Some #OpenSource information quotes I read about Microsoft use of #Unix (note: free software operating systems can be "Unix-like") on #Wikipedia - #wikipedia25 :

■ Xenix is a discontinued Unix operating system for various microcomputer platforms, licensed by Microsoft from AT&T Corporation. The first version was released in 1980, and Xenix was the most common Unix variant during the mid- to late-1980s.
■ Microsoft cofounder Paul Allen said MS-DOS and Xenix are "part of a family ... with a clear migration path", he added, promising binary compatibility of Xenix-compiled C software with MS-DOS, and interoperability [...]
■ In 1981, Microsoft said the first version of Xenix was "very close to the original Unix version 7 source" on the PDP-11, and later versions were to incorporate its own fixes and improvements.
■ [...] for a long time, Microsoft had the highest-volume [of the] AT&T Unix license.

Also, two (2) non- religious calendar types are Unix time and Terrestrial Time.
#religion #atheism #agnosticism #universe #date #timeperiod #epoch

Emergency declared in Española area due to crime

A state of emergency has been declared by New Mexico Gov. Michelle Lujan Grisham for areas across Rio…
#NewsBeep #News #Headlines #Crime #decision #drugtrafficking #Emergency #españolaarea #HomelandSecurity #localpartner #lujangrisham #police #publicsafetythreat #rioarribacounty #significantsurge #stability #state #timeperiod #UnitedStates #Us #USA
https://www.newsbeep.com/58766/

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