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Showing posts from January, 2015

Accelerating Frames: Cosmology Homework

Our cosmology course is well under way and it's  a lot of fun so far!  The class direction overall is towards describing the inflationary universe by means of quantum field theory, but this week we're focused on relativity.  We're allowed to work on our homework together, however, I'm spending most of my time in the lab this semester, so I'll be posting my homework notes here.  If you'd like to grab bits and pieces, make suggestions, or contribute, the whole shooting match will also be archived on github.

Our first homework contains a problem that involves accelerating reference frames.  The question is, given the transformation between the lab and the accelerating reference frame, figure out if the line element $ds^2$ is preserved.  There are a few interesting aspects to this problem.  First, while the transform looks similar to Rindler coordinates, it's not, (as ar as I can tell.)  Second, looking into Rindler coordinates a bit, they seem to maintain an i…

Lead Sample Purity and Tube Cutting: Lab Book 2015_01_22

For background on the experiment, please scroll down.
We can get a cheap piece of 3/4 inch diameter Pb from Rotometals.  Here are the details 3/4 17.14 per foot Nuclead has the same thing.  Check for purity and price. Also there’s Mayco. Next question, how pure does pure have to be?
Pb purity data: The following are reference articles about superconductors.  Each of them describes the purity of the Pb samples used.  The lower bar is set by the RMP article referenced below, as well as one other that mentions the use of ‘commercial’ grade material and the evidence of a transition to a superconducting state for this material.  The final reference from 1886 in the section defines commercial level material to have a quality not lower than 99% pure.  This information is being researched to determine what purity of sample we should use.  It would seem that a higher purity sample will produce fewer unexpected experimental results, as well as fewer alternative hypotheses to track down.  There is…

Mechanics I: Distance Traveled along a 1-D path

I ran into a rather abstruse question in today's first mechanics recitation.  The question gave the one dimensional position of a particle with respect to time as

$x = 10 - 4t + 2t^3$

It then asked for the distance traveled by the particle between t = 0 and t = 2.  The suggested answer, (from the prof in charge of TAs), was to plot the trajectory of the particle, thereby demonstrating the distance and displacement were different.  Here's the plot:


The idea is that you can see that the particle travelled form 10 to 8 and then back to 18, so the total distance is more than the displacement from 10 to 18 i.e. 8.
The question came up as to how to do this to get the exact answer.  Here goes
What we want to do is add up all the small, (read infinitesimal), distances travelled by the particle between time 0 and time 2.  The phrases, 'adding up', and infinitesimal provide the tip off that we'll do an integral, so:
Getting to the Integral the better way with the chain rule S…

Cosine Laws, Polyhedra, and Legendre Functions

I didn't make it into the lab today what with the holiday and all, but I did have time to read one of my favorite journals, American Mathematical Monthly from the +Mathematical Association of America .  The journal features a very interesting article[1] by Marshall Hampton[3] about cosine identities.  The article got me back to musing about solving for potentials with spherical symmetries and Legendre polynomials again[5].  I don't have time to work through this now, so I'm just recording my meandering thoughts here for future self, and anyone else that would like to take a look.

Hampton writes down the generalization of the law of cosines for polyhedra rather than just the plane, (pun intended), old triangle.  Here it is

$$0 = \sum_j\vec{n}_i\cdot\vec{n}_j\Delta_j = \Delta\left(i\right) - \sum_j c_{ij}\Delta_j$$

Where, $$c_{ij}$$ is the cosine between two faces of the polyhedra i and j, and $$\vec{n}_i$$ is a vector field normal to the i'th face.

Dr. Hampton states t…

Experimental Estimates and Deconstruction: Lab Book 2015_01_16

Took apart the apparatus at the bottom of the proposed Dewar stick.  This is stick that will eventually support all the required apparatus in liquid helium.  Pictures follow.  Per normal, if this is your first day on the site, scroll to the bottom for the experimental background.


The inside of the Dewar measure out at 1 and 1/8 inches.  That works out to about 1.25 cm.  Then, plugging that into an expression for the size of cylinder we can get to fit

Working with square cylinders Inscribe a square inside a circle circle radius 0.5625

The Day of Inconclusive Data: Lab Book 2015_01_15

In all likelihood, we’ll go with a resistive measurement, (to detect the superconducting state or lack thereof), using a small lead wire.  The following setup is from an Alfred Leitner video.We’ll use something even simpler, probably just a lead wire with a four point probe attached.  By the way, if you're new to the game, scroll to the bottom for background information.
We could sputter a line of lead onto a glass slide, but I don’t see the benefit yet. Working on finding out what it is.
Check this out later in the day per shaping samples: https://plus.google.com/116395125136223897621/posts/9c7dFfZ7ijT
I’m performing the check on the resistance of the primary coil as its cooled by liquid nitrogen.  The idea is that yesterday’s increase in output ignal with the superconductor ooled may have noly been do to the primary pulling more current as its resistance ramped down.  Here are a few pictures of the assembly being taken apart for the present work:





A one ohm resistor has been placed in…

Quench Detector Modifications: Lab Book 2015_01_13

If you're new to the experiment, scroll to the bottom for background that will catch you up with what's going on.
Liquid Nitrogen Fill The liquid nitrogen Dewar has been filled.  It’s initial and final weights were 146 and 170 pounds, respectively.  I’m guessing I must have read the initial weight incorrectly, but we may be paying too much for ’45 liters’ of nitrogen which might be closer to 13. liters/pound 0.5606 start 146 finish 170 net 24 Total Liters 13.4544
I’m doing the same measurement as yesterday with the sample immersed in liquid nitrogen.  There seems to be no change. 



I have however invented a reasonably good vibration detector for the vacuum pump across the room, see video here. I’m going to try a new configuration of