The Curated Internet

  This leads to Quantization of θ  because there are only specific values that become possible for  θ  when we impose that after n rotations it has to return back to its same starting point. ** And the following are those values: The problem of finding the values of θ for given values of n is more generally known as the N roots of unity.   We will leave it as an exercise, but the following animation plots all the possible the values of θ for integer values of n=2,3,4,.. from the above equation that we found:                             Source And those are the roots of the equation z^n  = z i.e if you start at these points on the unit circle and make n rotations you will get back to the same point that you started with. Have a good one! ** A more physical way to think about this is matching boundary conditions. ...

Below are some of the images from the observations made at the Lick Observatory using the Nickel Telescope during October 2018. (1) - M1 (Crab Nebula) (2) - M15 (3) - M16 (Eagle Nebula- Pillars of Creation) (4)&(5) - M17  (Omega Nebula) (6) - M42 (Orion Nebula) (7) - NGC6563 (Cat’s Eye Nebula) (8) - NGC7662 (Snowball Nebula)        

    Waaay back in 1054, a star in the constellation Tauras exploded in a dramatic fashion. The remnant of that massive explosion (supernova) is what we now call the Crab Nebula. But as a result of the explosion, the nebula is still expanding at a rate of over 1,000 kilometers per second. The above gif comparing the nebula in 1950 vs what it was in 1990 is one of the most epic ways to illustrate that fact.  

It can be casual to forget the magnificence of our planet and get lost in our tight-knit everyday lives. In the advent of a lunar eclipse (January 31, 2018 )  it is worth knowing that when it comes to eclipses, Earth holds a pristine status in our solar system.  To understand why, we need to shift our perspective a little bit and ask -”How would it be like if you were on Io (one of the moons of Jupiter) ?”                                            Image source : Deviant art The most startling thing about this experience would be that the Jupiter would appear 36 times larger than the full moon (from earth).  That’s HUGE!   Also since the moons of Jupiter lies in the same plane, you would be witnessing an eclipse every 42 hours …                   Moons - Io, Ganymede, and Callisto in solar eclipse In ...

  When one searches for Fourier series animations online, these amazing gifs are what they stumble upon. They are absolutely remarkable to look at. But what are the circles actually doing here?   Vector Addition Your objective is to represent a square wave by combining many sine waves. As you know, the trajectory traced by a particle moving along a circle is a sinusoid:   This kind of looks like a square wave but we can do better by adding another harmonic   We note that the position of the particle in the two harmonics can be represented as a vector that constantly changes with time like so:   And being vector quantities, instead of representing them separately, we can add them by the rules of vector addition and represent them a single entity i.e:                                                     ...

  The legend goes something like this: Gauss’s teacher wanted to occupy his students by making them add large sets of numbers and told everyone in class to find the sum of 1+2+3+ …. + 100. And Gauss, who was a young child (age ~ 10) quickly found the sum by just pairing up numbers:   Using this ingenious method used by Gauss allows us to write a generic formula for the sum of first n positive integers as follows:  

In this set of images the smoke from an incense stick is seen making a transition from a smooth laminar flow (very close to the incense stick) to a turbulent flow (far away from the incense stick) while passing through the Kelvin-Helmholtz instability (rolling vortices).

  Space shuttles are launched almost at all times from West to East to take advantage of the earth’s rotation How does earth’s rotation affect shuttles ? Earth is a spherical body rotating with some angular velocity. And as a result of this, the equator is rotating at a higher velocity than the poles. By launching a space shuttle from the equator you are getting a ‘speed boost’.     This means that if a shuttle is launched from the pole, it has to accelerate from 0 to 17000mph to reach orbital velocity. But if a shuttle is launched from the equator, it only needs to accelerate from 1025 to 17000mph. (that 1025mph initial velocity is given by the earth free of charge) This saves valuable amount of fuel required for propulsion     Polar Orbits Not all rockets are launched from the west to east and the direction is determined by the purpose of its payload. The satellites that are used for mapping for instance follow a Polar Orbit i.e they move from north to ...

Bernoulli’s principle, Bernoulli’s number, Bernoulli’s polynomial, Bernoulli’s differential equation and the list goes on. If you were like me and thought that Bernoulli was the name of a single guy, you are soo wrong. They were a family ! Over the course of three generations, the Bernoullis produced eight mathematically gifted academics who, between them, contributed to the foundations of applied mathematics and physics.  Some of the most popular ones of the family are: 1. Daniel Bernoulli ( Inventor of the Bernoulli’s equation ) 2. Jacob Bernoulli. ( The Bernoulli Numbers )  3. Johann Bernoulli. ( One of the early adopters of Infinitesimal Calculus).  And the rest contributed extensively to the fields of math and physics. If you are want to know more about the history of the Bernoulli Family click here. 

The acoustic sweet spot is defined as the listening position equidistant to each of the two front channels as they are from each other, so the arrival time of the sound is equal at your ears.   It was brought in the vogue of the public by the big bang theory where Sheldon cooper tries to find the sweet spot in a theater.    Why is it important? It is called the ‘sweet spot’ for a real good reason.  In a motion picture, an image is considered to be ‘good’ if the location of the performers can be clearly located. This is known as stereo imaging and it adds realism to the image.  The only person who hears this perfectly is the one who is in the sweet spot. ( no wonder Sheldon is obsessed with his spot! )   At this juncture, it is highly recommended that you check out the virtual barber shop to experience the acoustic sweet spot for yourself.  The virtual barber shop places you in the sweet spot and abuses sound technology to bring you this ...