### High Schoolers rough out a version of Cavendish's Experiment

Faced with students that didn't quite believe Newton's law of gravitation, Anthony Rennekamp
of Bishop O'Connell High School in Arlington, Virginia, he did what any physicist would do; he had his students build an experiment.  They created a version of Cavendish's experiment using two yard sticks, two one kilogram weights, and a few 50 pound dumbbells.  You can find the write-up of the experiment on the Physics Today website[1].  Mr. Rennekamp and his students also made a video of the experiment.  Given the slow velocities with which the masses move in this experiment, the class reasoned that speeding up a video was the only practical way to view the results.  They published their experimental results video on youtube, (you can watch it below).  There was some online disbelief that the experiment could work as well as it apparently did.  I thought it would be fun to just run a few back-of-the-envelope calculations here and see if their results are reasonable.

The equation for the gravitational attraction between two masses is

$F_g = G\dfrac{m_1m_2}{r^2}$

Where $G = 6.67 \times 10^{-11} m^3/kg\cdot s^2$ is the gravitational constant, $m_1$ is the mass of the first object in kg, $m_2$ is the mass of the second object, and $r$ is the distance between the objects in meters.  As you can see above, as the objects move closer to each other the force due to gravity will increase as $r$ decreases.  Let's ignore that though, and just make an estimate with the force calculated using the initial distance between the two masses, (the mass suspended on the end of the yardstick and the dumbbell).  The distance looks like 7 cm on the video, so we get:

$F_g = 6.67 \times 10^{-11} m^3/kg\cdot s^2 \dfrac{1 kg \cdot 22.68 kg}{0.07m^2} = 3.08 \times 10^{-7} N$

Then, using that force, we can get the time it should take the $1 kg$ mass to travel the $7 cm$:

$a = F/m = 1.51\times 10^{-7} m/s^2$

$distance = 1/2 a t^2$
$t = \sqrt{\dfrac{2*distance}{a}}$
$t = 673 seconds = 11.22 minutes$

While this is only an estimate, (it ignores the dependence of the force on distance; it also ignores the second set of masses), it comes out in the same ballpark as the effect measured by Mr. Rennekamp's class who found that it took 10 minutes for the hanging mass to swing to the dumbbell.  As an extra test, the class placed the dumbbells on the opposite sides of the yardstick and found that the yardstick twisted in the opposite direction as expected.  Pretty Cool!

References:
1.  Write-up of the experiment
http://scitation.aip.org/content/aip/magazine/physicstoday/news/10.1063/PT.5.2025

### 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…

### The Valentine's Day Magnetic Monopole

There's an assymetry to the form of the two Maxwell's equations shown in picture 1.  While the divergence of the electric field is proportional to the electric charge density at a given point, the divergence of the magnetic field is equal to zero.  This is typically explained in the following way.  While we know that electrons, the fundamental electric charge carriers exist, evidence seems to indicate that magnetic monopoles, the particles that would carry magnetic 'charge', either don't exist, or, the energies required to create them are so high that they are exceedingly rare.  That doesn't stop us from looking for them though!

Keeping with the theme of Fairbank[1] and his academic progeny over the semester break, today's post is about the discovery of a magnetic monopole candidate event by one of the Fairbank's graduate students, Blas Cabrera[2].  Cabrera was utilizing a loop type of magnetic monopole detector.  Its operation is in concept very simpl…

### 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…