## On the definition of  Angular Momentum

Why
is it mr^2 omega and not some other weird formula that is conserved? Why not mr^3 omega or mr^2 omega^2 ?

This is a great question. And to be honest, there is no intuitive answer as to why it is defined this way or that.

## What makes the definition special?

Conservation laws can be understood better through the Lagrangian formulation of classical mechanics.

That’s the conservation of momentum for a free particle. It means that this quantity mv remains constant with time (not m2v, not m2v2 ,just mv).

And similarly for a rotating body, one can find that the quantity that remains constant wrt time is the angular momentum.

And that’s the best rationale using modern physics that can be provided for why Angular momentum takes the form that it does.

Any other form would just not be conserved. Sure, you can construct a Lagrangian that would give you the form that you need but that would not  represent anything physical !

** If you have not heard about Lagrangian formulation of classical mechanics, the wiki article on Principle of Least action is a really good place to start..

The principle of Least/Stationary action remains central in modern physics and mathematics, being applied in thermodynamics, fluid mechanics, the theory of relativity, quantum mechanics, particle physics, and string theory.

If unit vectors always scared you for some reason, this neat little trick  from The story of i by Paul Nahin involving complex numbers is bound to be a solace.

It allows you find the tangential and radial components of acceleration through simple differentiation. How about that!

Have a good one!

** r = r(t),  θ =  θ(t)

# FYP book of the month: June

The Code Book tells the story of the most powerful intellectual weapon ever known: secrecy.

Continued fractions are a fascinating way to represent never ending, never-repeating decimal expansions like pi or e.

As a fun exercise on this Friday evening, take out your smart phone and open the calculator app and try this :

What do you get? Do you recognize this number? If you did, then try answering why it mysteriously popped out of nowhere to represent this continued fraction.

Have fun!

# Notes from a discussion about Hands

Recently, I found myself in the middle of a discussion with some close friends about the physics and maths that one can associate with our hands. Here are some notes from that discussion.

We apologize in advance if this post seems too factual and a bit disoriented, we couldn’t find a better way to share this information.

## Hand Protractor

If you stretch out your hand, you will notice that the fingers on your hand can be calibrated to be used as a protractor like so:

## Count from 0 to 1023 **

Want to count till 1023 just on your hands? Well, just assign each finger a binary value and chug along:

Also: Check out the binary dance by Vihart

## Rock, paper, scissor, Lizard,Spock

A simple of rock,paper, scissor is by itself a lot of fun and is a playground for game theorists and mathematicians

But adding new gestures to the game reduces the odds of getting a tie but increases the complexity of the game. One popular five-weapon expansion is “rock-paper-scissors-Spock-lizard”, invented by Sam Kass and Karen Bryla.

## The golden truth

This picture gets thrown everywhere on the Internet to illustrate the divine connection between hands and the golden ratio.

BUT in a study using X-rays conducted at St. Luke’s Medical Center, Chicago, it was found that only 1 of 12 bone length ratios contained the ratio phi in the 95% confidence interval.

And it was concluded that:

The application of the Fibonacci sequence to the anatomy of the human
hand, although previously accepted, is a relationship that is not
supported mathematically.

## Drip system on a farm land?

What looks like a drip irrigation system on a farm land is actually a close up of your hands when sweating.

The reason why this visualization is fascinating is because modern techniques (like full field optical coherence tomography (FF-OCT) )in Forensics can extract sweat pores information from the fingerprint offering more information about the perpetrator crucial to the investigation.

## Snapping fingers and cracking knuckles

When you snap your fingers, in order to get the satisfying full “snap” sound , the fast-moving second finger
must hit both the palm and a small portion of the third finger.

If it
hits only the palm and not the third(ring) finger, there will be a significant
reduction in the total ‘snap’ sound. Try it out!

When you get tired of snapping your fingers, try cracking your knucles. The sharp sound that you hear is caused by the collapse of a cavitation bubble (see image above) in the sinuovial fluid present at the joint.

## Fingers, shadows and the transit of Venus

If you are wondering how on earth can all of these words be possibly related, then try answering this question:

When you bring two of your fingers closer in the back drop of a light source, long before your fingers actually touch, the edges magically seem to touch each other.

Once you hypothesize a solution for this, head over here to understand what’s actually happening.

That’s about it! Hopefully not everything mentioned in this post seemed too trivial to you. Each topic mentioned in this post deserves a post on its own and is a lot of fun to explore.

We encourage to play around, have fun and have a great weekend ahead.

** The structure of sign language is yet another thing that is absolutely fascinating to explore. Click here and here  to quench your thirst for knowledge

# Introducing FYP’s book club

We are starting a new segment on the blog where we recommend one or two books in Math or Physics that everyone can read.

You are absolutely welcome to share your comments and reviews here once you are done. Also, if you would like us to check a book out, do let us know too!

Have a good one!

# Using Complex numbers in Classical Mechanics

When one is solving problems on the two dimensional plane and you are using polar coordinates, it is always a challenge to remember what the velocity/acceleration in the radial and angular directions ($v_r , v_{\theta}, a_r, a_{\theta}$) are. Here’s one failsafe way using complex numbers that made things really easy :

$z = re^{i \theta}$

$\dot{z} = \dot{r}e^{i \theta} + ir\dot{\theta}e^{i \theta} = (\dot{r} + ir\dot{\theta} ) e^{i \theta}$

From the above expression, we can obtain $v_r = \dot{r}$ and $v_{\theta} = r\dot{\theta}$

$\ddot{z} = (\ddot{r} + ir\ddot{\theta} + i\dot{r}\dot{\theta} ) e^{i \theta} + (\dot{r} + ir\dot{\theta} )i \dot{\theta} e^{i \theta}$

$\ddot{z} = (\ddot{r} + ir\ddot{\theta} + i\dot{r}\dot{\theta} + i \dot{r} \dot{\theta} - r\dot{\theta}\dot{\theta} )e^{i \theta}$

$\ddot{z} = (\ddot{r} - r(\dot{\theta})^2+ i(r\ddot{\theta} + 2\dot{r}\dot{\theta} ) )e^{i \theta}$

From this we can obtain $a_r = \ddot{r} - r(\dot{\theta})^2$ and $a_{\theta} = (r\ddot{\theta} + 2\dot{r}\dot{\theta})$ with absolute ease.

Something that I realized only after a mechanics course in college was done and dusted but nevertheless a really cool and interesting place where complex numbers come in handy!

## Buddhabrot

The Buddhabrot is an interesting fractal rendering technique for displaying the Mandelbrot Set.

Its name reflects its pareidolic resemblance to classical depictions of Gautama Buddha, seated in a meditation pose with a forehead mark (tikka) and traditional topknot (ushnisha).

** Pareidolia is a psychological phenomenon in which the mind responds to an image or a sound, by perceiving a familiar pattern where none exists. Check out more examples of pareidolia here

If one remembers this particular episode from the popular sitcom ‘Friends’ where Ross is trying to carry a sofa to his apartment, it seems that moving a sofa up the stairs is ridiculously hard.

But life shouldn’t be that hard now should it?

The mathematician Leo Moser posed in 1966 the following curious mathematical problem: what
is the shape of largest area in the plane that can be moved around a
right-angled corner in a two-dimensional hallway of width 1?
This question became known as the moving sofa problem, and is still unsolved fifty years after it was first asked.

The most common shape to move around a tight right angled corner is a square.

And another common shape that would satisfy this criterion is a semi-circle.

But
what is the largest area that can be moved around?

Well, it has been
conjectured that the shape with the largest area that one can move around a corner is known as “Gerver’s
sofa”. And it looks like so:

## Wait.. Hang on a second

This
sofa would only be effective for right handed turns. One can clearly
see that if we have to turn left somewhere we would be kind of in a tough
spot.

Prof.Romik from the University of California, Davis has
proposed this shape popularly know as Romik’s ambidextrous sofa that
solves this problem.

Although Prof.Romik’s sofa may/may not be the not the optimal solution, it is definitely is a breakthrough since this can pave the way for more complex ideas in mathematical analysis and more importantly sofa design.

Have a good one!

Once when lecturing in class Lord Kelvin used
the word ‘mathematician’ and then interrupting himself asked his class:
Do you know what a mathematician is?’

Stepping to his blackboard he
wrote upon it the above equation.

Then putting his finger on what he had written, he turned to
his class and said, ‘A mathematician is one to whom that is as obvious
as that twice two makes four is to you.

## ** Two interesting ways to arrive at the Gaussian Integral

Woah… The backlash that Lord Kelvin got after this post was just phenomenal.

There are many ways to obtain this integral (click here to know about other methods) , but here are two interesting ways to arrive at the Gaussian Integral which you may/may not have seen and may/may not be easy to follow.

## Gamma Function to the rescue

If you know about factorials (5!= 5.4.3.2.1), you know that they make sense only for integers.

But  Gamma function
extends this to non-integers values.  This integral form allows you to
calculate factorial values such as (½)!, (¾)! and so on.

The same can be used to evaluate the Gaussian Integral as follows:

## Differentiating under the Integral sign

In this technique known as ‘Differentiating under the integral sign’, you choose an integral whose boundary values are easy integrals to evaluate.

Here I(0) and I(∞); and differentiate with respect to a parameter β instead of the variable x to obtain the result.