#calculus
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@pustam_egr@mathstodon.xyzA 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
Some people use math just like they use people, to lie, cheat or steal. But math itself never lies, or cheats, or steals from me, thats why I love math.
My graph of the day. Inspired by the fact that
\[ \sum_{k=1}^{\infty}{\sin k} \]
is bounded but diverges, I've been exploring other similar sums. Here's the start of
\[ \sum_{k=1}^{\infty}{\tan k} \]
It's not shocking that it looks like \(y = \ln(\|\cos x\| ) \), but I definitely wonder if at some point it blows up down the line!
I've made my lecture notes for applied calculus 2 publicly available. The content of that course is mostly "multidimensional real functions", including all the practical formulas for computing surfaces and volumes, slopes, optimization etc. The lecture notes contain a lot of practice problems with solutions. Maybe they're helpful for someone either learning or teaching #calculus and #math
https://paulbalduf.com/teaching/
It's really fucking aggravating when you try to compute an integral, get to a solution, and then when you look at the instructor's solution, you find him using a formula that WAS NEVER INTRODUCED BEFORE JUST NOW!
The Missing Introduction to Calculus for AI https://medium.com/effortless-programming/the-missing-introduction-to-calculus-for-ai-3d5e8df6efa3 #AI #calculus #NeuralNetworks
The Missing Introduction to Calculus for AI https://medium.com/effortless-programming/the-missing-introduction-to-calculus-for-ai-3d5e8df6efa3 #AI #calculus #NeuralNetworks
Imaginary numbers aren't just mathematical curiosities. They simplify #trigonometry, #calculus, and #geometry, opening up new possibilities in #math and #science. A mathematician explains: https://theconversation.com/taking-a-leap-of-faith-into-imaginary-numbers-opens-new-doors-in-the-real-world-through-complex-analysis-233965
A few weeks back I was complaining about this meme and we had a good discussion about it on here. I ended up making a lesson plan about taxes and my students really liked it so I will share it. (It is for students with a basic understanding of integration. But, you could probably rework it to skip that bit.)
https://drive.google.com/file/d/14xaQ1hMpqy0A68MAbxN49zK4HxK8m2R6/view?usp=sharing
(And if you have feedback let me know!)
As somebody who suffered the torture of the maths part of an Electronics Engineering degree, I can honestly say this is the only time a Laplace Transform has made me smile.
#electronicsengineering #calculus #LaplaceTransforms #engineeringjokes
Bringing Vector Calculus to Life – With Your Support! 
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Bringing Vector Calculus to Life – With Your Support!
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Vector fields in #geogebra, #desmos and #math3d
Coming soon!
https://www.patreon.com/jcponce
∞Tthanks for your support! #mathematics #visualization #vector #calculus
It's almost "e day"!!! (2.7)
Get ready for Friday with this blog post: "Exploring e" gives some ideas for learning about the Euler number in math class.
#MTBoS #iTeachMath #ClassroomMath #MathEd #MathsEdChat #calculus #precalculus
Image credit: MathIsFun.com Sarah (@mathequalslove)1…
Reflections and Tangents
