Wright State University Lake Campus/2019-9/Gridiron pendulum

Anybody interested in doing a project based on the gridiron pendulum? See: *w:Gridiron pendulum * We could start with this simple demo and apply differential calculus to better model the system. It would make a good symposium project.

Let $x$ be the length of the red bar and $y$ be the length of the green one. Assume both expand when heated, and model this with a second order improvement of the linear theory.

$x=x_{0}+\alpha _{x}\Delta T+{\tfrac {1}{2}}\beta _{x}(\Delta T)^{2}$

$y=y_{0}+\alpha _{y}\Delta T+{\tfrac {1}{2}}\beta _{y}(\Delta T)^{2}$

Consider how $L$ varies with $\Delta T$, where

$L=2x-y$

The pictorial model shown in the image is convincing, especially if you use red and green dice to create a physical model. But if you can get used to the calculus you will realize that the mathematical model is simpler and more versatile.

This would be especially fun if we could do one of the following:

1. Find the non-linear (beta) terms for two metals and calculate the resulting error.
2. Find a cheap and easy way to make a physical lab for a typical high school or college lab experiment. Maybe use two different kinds of rubber bands and a hairdryer? See also:

w:Thermal_expansion#Thermal_expansion_coefficients_for_various_materials

Lost At Sea: The Search for Longitude

• https://www.youtube.com/watch?v=Zhkr3lbcul0
• https://www.pbs.org/wgbh/nova/longitude/