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.
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.
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.
New #physics #classicalmechanics video - solving the pendulum problem three ways: Newtonian, Lagrangian, Hamiltonian #iteachphysics
New #physics #classicalmechanics video - solving the simple harmonic oscillator with #hamiltonian mechanics. #python and phase space diagram included #iteachphysics
A cycloidal pendulum - one suspended from the cusp of an inverted cycloid - is isochronous, meaning its period is constant regardless of the amplitude of the swing. Please find the proof using energy methods: Lagrange's equations (in the images attached to the reply).
Background:
The standard pendulum period of \(2\pi\sqrt{L/g}\) or frequency \(\sqrt{g/L}\) holds only for small oscillations. The frequency becomes smaller as the amplitude grows. If you want to build a pendulum whose frequency is independent of the amplitude, you should hang it from the cusp of a cycloid of a certain size, as shown in the gif. As the string wraps partially around the cycloid, the effect decreases the length of the string in the air, increasing the frequency back up to a constant value.
In more detail:
A cycloid is the path taken by a point on the rim of a rolling wheel. The upside-down cycloid in the gif can be parameterized by \((x, y)=R(\theta-\sin\theta, -1+\cos\theta)\), where \(\theta=0\) corresponds to the cusp. Consider a pendulum of length \(L=4R\) hanging from the cusp, and let \(\alpha\) be the angle the string makes with the vertical, as shown (in the proof).
New #physics #classicalmechanics video - finding the moment of inertia of a hollow and solid sphere.
New #classicalmechanics video - finding normal modes of oscillation for the double pendulum. #python included of course
An Article in the Annual Review of Condensed Matter Physics on Turbulence by KR Sreenivasan and J Schumacher
https://www.annualreviews.org/content/journals/10.1146/annurev-conmatphys-031620-095842
What is the turbulence problem, and when can we say it’s solved? 
This deep dive by Sreenivasan & Schumacher explores the math, physics, and engineering challenges of turbulence—from Navier-Stokes equations to intermittency and beyond. A must-read for anyone fascinated by chaos, complexity, and the unsolved mysteries of fluid dynamics! 
A summary of the talk presented by KR Sreenivasan in December 2023 at the International Center for Theoretical Sciences (ICTS-TIFR) in Bengaluru, as part of a program on field theory and turbulence.
https://www.youtube.com/watch?v=fwVSBYh-KC4
"Field Theory and Turbulence" program link: https://www.icts.res.in/discussion-meeting/ftt
#FluidDynamics #Physics #NavierStokes #UnsolvedMystery #Mechanics #Dynamics #FluidMechanics #Science #Chaos #TurbulentMotion #Randomness #Chaotic #Fluid #ClassicalMechanics
#Turbulence
New #physics #classicalMechanics video - finding the normal modes for coupled oscillators using the eigenvalue problem (and #python of course)
This video by Up and Atom (also available on YouTube) is a great intro to John Norton’s 2008 paper ‘The dome: an unexpectedly simple failure of determinism’:
https://nebula.tv/videos/upandatom-the-dome-paradox-a-loophole-in-newtons-laws/
The paper itself: https://doi.org/10.1086/594524
It's the end of the semester, so I put together all my #physics videos and worksheets for 1st semester #classicalmechanics
https://rhettallain.com/classical-mechanics-course-material-and-videos/
New #classicalmechanics #physics video - three ways to solve for the motion of a pendulum: Newtonian, Lagrangian, and Hamiltonian.
New #physics #classicalmechanics video - a mass slides down a frictionless parabola. Will it ever lose contact with the surface? Solved using #lagrangian and #python
New #physics video - using Lagrange Multipliers to find forces of constraint.
#classicalmechanics #lagrangian
Still working on #classicalmechanics videos. Here's the Lagrangian solution to a block on a moveable wedge. #physics
Modeling the #Foucault pendulum with #python to show the rotation of the Earth. #physics #classicalMechanics
New #python #classicalmechanics #physics video - modeling the motion of a thrown ball in a rotating space station.
New #physics video - free fall motion with the Coriolis force (#python code included). #classicalmechanics
This is an interesting problem in #ClassicalMechanics and exercised luminaries like #Newton and #Euler. I think the latter's use of the #CalculusOfVariations is a stroke of genius.
