Khurram Wadee ✅May 19

In elementary #mechanics we are taught about the #SimplePendulum, which is modelled as a point #mass hanging via a rigid #rod or #string of a given length, under uniform #gravity. This simple model is also useful in introducing #OrdinaryDifferentialEquations and helps us to understand #SimpleHarmonicMotion.

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Show threadKhurram Wadee ✅
One of the interesting properties such a #pendulum has is that its #Period is only a function of its length and the strength of gravity, no matter the size of the amplitude of the oscillation.
May 19 at 1:32pmWeb
Show threadKhurram Wadee ✅May 19

The above is only strictly true for “small” amplitudes. If the #pendulum is subject to large amplitudes, it is no longer governed by the simple linear differential equations alluded to above. Given that the rod remains rigid and no energy is dissipated, the pendulum #equations may still be solved exactly, though they are now #nonlinear.

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Show threadKhurram Wadee ✅May 19
Of course this was known to many famous #mathematicians, like #Euler and #Jacobi and they solved the equations in terms of #EllipticIntegrals. I find it satisfying to be able to visualize the #dynamics of the this system, using #FreeSoftware, such as #WxMaxima.
Show threadKhurram Wadee ✅May 19

Here we see three identical pendulums, oscillating independently. The red and purple ones are vibrating with small amplitudes and so their periods are nearly the same. But the blue one is undergoing what would be considered to be very large amplitude oscillations and has a significantly longer period. In fact, as the amplitude approaches π radians, the period increases without bound and approaches infinity.

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Show threadKhurram Wadee ✅May 19
Here’s an example of a pendulum, swinging up to 99.99% up from the six o’clock position to the twelve.
Show threadDaniel LakelandMay 19
@mkwadee
Love it when i see people using Maxima in the wild. Definitely people are sleeping on it relative to what it ought to be. In my view everyone should learn about it in High School. Proprietary software or special purpose calculator hardware etc is no substitute for a piece of free software that is as powerful as Maxima and yet can run on an RPi costing less than a TI calculator.
Show threadKhurram Wadee ✅May 19
@dlakelan Many years ago, I used to use Mathematica and got quite adept at it. But it cost a lot and then I found out that #foss equivalents existed. I can do all I want with #Maxima.
Show threadDaniel LakelandMay 19
@mkwadee
I remember the day it was released under GPL, I think I found out about it reading Slashdot.org. Thank you Bill Schelter!