Relativator (Circular Slide-Rule) – Simulated with Desmos

The Relativator (revisited)

This is an update of my 2006 post (reconstructed in 2014) Relativator: The circular slide-rule for physicists.
This is a circular slide-rule for doing relativistic calculations for elementary particle physics that I learned about from
– an article by Elizabeth Wade ( “Artifact: Relativator”, Symmetry (FNAL/SLAC), 01/01/06,
https://www.symmetrymagazine.org/article/december-2005january-2006/artifact-relativator  https://www.symmetrymagazine.org/sites/default/files/legacy/pdfs/200512/artifact_relativator.pdf ),
which is based on a blog post and photos posted by
– Peter Steinberg
(“The Relativator”, Quantum Diaries, March 12, 2005,  https://qd.typepad.com/5/2005/03/the_relativator.html )
(“artifact:relativator”, Entropy Bound, Jan. 4, 2006, https://entropybound.blogspot.com/2006/01/artifact-relativator.html )

The Relativator was sold by (as printed) Atomic Laboratories, Inc. 3086 Claremont Ave, Berkeley 5, California , which seems to be a division of Cenco Instruments (Central Scientific Company).

The Relativator in Desmos

• So, I present…robphy – relativator virtualized using Desmos 2025
  https://www.desmos.com/calculator/lii1mjbpoo

  

How it was constructed

• Inspired by my GeoGebra simulation of a Vernier Caliper ( https://www.geogebra.org/m/DemUu87n ),
  I thought that I could use similar methods to make a simulator for the Relativator. Over the years, I learned how to construct (by calculation) the tick marks on the Relativator,
  based on photographs from the article and photos referenced above.
  Admittedly, the calculations, the structure, and the color-choices of the Desmos simulation are not optimized.
  My primary goal has been functionality, based on the photographs that were made available.

The meaning of the scales

• The three outermost concentric scales are
  – $\beta=v/c$, the dimensionless-velocity-fraction (practically $0<\beta<1$; actually $10^{-6}\leq\beta \leq 0.999999995$)
  – $\gamma\beta$  (for relativistic momentum) , where $\gamma=\frac{1}{\sqrt{1-\beta^2}}$ is the time-dilation factor ($10^{-6}\leq\gamma\beta \leq 10^4$)
  – $(\gamma-1)$  (for relativistic kinetic energy), with range $10^{-6}\leq (\gamma-1) \leq 10^4$ and a notch at $(\gamma-1) =10^0=1$
  These scales are fixed relative to each other.
  For example, along radial line through $\beta=0.99$,
  you can read-off $\gamma\beta=\frac{\beta}{\sqrt{1-\beta^2}}\approx 7.018$ and $(\gamma-1)=\frac{\beta}{\sqrt{1-\beta^2}}-1\approx 6.089$ (as seen above).
  In the Desmos simulation, this is the $Z_{tuneTo099}$ preset.
• The inner scale (which sits on a concentric disk) are particle rest-masses in electron Volts$/c^2$ , ranging from $1\rm\ eV/c^2$ to $10^{10}\rm\ eV/c^2$.

How I think it works

• We will always keep the inner disk and the mass-scale fixed, and rotate the set of outer scales.
• Given $m=1\rm\ eV/c^2$ , by dragging the orange star, we line up (as shown) the outer black notch on the $(\gamma-1)$ scale with $m=1\rm\ eV/c^2=10^{-6}\rm MeV/c^2$, which is at the inner black notch on the mass-scale.
  
• Then, further given $\beta=0.99$, drag the green star to move the hairline to $\beta=0.99$.
  Now, you can determine the associated relativistic-kinetic energy as $m (\gamma -1)c^2 \approx 6\rm\ eV \approx 6\times 10^{-6}\rm\ MeV$ by reading the innermost scale. (One could zoom-in to get a better reading.  In principle, one could also calculate a refined scale for more accuracy.)
• If instead, you were further given $\gamma\beta=7.01$, you would have moved the hairline to $\gamma\beta=7.01$ (here, practically the same hairline location), then read off the relativistic kinetic energy.
• If instead, you were further given $\gamma=7.089$, you would have moved the hairline to $(\gamma-1)=7.089-1=6.089$ (here, practically the same hairline location), then read off the relativistic kinetic energy.
• If instead, you were further given $K_{rel}=10\rm\ eV$, you would have moved the hairline to line up with innermost scale at $10\rm\ eV$ , then read off the associated velocity: $\beta \approx 0.995$.
  
• For an electron, use $m=0.511\rm\ MeV/c^2=0.511\times 10^6\rm\ eV/c^2$, as shown in the first image of the Desmos simulation.
• Now, here’s a puzzle.
  Suppose you have a particle with relativistic kinetic energy $10\rm\ MeV$ with velocity $\beta =0.6$.
  What is the rest-mass of the particle?
  With the green star, tune the hairline to $10\rm\ MeV$ on the innermost scale.
  With the orange star, move the outer-scales until $\beta =0.6$ appears at the hairline.
  Note where the outer black notch from the $(\gamma-1)$-scale points to on the innermost mass-scale. That determines the rest-mass.