Monday, March 23, 2015

Gravity Probe B Notes: Projecting Vectors via the Dot Product and the Importance of High School Trig

I'm in the process of reading Schiff's Gravity Probe B inception paper[1].  Gravity Probe B was the satellite borne experiment that detected the Earth's gravitomagnetic field, but that's not what I'll be talking about today.  This post is more about a math trick/pattern.  It's a mathematical pattern that comes up pretty frequently in physics, so I figured it was worth a few notes here.  The first picture below shows the equation for the torque on a spinning object due to a spherical source of gravity, (like the Earth), with a bit of its attendant explanation by Schiff.  My notes can be seen to the left:

The cool part I'm going to focus on today is one of the smallest expressions within the rather ginormous equation 3, (also shown in picture 2):


I've run into structures like this in the past and it took me awhile to realize what they did.  Likewise for some of my classmates.  The short version of the story is that this operation gives the projection of the omega vector along the r vector's direction.  It's all done with dot products, and consequently, high school trigonometry.  Here's a picture:

We've got an arbitrary omega and r vector.  The cosine operation shown gives the horizontal component of the omega vector with respect to the r vector.  This is just plain old trig.  The vector equation above accomplishes the same thing with a different notation.  First, we write down the dot product as, (in the G+ version, the following four equations can be found in the pictures as well),

$$\left(\vec{\omega}\cdot\vec{r}\right) = |\vec{\omega}||\vec{r}|cos\theta$$

and we're most of the way there.  The issue is that the dot product carries along a factor equal to the length of the r vector.  That's not what we want, so we divide it out giving:

$$\left(\vec{\omega}\cdot\vec{r}\right)/r = |{\omega}|cos\theta$$

and we're done... except we're not.  In the expression from Schiff's paper we have a vector that points in the r direction.  To get this, we can just multiply the above numeric result times the r vector

$$\left(\vec{\omega}\cdot\vec{r}\right)\vec{r}/r = |{\omega}|cos\theta\; \vec{r}$$

There's only one last issue left.  The r vector points in the correct direction, but it has the magnitude of r attached to it.  Consequently, when we multiplied by it, we wound up with our result being r times bigger than it should be again.  So... we just divide out another factor of the length of the r vector to get

$$\left(\vec{\omega}\cdot\vec{r}\right)\vec{r}/r ^2= |{\omega}|cos\theta\; \hat{r}$$

and we finally have the component of the omega vector pointing in the r direction!  The r vector with a hat instead of an arrow over it indicates that the vector points in the correct direction, but that its length is one.

1.  Schiff's paper describing the original ideas behind Gravity Probe B, the satellite experiment that detected the Earths' gravitomagnetic field
Schiff, L.I., "Possible New Experimental Test of General Relativity Theory", Physical Review Letters, 4, (1960), 215

Friday, March 20, 2015

Superconducting Electrons as a Frictionless Superfluid

While doing research for an article I'm writing about Janet Tate and her Gravity Probe B experiment[4], I found a few cool things regarding superconductors, frictionless bearings, and the Egg of Columbus experiment this morning.

The Egg of Columbus demonstration was first performed by Nicola Tesla in 1893 at the World's Columbian Exposition[1].  Here's a brief video from MIT showing a modern day version of the demonstration[2]:

The MIT site[2] describes the apparatus as follows:

"A toroid with three different wire windings is connected to 220 VAC 3-phase voltage. The voltage phase of each of the three windings lags 120 degrees behind the next, creating a changing induced magnetic field. The changing field causes metal objects to rotate when placed inside.
Motors using this principle are very common. In fact, power lines are often seen in sets of three because they are carrying three phases. For more information on 3-phase voltage,"
Alfred Leitner made use of a similar apparatus to demonstrate one of the properties of liquid helium.  Interstingly he points out that the rotating cylinder in the liquid helium Dewar is made of copper,(a non-superocnducting matieral).

Today's interesting find has to do with what happens when you replace the copper with a superconductor, a lead sphere in this case.  I. Simon reported[3] that while their Egg of Columbus style apparatus worked just fine, spinning a  lead sphere at room temperature, when the lead was cooled to its superconducting state, the sphere would no longer spin!  As long as care was taken not to trap residual magnetic fields in the superconductor as it cooled, the transverse magnetic field of the stator coils was unable to effect the cylinder in the least!

1. Tesla's Egg of Columbus on Wikipedia
2.  MIT demonstration of the Egg of Columbus
3.  Simon, I, "Forces Acting on Superconductors in Magnetic Fields", Journal of Applied Physics, 24, (1953), p. 19
4.  Tate, J, "Precise Determination of the Cooper-Pair Mass", Physical Review Letters, 62, (1989), 845

Friday, March 13, 2015

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!

1.  Write-up of the experiment

Wednesday, March 4, 2015

The Pink Cloud from Outer Space (Video Coverage)

Michael Heiland serendipitously took this phenomenal time lapse video of the pink space cloud reported over Arizona last week on 2/25/2015. The cloud was formed by the Air Force Research Lab's rocket-launched ionospheric research experiment .  The video was taken from Michael's perch on Mount Lemmon northeast of Tucson.

Based on the timing of this video showing the appearance of the cloud pretty much coincident with sunrise, the two science questions remain unanswered.

Did the substance released in the experiment react chemically with the sparse oxygen in the ionosphere causing a glow in the process, as in the first Smoke Puff experiment back in 1956[2]?

Or, was sunlight responsible for ionizing the substance in the same manner as the phosphorous payload released in the Smoke Puff 2 experiment[3]?

+Michael Heiland is a bit of a phenomenon himself.  He became famous for a gorgeous time lapse video of the Phoenix valley he made as a high school senior.  More examples of his photography and videography can be found at[4]

1.  Pink Clouds and Science Reruns

2.  Journal of Chemical Physics coverage of the 1956 experiments, (apologies for the paywall)

3.  1958 Popular Mechanics article describing the first series of experiments


5.  More coverage of the 1950s experiment:

Friday, February 27, 2015

Pink Clouds and Science Reruns

A pink cloud was reported in the early morning, (pre-sunrise), sky over Arizona on Wednesday[1]. NASA and the DOD soon thereafter took credit for the cloud.  They had launched a rocket into the ionosphere where it released a vapor that created the cloud.  The purpose of the experiment was to study the effects of the vapor on the ionosphere itself.  The article, referenced above from ABC, said:

"The experiment, which also involved using ground stations to take measurements of the ionosphere, was intended to develop scientific explanations for ionospheric disturbances and their effects on modern technology, officials said."

This has all been done before[2] as it turns out!  In 1956 the Air Force launched two missiles from White Sands Missile Range with payloads of nitric oxide.  The gas released in the ionosphere created a glowing cloud described as being 'yellow-red'[3] in color.  They were studying the ionosphere as well, which, back in 1958, was described with a bit more panache [4]:

"In this electronic age, everybody knows that the ionosphere is an electrified upper atmosphere region that bounces off radio waves around the globe."
The 1958 definition of the ionosphere also nicely explains what the experiment in both cases was looking for: how radio wave propagation was effected, (also shown in the following diagram from the same 1958 piece):

The folks at Mysterious Universe[5] seem a little peeved about the DoD's failure to reveal what the specific released vapor was, but while the vapor could have been a number of things, it's interesting to note the color of light emitted by ionized nitrogen:

1.  ABC News report:

2.  Coverage of the earlier experiment here:

3.  Journal of Chemical Physics coverage of the 1956 experiments, (apologies for the paywall)

4.  1958 Popular Mechanics article describing the first series of experiments

5.  Mysterious Universe