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#classicalmechanics

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Pustam | पुस्तम | পুস্তম🇳🇵

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 2πL/g or frequency 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(θ−sin⁡θ,−1+cos⁡θ), where θ=0 corresponds to the cusp. Consider a pendulum of length L=4R hanging from the cusp, and let α be the angle the string makes with the vertical, as shown (in the proof).

GIF
#Pendulum#Cycloid#Period
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Pustam | पुस्तम | পুস্তম🇳🇵

An Article in the Annual Review of Condensed Matter Physics on Turbulence by KR Sreenivasan and J Schumacher
annualreviews.org/content/jour

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.
youtube.com/watch?v=fwVSBYh-KC

"Field Theory and Turbulence" program link: icts.res.in/discussion-meeting


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