Skip to main content

Lab Book 2014_05_16 Fiberglass Dewar Leaks and more on Relativistic Projectile Range

Lab Book 2014_05_16     Hamilton Carter

Summary
There's a minor setback.  The fiberglass Dewar has a leak.  On the theory side of things, the relativistic range equation is shown to be proper x velocity times the rapidity of the y component of velocity divided by the acceleration due to gravity which is about what you'd expect it to look like if you first looked at the classical result and then squinted.  Work is being done to determine what, if anything, to make of the result.  Notes and a brief Mathematica file are included.

Leak testing the fiberglass Dewar today.  I’m also looking through the second Tehran paper.


There is a leak at the Teflon joint that is away from the Dewar.  It’s a rather small leak that can’t be detected using the roughing pump gauge.


I've very slightly opened the Dewar valve and begun pumping on the large volume.  After about five minutes, the vacuum was back down to 3 * 10^-1.

It appears that the fiberglass Dewar seals did not reseat. I opened the valve a little more which I expected to improve the vacuum pressure if we didn't have a leak since it would increase the available pumping speed.  The pressure got worse and stayed worse.   The mechanical pump is not pulling below 9*10^-1 after about an hour of pumping.

Theory Work
I showed that Shahin and MacColl’s range equations are equivalent.  The relativistic range equations from Shahin’s paper and my extension, (from yesterday),are


and


I was wrong yesterday.  The factors to the left of the log are constants, but it’s interesting to vary beta.  When it goes to 1, the expression goes to infinity.  There’s one other interesting issue though.  The range expression can also be written as:


Written this way, it’s the horizontal component of proper velocity times the vertical component of rapidity dived by the acceleration of gravity. The product of the speed of light times the veritical component of rapidity divided by g gives a proper time that’s equal to the hang time of the projectile.  There’s a second expression for the vertical hang time agreed upon by both MacColl and Shahin.  The expression is


converting this to proper time, we get


There’s probably a mistake here somewhere, but I’m not seeing it today.

Shahin and MacColl’s range equations were compared [mathematica file] .  They’re quite different notationally, but they are mathematically equivalent.


More raw notes:











Labels:

Comments

Post a Comment

Please leave your comments on this topic:

Popular posts from this blog

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

Now available as a Kindle ebook for 99 cents ! Get a spiffy ebook, and fund more physics 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 ...
10 comments
Read more

The Alcubierre Warp Drive Tophat Function and Open Science with Sage

I transferred yesterday's Mathematica file with the Alcubierre warp drive[2] line element and space curvature calculations to the  +Sage Mathematical Software System  today, (the files been  added to the public repository [3]).  If you haven't used Sage before, it's a Python based software package that's similar in functionality to Mathematica.  Oh, and it' free.  I also worked a little more on understanding the theory, but frankly, I made far more progress with the software than the theory.  What follows will be a little more of the Alcubierre theory, plus, a cool Sage interactive demo of one of the Alcubierre functions[1], as well as a bit about my first experience with using Sage. Theory The theory is fun, but it's moving slowly.  Here's the chalk board from this morning's discussion Alcubierre setup the derivation using something called the 3+1 formalism which means we consider space to be flat, (in this case), slices that are labelled ...
3 comments
Read more

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...
1 comment
Read more