#mechanics
Forgive the recent apparent obsession (I'd call it a fascination) with the #cycloid but I've just discovered something I'd not heard of before. It is also called a #TautochroneCurve or #Isochrone curve, which means that a particle starting from any location on the curve will get to the #MinimumPoint at precisely the same time as a particle starting at any other point.
Here's an #animation I wrote today in #Maxima which illustrates the property.
#Dynamics #Kinematics #Mathematics #AppliedMathematics #Mechanics #ClassicalMecanics #WxMaxima #FreeSoftware #MyWork #CCBYSA
A couple of weeks ago, I posted an #animation of a point on a circle generating a #cycloid.
If you turn the curve "upside down", you get the #BrachistochroneCurve. This curve provides the shortest travel time starting from one cusp to any other point on the curve for a ball rolling under uniform #gravity. It is always faster than the straight-line travel time. 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.
Anyway, the #animation took a bit of thought as it requires a bit of #Mechanics, some #Integration and is made a bit more tricky as the curve is multi-valued and so you need to treat different branches separately. The #AnimatedGif was produce with #WxMaxima.
For some reason that I can't fathom, I'm not finding it possible to upload the graphic and so if you want to see it, please use the link below.
https://drive.google.com/file/d/1edcZnZ_uiaQOQrFB03XHslNUVCmxcSut/view?usp=drive_link
#MyWork #CCBYSA #Mathematics #Maths #AppliedMathematics #Physics #Calculus
Imagine a circular wheel rolling, without skidding, on a flat, horizontal surface. The #locus of any given point on its #circumference is called a #cycloid. It is a #periodic #curve over a length equivalent to the #circle's circumference and has #cusps whenever the point is in contact with the surface (i.e. the two sides of the curve are tangentially vertical at that point).
Interestingly, it is also the curve that solves the #Brachistochrone problem, which means that starting at a cusp on the inverted curve (maximum height), a frictionless ball will roll under uniform gravity in minimum time from the start to any other point on the curve, even beating the straight line path.
#Mathematics #Geometry #Maths #AppliedMathematics #Mechanics #Kinematics #Dynamics #Physics #MyWork #CCBYSA #WxMaxima
https://www.youtube.com/watch?v=D_tOt9w7yyE
#Mind-bending #Secrets #Of The #Universe Explained Simply ft. #Abhijit ...
14 May 2021 ... ... #pluto in 1970s 21:55- How science figured Gas giant planets are dangerous 24:05- What are #Quantum #mechanics 26:34- #Ranveer on Xmen movie's .
This series is well produced and quite understandable
Source: https://youtube.com/watch?v=aPHelsoq4pg
Science of #Bliss and #Kundalini - Part 1 - #Biological #Mechanics with #DanWinter (1 of 6)
#MikeRutherford 10/2/1950 #Birth English musician. He is a founding member of Genesis.
https://www.youtube.com/watch?v=5hr64MxYpgk
#Mike + The #Mechanics - The #Living #Years (Official #Video)
I was explaining to my wife how the #velocity #components of an idealized #projectile differ. She was having some problems understanding that only the #vertical component changes and that the #horizontal component remains constant. In the end, I knocked up this little animation to tr and explain pictorially (the program in #WxMaxima is actually interactive. so you can stop it, step through it or run it backwards if that helps).
#Maths #Mathematics #Mechanics #ConstantAcceleration #Vectors #Dynamics #MyWork #CCBYSA #FreeSoftware
It's straightforward to calculate the #components of a #2D #vector given that you have a pre-defined set of #axes. Often, we talk about #cartesian #coordinates. If the vector changes, whether in #magnitude or #direction, we can calculate the updated components too. The converse question is what if the the vector is held constant and you change or #rotate the axes? The components in the new coordinate system can be found with a little bit of #trigonometry and #algebra. As vectors are #first-order #tensors, they are much simpler to calculate than quantities like stress for which I've also made an animation for if you search for the hashtag #tensors.
In this animation, you can see the components changing as the axes go through a full rotation.
In #SolidMechanics, #stress has some counter-intuitive properties, particularly when you rotate the coordinate system. The values of normal and shear stress transform according to the rules of #second-order #tensors, which is a step above #vectors (first-order tensors). To avoid always having to use #matrix #algebra to find the components, #engineers and #mathematicians have long used the Mohr's circle to evaluate them. This is a very handy tool and allows you to find the new values by using circular geometry instead.
Here, I've used #WxMaxima to create this #animation.
#Mechanics #Mathematics #Engineering #MohrsCircle #FreeSoftware
In the #2D #elasticity, #equilibrium of #stresses can be represented on an infinitesimal rectangular element with components of both #DirectStress and #ShearStress generally acting on all four edges. If you were to rotate the rectangle, the stresses change in a precisely orchestrated fashion. In the orientation where the shear stress components vanish, we get what are called #PrincipalStresses and they and their directions can be ascertained precisely through #eigenvalue analysis. This little graphical demonstration shows one example of such a state. The #AnimatedGif was produced a routine written in #WxMaxima.
#Mathematics #TheoryOfElasticity #Mechanics #SolidMechanics #Engineering #MyWork #CCBYSA #WorkInProgress
Noam Chomsky - The 5 Filters of the Mass Media Machine
https://www.youtube.com/watch?v=34LGPIXvU5M
' #AmyGoodman narrates a short video piece based on #NoamChomsky’s #book “ #ManufacturingConsent”, a seminal work on #mainstream #journalism and its role in the #mechanics of #power.' According to #Chomsky, media operate through #5 #filters: #ownership, #advertising, the #media #elite, #flak and the #common #enemy. #analysis #propaganda #machine #msm #profit #freedom #democracy
