#classicalmechanics — Public Fediverse posts
Live and recent posts from across the Fediverse tagged #classicalmechanics, aggregated by home.social.
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In 1851, a French physicist suspended a metal ball from a 67-metre wire, and it became one of the first direct proofs that the Earth rotates |
Most of us accept that the Earth rotate…
#NewsBeep #News #US #USA #UnitedStates #UnitedStatesOfAmerica #Physics #1851physicsexperiment #classicalmechanics #earthrotationexperiment #earth'srotationdemonstration #famousscienceexperiments #howtheearthrotates #léonfoucault #panthéonparis #prooftheearthrotates #Science
https://www.newsbeep.com/us/746643/ -
In 1851, a French physicist suspended a metal ball from a 67-metre wire, and it became one of the first direct proofs that the Earth rotates |
Most of us accept that the Earth rotate…
#NewsBeep #News #US #USA #UnitedStates #UnitedStatesOfAmerica #Physics #1851physicsexperiment #classicalmechanics #earthrotationexperiment #earth'srotationdemonstration #famousscienceexperiments #howtheearthrotates #léonfoucault #panthéonparis #prooftheearthrotates #Science
https://www.newsbeep.com/us/746643/ -
In 1851, a French physicist suspended a metal ball from a 67-metre wire, and it became one of the first direct proofs that the Earth rotates |
Most of us accept that the Earth rotates because we have been taught i…
#NewsBeep #News #Physics #1851physicsexperiment #AU #Australia #classicalmechanics #earthrotationexperiment #earth'srotationdemonstration #famousscienceexperiments #howtheearthrotates #léonfoucault #panthéonparis #prooftheearthrotates #Science
https://www.newsbeep.com/au/782862/ -
In 1851, a French physicist suspended a metal ball from a 67-metre wire, and it became one of the first direct proofs that the Earth rotates |
Most of us accept that the Earth rotates because we have been taught i…
#NewsBeep #News #Physics #1851physicsexperiment #AU #Australia #classicalmechanics #earthrotationexperiment #earth'srotationdemonstration #famousscienceexperiments #howtheearthrotates #léonfoucault #panthéonparis #prooftheearthrotates #Science
https://www.newsbeep.com/au/782862/ -
In 1851, a French physicist suspended a metal ball from a 67-metre wire, and it became one of the first direct proofs that the Earth rotates |
Most of us accept that the Earth rotates because we have been taught it s…
#NewsBeep #News #Physics #1851physicsexperiment #CA #Canada #classicalmechanics #earthrotationexperiment #earth'srotationdemonstration #famousscienceexperiments #howtheearthrotates #léonfoucault #panthéonparis #prooftheearthrotates #Science
https://www.newsbeep.com/ca/783968/ -
https://www.europesays.com/uk/1072224/ In 1851, a French physicist suspended a metal ball from a 67-metre wire, and it became one of the first direct proofs that the Earth rotates | #1851PhysicsExperiment #ClassicalMechanics #EarthRotationExperiment #Earth'sRotationDemonstration #FamousScienceExperiments #HowTheEarthRotates #LéonFoucault #PanthéonParis #Physics #ProofTheEarthRotates #Science #UK #UnitedKingdom
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In 1851, a French physicist suspended a metal ball from a 67-metre wire, and it became one of the first direct proofs that the Earth rotates |
Most of us accept that the Earth rotates because we have been taug…
#NewsBeep #News #Physics #1851physicsexperiment #classicalmechanics #earthrotationexperiment #earth'srotationdemonstration #famousscienceexperiments #howtheearthrotates #léonfoucault #panthéonparis #prooftheearthrotates #Science #UK #UnitedKingdom
https://www.newsbeep.com/uk/676844/ -
https://www.europesays.com/ie/571666/ In 1851, a French physicist suspended a metal ball from a 67-metre wire, and it became one of the first direct proofs that the Earth rotates | #1851PhysicsExperiment #ClassicalMechanics #EarthRotationExperiment #Earth'sRotationDemonstration #Éire #FamousScienceExperiments #HowTheEarthRotates #IE #Ireland #LéonFoucault #PanthéonParis #Physics #ProofTheEarthRotates #Science
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Noether’s Theorem: Symmetry→Conservation Law.
Symmetries of the action imply conserved quantities per Noether's theorem.
A path q(t) is shifted by δq = φ but smoothed in transition layers τ wide at t0 and t1, where Δq̇ ≈ φ/τ. Action changes there approximate ΔS ≈ (∂L/∂q̇) Δq̇ τ ≈ −(∂L/∂q̇ φ)(t1) and ≈ +(∂L/∂q̇ φ)(t0).
Symmetry sets total ΔS = 0, hence (∂L/∂q̇ φ) is conserved from t0 to t1.Momentum conservation's derivation from space translation symmetry and energy conservation from time translation symmetry in mechanics and physics is done by it.
#NoethersTheorem #Physics #TheoreticalPhysics #ClassicalMechanics #AnalyticalMechanics #LagrangianMechanics #Symmetry #ConservationLaws #ActionPrinciple #CalculusOfVariations #Mathematics #STEM #Science #PhysicsEducation #LearnPhysics #QuantumPhysics #FieldTheory #GeneralizedCoordinates #MomentumConservation #EnergyConservation #AngularMomentum #TheoreticalScience #PhysicsNotes #Infographic #ScienceCommunication #EducationalContent #EMNoether #MathPhysics #PhysicsStudents #STEMEducation
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Noether’s Theorem: Symmetry→Conservation Law.
Symmetries of the action imply conserved quantities per Noether's theorem.
A path q(t) is shifted by δq = φ but smoothed in transition layers τ wide at t0 and t1, where Δq̇ ≈ φ/τ. Action changes there approximate ΔS ≈ (∂L/∂q̇) Δq̇ τ ≈ −(∂L/∂q̇ φ)(t1) and ≈ +(∂L/∂q̇ φ)(t0).
Symmetry sets total ΔS = 0, hence (∂L/∂q̇ φ) is conserved from t0 to t1.Momentum conservation's derivation from space translation symmetry and energy conservation from time translation symmetry in mechanics and physics is done by it.
#NoethersTheorem #Physics #TheoreticalPhysics #ClassicalMechanics #AnalyticalMechanics #LagrangianMechanics #Symmetry #ConservationLaws #ActionPrinciple #CalculusOfVariations #Mathematics #STEM #Science #PhysicsEducation #LearnPhysics #QuantumPhysics #FieldTheory #GeneralizedCoordinates #MomentumConservation #EnergyConservation #AngularMomentum #TheoreticalScience #PhysicsNotes #Infographic #ScienceCommunication #EducationalContent #EMNoether #MathPhysics #PhysicsStudents #STEMEducation
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Noether’s Theorem: Symmetry→Conservation Law.
Symmetries of the action imply conserved quantities per Noether's theorem.
A path q(t) is shifted by δq = φ but smoothed in transition layers τ wide at t0 and t1, where Δq̇ ≈ φ/τ. Action changes there approximate ΔS ≈ (∂L/∂q̇) Δq̇ τ ≈ −(∂L/∂q̇ φ)(t1) and ≈ +(∂L/∂q̇ φ)(t0).
Symmetry sets total ΔS = 0, hence (∂L/∂q̇ φ) is conserved from t0 to t1.Momentum conservation's derivation from space translation symmetry and energy conservation from time translation symmetry in mechanics and physics is done by it.
#NoethersTheorem #Physics #TheoreticalPhysics #ClassicalMechanics #AnalyticalMechanics #LagrangianMechanics #Symmetry #ConservationLaws #ActionPrinciple #CalculusOfVariations #Mathematics #STEM #Science #PhysicsEducation #LearnPhysics #QuantumPhysics #FieldTheory #GeneralizedCoordinates #MomentumConservation #EnergyConservation #AngularMomentum #TheoreticalScience #PhysicsNotes #Infographic #ScienceCommunication #EducationalContent #EMNoether #MathPhysics #PhysicsStudents #STEMEducation
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Noether’s Theorem: Symmetry→Conservation Law.
Symmetries of the action imply conserved quantities per Noether's theorem.
A path q(t) is shifted by δq = φ but smoothed in transition layers τ wide at t0 and t1, where Δq̇ ≈ φ/τ. Action changes there approximate ΔS ≈ (∂L/∂q̇) Δq̇ τ ≈ −(∂L/∂q̇ φ)(t1) and ≈ +(∂L/∂q̇ φ)(t0).
Symmetry sets total ΔS = 0, hence (∂L/∂q̇ φ) is conserved from t0 to t1.Momentum conservation's derivation from space translation symmetry and energy conservation from time translation symmetry in mechanics and physics is done by it.
#NoethersTheorem #Physics #TheoreticalPhysics #ClassicalMechanics #AnalyticalMechanics #LagrangianMechanics #Symmetry #ConservationLaws #ActionPrinciple #CalculusOfVariations #Mathematics #STEM #Science #PhysicsEducation #LearnPhysics #QuantumPhysics #FieldTheory #GeneralizedCoordinates #MomentumConservation #EnergyConservation #AngularMomentum #TheoreticalScience #PhysicsNotes #Infographic #ScienceCommunication #EducationalContent #EMNoether #MathPhysics #PhysicsStudents #STEMEducation
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Noether’s Theorem: Symmetry→Conservation Law.
Symmetries of the action imply conserved quantities per Noether's theorem.
A path q(t) is shifted by δq = φ but smoothed in transition layers τ wide at t0 and t1, where Δq̇ ≈ φ/τ. Action changes there approximate ΔS ≈ (∂L/∂q̇) Δq̇ τ ≈ −(∂L/∂q̇ φ)(t1) and ≈ +(∂L/∂q̇ φ)(t0).
Symmetry sets total ΔS = 0, hence (∂L/∂q̇ φ) is conserved from t0 to t1.Momentum conservation's derivation from space translation symmetry and energy conservation from time translation symmetry in mechanics and physics is done by it.
#NoethersTheorem #Physics #TheoreticalPhysics #ClassicalMechanics #AnalyticalMechanics #LagrangianMechanics #Symmetry #ConservationLaws #ActionPrinciple #CalculusOfVariations #Mathematics #STEM #Science #PhysicsEducation #LearnPhysics #QuantumPhysics #FieldTheory #GeneralizedCoordinates #MomentumConservation #EnergyConservation #AngularMomentum #TheoreticalScience #PhysicsNotes #Infographic #ScienceCommunication #EducationalContent #EMNoether #MathPhysics #PhysicsStudents #STEMEducation
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⚙️ Weight lifting and physics are essentially the same thing. Most people just never think of it that way.
Newton's laws, torque, lever arms, elastic energy. Your body is a system of levers, and every good lift is a small lesson in classical mechanics. 🏋️
This explainer breaks it down really well. Worth a few minutes of your Monday.
🔗 https://buff.ly/gEr6nQj -
⚙️ Weight lifting and physics are essentially the same thing. Most people just never think of it that way.
Newton's laws, torque, lever arms, elastic energy. Your body is a system of levers, and every good lift is a small lesson in classical mechanics. 🏋️
This explainer breaks it down really well. Worth a few minutes of your Monday.
🔗 https://buff.ly/gEr6nQj -
⚙️ Weight lifting and physics are essentially the same thing. Most people just never think of it that way.
Newton's laws, torque, lever arms, elastic energy. Your body is a system of levers, and every good lift is a small lesson in classical mechanics. 🏋️
This explainer breaks it down really well. Worth a few minutes of your Monday.
🔗 https://buff.ly/gEr6nQj -
⚙️ Weight lifting and physics are essentially the same thing. Most people just never think of it that way.
Newton's laws, torque, lever arms, elastic energy. Your body is a system of levers, and every good lift is a small lesson in classical mechanics. 🏋️
This explainer breaks it down really well. Worth a few minutes of your Monday.
🔗 https://buff.ly/gEr6nQj -
⚙️ Weight lifting and physics are essentially the same thing. Most people just never think of it that way.
Newton's laws, torque, lever arms, elastic energy. Your body is a system of levers, and every good lift is a small lesson in classical mechanics. 🏋️
This explainer breaks it down really well. Worth a few minutes of your Monday.
🔗 https://buff.ly/gEr6nQj -
Can a uniformly dense sphere in a vacuum rotate on two axes at once? Dive into the physics of rotating bodies, angular momentum, and the limits of motion in classical mechanics.
#RotationalPhysics #AngularMomentum #ClassicalMechanics #ScienceMystery
https://www.scientificworldinfo.com/2026/04/can-uniformly-dense-sphere-in-vacuum-rotate-on-two-axes.html -
Structure and Interpretation of Classical Mechanics
https://tgvaughan.github.io/sicm/toc.html
#HackerNews #StructureAndInterpretation #ClassicalMechanics #Physics #Education #ComputationalPhysics
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Structure and Interpretation of Classical Mechanics
https://tgvaughan.github.io/sicm/toc.html
#HackerNews #StructureAndInterpretation #ClassicalMechanics #Physics #Education #ComputationalPhysics
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Structure and Interpretation of Classical Mechanics
https://tgvaughan.github.io/sicm/toc.html
#HackerNews #StructureAndInterpretation #ClassicalMechanics #Physics #Education #ComputationalPhysics
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Structure and Interpretation of Classical Mechanics
https://tgvaughan.github.io/sicm/toc.html
#HackerNews #StructureAndInterpretation #ClassicalMechanics #Physics #Education #ComputationalPhysics
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Structure and Interpretation of Classical Mechanics
https://tgvaughan.github.io/sicm/toc.html
#HackerNews #StructureAndInterpretation #ClassicalMechanics #Physics #Education #ComputationalPhysics
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Faculty of Mechanical Engineering
VinUniversitySee the full job description on jobRxiv: https://jobrxiv.org/job/vinuniversity-27778-faculty-of-mechanical-engineering/
#classicalmechanics #Computationalmechanics #ScienceJobs #hiring #research
https://jobrxiv.org/job/vinuniversity-27778-faculty-of-mechanical-engineering/?fsp_sid=4057 -
Faculty of Mechanical Engineering
VinUniversitySee the full job description on jobRxiv: https://jobrxiv.org/job/vinuniversity-27778-faculty-of-mechanical-engineering/
#classicalmechanics #Computationalmechanics #ScienceJobs #hiring #research
https://jobrxiv.org/job/vinuniversity-27778-faculty-of-mechanical-engineering/?fsp_sid=4057 -
Faculty of Mechanical Engineering
VinUniversitySee the full job description on jobRxiv: https://jobrxiv.org/job/vinuniversity-27778-faculty-of-mechanical-engineering/
#classicalmechanics #Computationalmechanics #ScienceJobs #hiring #research
https://jobrxiv.org/job/vinuniversity-27778-faculty-of-mechanical-engineering/?fsp_sid=4057 -
Faculty of Mechanical Engineering
VinUniversitySee the full job description on jobRxiv: https://jobrxiv.org/job/vinuniversity-27778-faculty-of-mechanical-engineering/
#classicalmechanics #Computationalmechanics #ScienceJobs #hiring #research
https://jobrxiv.org/job/vinuniversity-27778-faculty-of-mechanical-engineering/?fsp_sid=4057 -
Faculty of Mechanical Engineering
VinUniversitySee the full job description on jobRxiv: https://jobrxiv.org/job/vinuniversity-27778-faculty-of-mechanical-engineering/
#classicalmechanics #Computationalmechanics #ScienceJobs #hiring #research
https://jobrxiv.org/job/vinuniversity-27778-faculty-of-mechanical-engineering/?fsp_sid=4057 -
Faculty of Mechanical Engineering
VinUniversitySee the full job description on jobRxiv: https://jobrxiv.org/job/vinuniversity-27778-faculty-of-mechanical-engineering/
#classicalmechanics #Computationalmechanics #ScienceJobs #hiring #research
https://jobrxiv.org/job/vinuniversity-27778-faculty-of-mechanical-engineering/?fsp_sid=2692 -
Faculty of Mechanical Engineering
VinUniversitySee the full job description on jobRxiv: https://jobrxiv.org/job/vinuniversity-27778-faculty-of-mechanical-engineering/
#classicalmechanics #Computationalmechanics #ScienceJobs #hiring #research
https://jobrxiv.org/job/vinuniversity-27778-faculty-of-mechanical-engineering/?fsp_sid=2692 -
Faculty of Mechanical Engineering
VinUniversitySee the full job description on jobRxiv: https://jobrxiv.org/job/vinuniversity-27778-faculty-of-mechanical-engineering/
#classicalmechanics #Computationalmechanics #ScienceJobs #hiring #research
https://jobrxiv.org/job/vinuniversity-27778-faculty-of-mechanical-engineering/?fsp_sid=2692 -
Faculty of Mechanical Engineering
VinUniversitySee the full job description on jobRxiv: https://jobrxiv.org/job/vinuniversity-27778-faculty-of-mechanical-engineering/
#classicalmechanics #Computationalmechanics #ScienceJobs #hiring #research
https://jobrxiv.org/job/vinuniversity-27778-faculty-of-mechanical-engineering/?fsp_sid=2692 -
Faculty of Mechanical Engineering
VinUniversitySee the full job description on jobRxiv: https://jobrxiv.org/job/vinuniversity-27778-faculty-of-mechanical-engineering/
#classicalmechanics #Computationalmechanics #ScienceJobs #hiring #research
https://jobrxiv.org/job/vinuniversity-27778-faculty-of-mechanical-engineering/?fsp_sid=2692 -
Faculty of Mechanical Engineering
VinUniversitySee the full job description on jobRxiv: https://jobrxiv.org/job/vinuniversity-27778-faculty-of-mechanical-engineering/
#classicalmechanics #Computationalmechanics #ScienceJobs #hiring #research
https://jobrxiv.org/job/vinuniversity-27778-faculty-of-mechanical-engineering/?fsp_sid=1712 -
Faculty of Mechanical Engineering
VinUniversitySee the full job description on jobRxiv: https://jobrxiv.org/job/vinuniversity-27778-faculty-of-mechanical-engineering/
#classicalmechanics #Computationalmechanics #ScienceJobs #hiring #research
https://jobrxiv.org/job/vinuniversity-27778-faculty-of-mechanical-engineering/?fsp_sid=1712 -
Faculty of Mechanical Engineering
VinUniversitySee the full job description on jobRxiv: https://jobrxiv.org/job/vinuniversity-27778-faculty-of-mechanical-engineering/
#classicalmechanics #Computationalmechanics #ScienceJobs #hiring #research
https://jobrxiv.org/job/vinuniversity-27778-faculty-of-mechanical-engineering/?fsp_sid=1712 -
Faculty of Mechanical Engineering
VinUniversitySee the full job description on jobRxiv: https://jobrxiv.org/job/vinuniversity-27778-faculty-of-mechanical-engineering/
#classicalmechanics #Computationalmechanics #ScienceJobs #hiring #research
https://jobrxiv.org/job/vinuniversity-27778-faculty-of-mechanical-engineering/?fsp_sid=1712 -
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.
#MyWork #CCBYSA #Mathematics #ClassicalMechanics #AppliedMathematics #Physics #Animation
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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.
#MyWork #CCBYSA #Mathematics #ClassicalMechanics #AppliedMathematics #Physics #Animation
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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.
#MyWork #CCBYSA #Mathematics #ClassicalMechanics #AppliedMathematics #Physics #Animation
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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.
#MyWork #CCBYSA #Mathematics #ClassicalMechanics #AppliedMathematics #Physics #Animation
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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.
#MyWork #CCBYSA #Mathematics #ClassicalMechanics #AppliedMathematics #Physics #Animation
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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).#Pendulum #Cycloid #Period #Frequency #SHM #TimePeriod #CycloidalPendulum #Lagrange #Cusp #Energy #KineticEnergy #PotentialEnergy #Lagrangian #Length #Math #Maths #Physics #Mechanics #ClassicalMechanics #Amplitude #CircularFrequency #Motion #Vibration #HarmonicMotion #Parameter #ParemeterizedEquation #GoverningEquations #Equation #Equations #DifferentialEquations #Calculus
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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.
#MyWork #CCBYSA #Mathematics #ClassicalMechanics #AppliedMathematics #Physics #Animation
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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.
#MyWork #CCBYSA #Mathematics #ClassicalMechanics #AppliedMathematics #Physics #Animation
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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-031620-095842What 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 -
The (Newtonian) #Physics or #ClassicalMechanics of #ChangeManagement according to the #kihbernetic worldview:
The #Force required to put the change in motion is proportional to its #Mass or how large the change is; because the bigger the change the more #Friction it creates. The #Gravity, or how immediate the necessity to change is, will also increase the friction of the #ResistanceToChange.
However, once the change is put in motion, the #Acceleration produced by the mass’s #Inertia will start “pulling” the change by itself and only minimal force will need to be applied to #Control that the change is moving in the right direction and/or with the right speed of change.
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Spheres in a Space with Trillions of Dimensions
I don't venture into speculative science writing - this is just about classical statistical mechanics; actually about a special mathematical aspect. It was one of the things I found particularly intriguing in my first encounters with statistical mechanics and thermodynamics a long time ago - a curious feature of volumes. I was mulling upon how to 'briefly motivate' the calculation below in a comprehensible way, a task I might have failed at years ago already, when I tried to use […]https://elkement.art/2017/06/17/spheres-in-a-space-with-trillions-of-dimensions/
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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).#Pendulum #Cycloid #Period #Frequency #SHM #TimePeriod #CycloidalPendulum #Lagrange #Cusp #Energy #KineticEnergy #PotentialEnergy #Lagrangian #Length #Math #Maths #Physics #Mechanics #ClassicalMechanics #Amplitude #CircularFrequency #Motion #Vibration #HarmonicMotion #Parameter #ParemeterizedEquation #GoverningEquations #Equation #Equations #DifferentialEquations #Calculus