the tube and there will be no relative motion between the two.  At this point, flip the tube over 180 degrees.  The water will tend to keep rotating in the direction it was going in while the tube will now be moving in the opposite direction.  By measuring the speed of the water, you get a measure of the Earth's rotational speed.

Big deal you say.  A Foucault pendulum can give the same results you say.  Well, as it turns out, it has a few drawbacks with respect to Compton's apparatus.  Consider the case where either apparatus wasn't sitting at the Earth's pole.  Then, you only get a measure of the Earth's rotation with respect to the axis of rotation perpendicular to the horizontal plane at your location.  With a pendulum, you're done.  You can show that the Earth  is rotating, and knowing your location on the Earth, and making an assumption about the Earth's axis of rotation, you can determine how quickly the Earth is rotating.  The annoying part, however, is making that assumption about the Earth's axis of rotation.  With Compton's apparatus, you simply make two more measurements.  Your second measurement is made with the plane of the ring perpendicular to the horizontal plane at your location.  Then, you merely rotate the tube 90 degrees while still keeping it's plane perpendicular to horizontal.  You now have three perpendicular components of the Earth's rotational speed and can re-construct both the Earth's total rotational speed and the direction of the axis of location.  Compton's results are reported in the Scientific American article and shown below in picture  3:

Compton's article in Scientific American:

References:
http://www.jstor.org/discover/10.2307/4085812?uid=3739920&uid=2&uid=4&uid=3739256&sid=21101730076791

2.  +American Physical Society's Physical Review article on Compton's apparatus
http://prola.aps.org.lib-ezproxy.tamu.edu:2048/abstract/PR/v5/i2/p109_1
DOI: 10.1103/PhysRev.5.109

3.  Science Magazine on Compton's apparatus
http://www.sciencemag.org/search?submit=yes&volume=37&firstpage=803&journal_search_volume_go.x=-258&journal_search_volume_go.y=-370&journal_search_volume_go=go&andorexactfulltext=and&andorexacttitleabs=and&andorexactfulltext=and&andorexacttitleabs=and

4.  Compton biography
http://www.sciencedirect.com.lib-ezproxy.tamu.edu:2048/science/article/pii/0360301681900900

5.  Free Scientific American article

### Cool Math Tricks: Deriving the Divergence, (Del or Nabla) into New (Cylindrical) Coordinate Systems

The following is a pretty lengthy procedure, but converting the divergence, (nabla, del) operator between coordinate systems comes up pretty often. While there are tables for converting between common coordinate systems, there seem to be fewer explanations of the procedure for deriving the conversion, so here goes!

What do we actually want?

To convert the Cartesian nabla

to the nabla for another coordinate system, say… cylindrical coordinates.

What we’ll need:

1. The Cartesian Nabla:

2. A set of equations relating the Cartesian coordinates to cylindrical coordinates:

3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system:

How to do it:

Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables.

The chain rule can be used to convert a differential operator in terms of one variable into a series of differential operators in terms of othe…

### Lab Book 2014_07_10 More NaI Characterization

Summary: Much more plunking around with the NaI detector and sources today.  A Pb shield was built to eliminate cosmic ray muons as well as potassium 40 radiation from the concreted building.  The spectra are much cleaner, but still don't have the count rates or distinctive peaks that are expected.
New to the experiment?  Scroll to the bottom to see background and get caught up.
Lab Book Threshold for the QVT is currently set at -1.49 volts.  Remember to divide this by 100 to get the actual threshold voltage. A new spectrum recording the lines of all three sources, Cs 137, Co 60, and Sr 90, was started at approximately 10:55. Took data for about an hour.
Started the Cs 137 only spectrum at about 11:55 AM

Here’s the no-source background from yesterday
In comparison, here’s the 3 source spectrum from this morning.

The three source spectrum shows peak structure not exhibited by the background alone. I forgot to take scope pictures of the Cs137 run. I do however, have the printout, and…

### Unschooling Math Jams: Squaring Numbers in their own Base

Some of the most fun I have working on math with seven year-old No. 1 is discovering new things about math myself.  Last week, we discovered that square of any number in its own base is 100!  Pretty cool!  As usual we figured it out by talking rather than by writing things down, and as usual it was sheer happenstance that we figured it out at all.  Here’s how it went.

I've really been looking forward to working through multiplication ala binary numbers with seven year-old No. 1.  She kind of beat me to the punch though: in the last few weeks she's been learning her multiplication tables in base 10 on her own.  This became apparent when five year-old No. 2 decided he wanted to do some 'schoolwork' a few days back.

"I can sing that song... about the letters? all by myself now!"  2 meant the alphabet song.  His attitude towards academics is the ultimate in not retaining unnecessary facts, not even the name of the song :)

After 2 had worked his way through the so…