Now we at Sapioplasm focus not only on presenting news, but on breaking down the science behind it. Let’s try to demystify some of the terms used for describing launch. To quote from the post “The unique triumph of PSLV-C37” on ISRO’s home page:

“The satellites were inserted into a Sun-Synchronous orbit at 506 km above the earth, with an inclination of 97.46°.”

What is the sun-synchronous orbit?

So, a sun-synchronous orbit is a special case of the polar orbit, which is a type of Low Earth Orbit (LEO). Too much? Lets take it one step at a time.

The low earth orbit is where all manned space missions except the Apollo program have taken place. The International Space Station operates in the LEO. Objects in orbit between 160 to 2,000 km above the earth’s surface are said to be in LEO. Satellites in this orbit are still within the upper layers of the earth’s atmosphere, which is why they still face atmospheric drag, but it decreases with altitude. The benefits of using this orbit are numerous as far as navigation, weather and some communication satellites are concerned: Low orbital periods (between 88 to 127 mins to orbit the earth) for observing regions multiple times, low latency and high bandwidth for communication and ease of getting a closer look at clouds and weather patterns from above. Most of all, less fuel is needed, i.e lesser costs are incurred to launch satellites in this orbit than others like the Geosynchronous orbit (35,786 kms above the earth’s surface).

Of course, too much of anything is never good and frequent launches have led to accumulation of space debris in the LEO.

It is easy to guess what a polar or near-polar orbit is – the satellite tracks an orbit above the poles while the earth rotates. The inclination of this orbit with respect to the equator is then 90° (for strictly polar orbits). The earth’s gravity and rotation cause the areas the satellite covers on earth to be shifted to the west after each orbital period. This short video helps visualize how satellites in polar orbits map the earth, one swathe at a time.

Now, the earth is not a perfect sphere. Bulges around the equator cause the satellite’s polar orbit itself to rotate around the earth. This is called nodal regression and it depends on the orbit’s inclination and height. To quote an example from chap.4 of Visual satellite observing(http://www.satobs.org/faq/Chapter-04.txt), “ For 555 km (300 nm) altitude, 130 degrees inclination, the nodal regression is 4.7 degrees per day eastward.” Sun-synchronous satellites make use of nodel regression in order to cancel out the changes in the position of the sun relative to any point on earth, which is a result of the earth’s path around the sun. So what does this mean? Everday, when the satellite goes over a region on the earth, the position of the sun with respect to the satellite and the earth would stay the same.

The orbit is chosen at such an inclination that the satellite crosses regions on the earth at the same local solar time each day.

NASA’s Earth Observatory page about Three Classes of Orbit (http://earthobservatory.nasa.gov/Features/OrbitsCatalog/page2.php) clarifies this with an example: ” For the Terra satellite for example, it’s always about 10:30 in the morning when the satellite crosses the equator in Brazil. When the satellite comes around the Earth in its next overpass about 99 minutes later, it crosses over the equator in Ecuador or Colombia at about 10:30 local time.”

This is an advantage because every time that the satellite passes over the region, it will have the same illumination conditions, i.e the sun will shine on the earth from the same angle at least for the duration of that season. So satellites that take pictures would need bright conditions while long wave radiation measuring satellites would need complete darkness.

However, the downside to all these kinds of orbits is that constant monitoring of one spot on the earth is not possible, which is where the geosynchronous orbit used by many communication satellites comes in.

The sun-synchronous orbit has about about 95 to 100 degrees of inclination depending upon its altitude and gives a nodal

regression of 0.98 degrees eastwards.

So now we know exactly what scientists at ISRO mean when they say “Sun-Synchronous orbit at 506 km above the earth, with an inclination of 97.46°.”

Returning to the achievement in question. Apart from this being a record for the most number of satellites inserted in orbit till date, this move also required an immense amount of calculations as far as safe separation of these satellites is concerned. Inserting such a large number of satellites in one orbit meant that the angle of separation and timing was key to avoid collisions, as Dr. K. Sivan, Director of the Vikram Sarabhai Space Centre (VSSC) has said.

More details in this link: http://www.thehindu.com/sci-tech/science/How-ISRO-plans-to-launch-103-satellites-on-a-single-rocket/article17075073.ece

India’s PSLV has proven to be a reliable launch vehicle for the past few years, also launching the lunar probe, Chandrayaan in 2008 as well as the Mars Orbiter Mission, Mangalyaan in 2013.

ISRO will now focus on perfecting PSLV’s more powerful counterpart, the Geosynchronous launch vehicle(GSLV), which will be uses indigenously developed cryogenic technology…but more on that later.

*Russia held the record for most satellites launched so far after launching 37 satellites in June 2014 with its Dnepr rocket

External Links:

http://isro.gov.in/unique-triumph-of-pslv-c37

https://www.nasaspaceflight.com/2014/06/russian-dnepr-rocket-record-launch-37-satellites/

http://www.universetoday.com/85322/what-is-low-earth-orbit/

http://satellites.spacesim.org/english/anatomy/orbit/low.html

http://earthobservatory.nasa.gov/Features/OrbitsCatalog/page2.php

http://www.satobs.org/faq/Chapter-04.txt

https://marine.rutgers.edu/cool/education/class/paul/orbits2.html

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Okay, so what is the first thing that comes to mind when you see the number 2017. Prime number of course. Well, if you are a geek like us. Yes, this year is the prime number year, but it is not just any other prime number year, it is the **Sexy Prime Number** year. Confused? Lets take a look.

The largest known prime number as of beginning of 2017 is 2^74,207,281 − 1, a number with 22,338,618 digits. It is a Mersenne prime. Any prime number which can be written in the form 2^*n* − 1 is called a Mersenne prime. What about 2017? Well, 2017 can be written as 2018-1, but unfortunately 2018 is not a power of 2, so 2017 is not a Mersenne prime. But hey there are hundreds of prime number types. Some interesting types include Pythagorean prime, which is a prime number which can be expressed in the form 4n+1 where n is an integer. 2017 can be expressed in this form as 4×504+1=2017. So yes 2017 is a Pythagorean prime number.

One more type of prime number which we found fascinating was Happy primes. In order to understand Happy primes, we need to understand Happy numbers. Consider a number, say 19. Now square and add the individual digits – 1^2 + 9^2 = 82. Continue the process again and again and again, till you get 1.

1^2 + 9^2 = 82

8^2 + 2^2 = 68

6^2 + 8^2 = 100

1^2 + 0^2 + 0^2 = 1

Once you get one, you will see that if you repeat the same process again and again, you will still get one. If the chain ends at 1, that number is called a happy number. And the prime numbers which are happy are called happy primes. So 7, 13, 19, etc. are happy primes.

We tried doing it for 2017. It is not a happy prime, probably because 2016 year was depressing for it too, but something else turned up.

2017

2^2 + 0^2 + 1^2 + 7^2 =54

5^2 + 4^2 = 41

4^2+1^2 = 17

…(continue for a little long)

1^2 + 4^2 + 5^2 = 42

And we get 42, the answer to the life, the universe and everything. The best two digit number, although some argue that it’s 69.

Coming back to the introductory paragraph, we said that 2017 is a sexy prime. Well, sexy refers to the Latin word “sex” meaning “six.” So a pair of prime numbers which differ by 6 are called sexy primes. 2017 and 2011 are both prime numbers and the difference between them is six, so both are sexy prime numbers.

My happy new year tweet had a little amount of geekiness, with binary numbers. It was my tweet number 111, which is a repunit number. It is a number containing just the digit 1. And the prime number which is also a repunit (eg. 11) is called the repunit prime.

#HappyNewYear #Nerdgasm #Binary pic.twitter.com/X13iVh9xe0

— Arnav Nair (@arnavramdasnair) December 31, 2016

2017 is 306th Prime Number, and we hope this year will be of prime quality.

Links you may like

https://en.wikipedia.org/wiki/Mersenne_prime

https://en.wikipedia.org/wiki/Largest_known_prime_number

https://en.wikipedia.org/wiki/Sexy_prime

https://en.wikipedia.org/wiki/Repunit

https://en.wikipedia.org/wiki/Happy_number

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**Zeno’s Paradox**

Consider that the Greek war hero Achilles races a tortoise. Our hero is compassionate and decides to give the tortoise a head start. Who would win the race? Easy, you say. Achilles!

Erm…no. Not as the Greek philosopher Zeno imagined it, at least.

When Achilles starts the race, the tortoise has already covered some distance. Now, by the time Achilles gets to the point that the tortoise previously was, the tortoise has covered some more distance. “Okay”, says Achilles and bridges that gap too. By then, the tortoise has gone a little further…See where this is heading? The tortoise would always lead this race, even if it is by an infinitesimally small distance. HOW???

One reason this paradox is misleading is due to thinking in terms of discrete steps. Of course Achilles would rush past the tortoise in a matter of seconds in continuous time! This is like walking from a point A to point B while taking steps that are half as long as the previous one each time. (Apart from this being humanely impossible after the point of a few millimeters,) At some point of time there would hardly any perceivable distance left between the person and point B. Can we then say that he has reached point B?

An alternative explanation to this is the case of convergent series. Like cutting a pie into half, then splitting the remaining half into quarters, splitting the quarters into one-eights and so on. We’re splitting something finite like distance into infinite pieces. We know that the series converges from the fact that the pie was whole before we started cutting it in this bizarrely obsessive way. The sum of the infinite series 1/2 +1/4 +1/8 +1/16 +….. is, in fact, 1. So, in this race of infinite steps, at some point Achilles and the tortoise are bound to meet. And yet, that point, my friend, is infinitely far.

(Under special conditions, this paradox can also be divergent. See http://www.slate.com/articles/health_and_science/science/2014/03/zeno_s_paradox_how_to_explain_the_solution_to_achilles_and_the_tortoise.html for more)

Another explanation for this thought experiment is the endlessness of the sequence. With each step, the lead that the tortoise will have over Achilles goes on decreasing. It decreases to the extent that it is almost zero. This article about the paradox is detailed, but worth the read!

http://barang.sg/index.php?view=achilles&part=2

So, no matter what, Achilles will overtake the tortoise at some point of time. May be if we can even the odds by giving Achilles some more handicap. May be if we ask him to wear high heeled shoes during the race. Could it be his Achilles’ heel?

__Grandfather Paradox__

Imagine you have created a time machine. Now you go back in time and have this crazy idea – Let me kill my grandfather. So you aim the gun at your two year old grandfather and pull the trigger. So, your grandfather’s dead, which means he will never have a chance to create your father, which means you were not born, which means you did not create the time machine, which means you don’t go back in time and you don’t kill your grandfather and your grandfather lives and … Honestly, at what point in this sentence did your head start paining? This is the classic grandfather paradox.

So do you exist or not? What does it mean? One simple explanation is travelling back in time is not permitted by the laws of physics. If this hypothesis is correct, then life would be much simpler. You can’t go back in time and cannot cause any of such crazy nonsense.

But think about what Einstein has taught us, life is not simple. Of course we cannot say that this hypothesis is false, but we should search for other explanations in case this doesn’t turn out to be true.

Another explanation which would help us sleep at night is, the moment you do this crazy nonsense, the universe splits into two strands, an alternate timeline (Damn you Barry Allen). One is the one in which you are alive and the second one in which your grandfather’s dead at 2 and you don’t even exist.

That’s cool; I go in the past from one timeline and create an alternate universe. I could live with that. However, there could be a third possibility.

Now comes the crazy one for the crazy scientist in you, who decided to ruin his/her existence. The universe is in a superposed state i.e. Schrodinger’s Cat’s Grandfather State. Such kind of weird superposition thingies happen all the time at small, teeny tiny quantum level. Could this happen at big, macroscopic level? There’s only one way to find out. (Statutory Warning: The author of this article does not encourage murder. Please find an alternative way to test this hypothesis). If true, your grandfather is both simultaneously alive and dead; and you simultaneously exist and do not. Both the histories run in parallel.

For more on this check out the video made by minutephysics on this topic: https://www.youtube.com/watch?v=XayNKY944lY

Okay, if you need to take an aspirin, go ahead, because the next paradox is even crazier.

__Twin paradox__

Ah, Einstein, making life miserable since 1905. 1905, is regarded as the miracle year or “*Annus Mirabilis* ” of Einstein, as he published four amazing papers which revolutionized physics. One of these papers was the special theory of relativity (Not the Title of the Paper). We’ll not go into detail of relativity, but we’ll look at one aspect of it – The Time Dilation a.k.a. that weird time stuff in the movie “Interstellar”.

Einstein derived mathematically that as a person moves faster relative to observer, that person’s clock ticks slower than the one with the observer. However, if we look from the moving person’s point of view, the observer is moving and that person is at rest. So according to the observer, the moving person’s clock is ticking slower whereas from the moving person’s point of view, the observer’s clock ticks slower.

How can it be that two people see two different things? Who is correct? Answer: They both are. The order of events and the times depend upon which frame of reference you are present in. This is called “Relativity of Simultaneity.” Guys, we have not entered into paradox yet. Keep the aspirin handy in case you haven’t taken it already.

Okay so if we accept, although it is hard to accept even if it is true, that time is reference frame dependent, imagine identical twins, say Albert and Rupert Einstein, both standing near the launch site from where one of the twin, say Rupert, is going to start its interstellar journey to Proxima Centauri.

Not a huge distance, just a four light year trip, eight for a return trip. The twins are currently 30 years old. Rupert embarks upon this journey at almost the speed of light. He reaches a planet of our neighbour star system, collects some samples and comes back to earth again with near light speed. Since Rupert was travelling at almost the speed of light, Albert sees that the clock of Rupert slows down during the journey and he notices that the astronaut had hardly aged at all whereas Albert himself aged 8 years and had grey hair and a cholesterol problem. So far, so good. But from Rupert’s point of view, Earth itself is moving away from him at the speed of light as they reach the planet. And then Earth moves towards him at the speed of light as they head back home, near light speed to be precise. So according to him, Albert should age slower than him. But that doesn’t happen. Even he finds out that he is the younger twin. How is that possible? A TWIN PARADOX.

Fortunately, this paradox is resolved by a simple logic. Let’s take a look at the nature of two reference frames. Albert is in what’s called an inertial reference frame. Any reference frame which is not accelerating is called an inertial reference frame. Of course it is not a perfect inertial frame, if you take into consideration the rotation and the revolution of Earth, you will get some acceleration, but the acceleration is relatively small and can be neglected. But in case of Rupert, the frame which he is in first accelerates to attain speed of light and then decelerates to zero as it reaches the planet. And again on the journey back, we see the same.

Of course there is a period where Rupert’s frame is moving with a constant speed and is inertial, at that time dilation takes place as prescribed by Dr. Einstein and Rupert does see the clock of Albert ticking slower than his. But during the acceleration something different happens.

Remember when I told you that according to special theory of relativity, faster you go slower the time ticks. Well, it’s called “special” for a reason. It is applicable for only a special case of inertial frame of references. In order to study the accelerating frame of references as well, we need to go to more general theory – The General Theory of Relativity. If the movie interstellar has taught us anything, it is that more the gravitational pull, slower our clocks tick. And in general theory we have something called an equivalence principal which states that there is no difference between acceleration due to gravity and the acceleration in general, somewhat like that. So during the time of acceleration and deceleration of Rupert’s frame of reference, Rupert’s time runs slower, way slower and accounts for the solution to the paradox.

Another minutephysics video explains this beautifully, check that out using the following link: https://www.youtube.com/watch?v=0iJZ_QGMLD0

If you need a prequel to this video, which talks about the tools needed to solve this paradox, follow this link: https://www.youtube.com/watch?v=Bg9MVRQYmBQ

**The Birthday Paradox**

This paradox says that in a group of 23 people, there is a 50% chance that two of them will share the same birthday. This seems ridiculous at first, because there are 365 days in a year! Well, this is exactly why it is called a paradox. Lets get to the math to prove this-

Lets look at the probability that two people do NOT share a birthday. This means that the second person out of the two should have his/her birthday on one of the other 364 days of the year, making the probability 365/365 * 364/365 = 0.9972. This means that two people will not share birthdays 99.72% of the time.

As the number of people go on increasing, the probability that they do not share a birthday goes on decreasing.

For 3 people, the chance that they will not have the same birthday is 365/365 * 364/365 *363/365 or 0.9917, for 4 people, 0.9863, for 10 people, 0.8830, for 15, 0.7471 it and so on..

For 23 people, this probability comes down to 0.4927, which means there is a more than 50% (50.73% to be precise chance that any two people in the group share the same birthday. As people in the group go on increasing, the chances will increase even further… This implies that in a group of 57 people, there will be approximately 99% chance that birthdays will be shared! A great and in-depth article on this …is https://betterexplained.com/articles/understanding-the-birthday-paradox/

**The Infinite**** Hotel Paradox**

Another paradox/thought experiment on just how mind-boggling infinity is. Consider that there exists a hotel with infinite rooms where each room has a guest in it. Now, a guest turns up looking for a room. Can he be accommodated? Yes! Each guest is asked to move to room n+1 , thus freeing up the first room in the series of an infinite rooms. Next, infinite guests show up, all clamoring for rooms. It is possible to accommodate them too. How, you ask? Well, we need to free up infinitely many rooms. So each guest is asked to move to a room with a number that is twice of his current room. Let’s see if you’re following here. If there are 10 rooms, and guests in 1-10 move to rooms 2, 4, 6…20, we free up 10 rooms, right? Simple. So if we send the infinite number of guests to rooms up to 2 x infinity (= infinity???) we still free up infinite rooms. Err… But the best is yet to come. What if, infinitely many coaches with infinitely many passengers show up? We would want to vacate infinity x infinity (= infinity???) number of rooms. We will let you think. (Hint: The **prime** rule is to avoid double bookings)

Also, this video is a delight! https://www.youtube.com/watch?v=Uj3_KqkI9Zo

This Wikipedia post elaborates further https://en.wikipedia.org/wiki/Hilbert’s_paradox_of_the_Grand_Hotel

PS: In case you didn’t watch the video, the infinity here is countable infinity (which means only positive real numbers…No complex or negative numbered rooms, phew!)

Hope you have enjoyed reading this post. Let us know your feedback or if we made any boo boo, in the comments.

Thank you.

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-reddit comment (xsnac)

Mathematics is believed to be the language of physics. If you are into physics, mathematics is inevitable. Mathematics concerns itself with abstract ideas. And physicists use those abstract relationships to replace it with the physical things and get the quantitative relationship. And sometime, physicists demand a new tool to solve its problems and mathematics creates it for them.

In order to taste the flavor of mathematics and understand the affair between its concept and that of physics, one must try to think of some physical example that could fit in perfectly with that mathematical concept. For example, I used to have a problem with discontinuous functions. My maths teacher used to write three different relations for the same variable, and I used to think, where I would find something like this. Something behaves in certain way up to some point and then decides to change its behavior? So I tried to think of some example in the physical world which behaves in this weird fashion, and lo and behold, we have refraction.

__The Example__

Consider the figure…

A light ray originates from the optically denser medium, crosses the interface and enters the rarer medium. In this example I have considered the denser medium to be glass, with refractive index 1.5, and rarer medium to be air, with refractive index of about 1. As the light ray crosses the interface, the ray gets bent away from the normal. I have considered two angles θ_{1} and θ_{2} and I have to find the relationship between them.

__The Relation__

So we have three different relationships to form one function of θ_{2}. Up to the critical angle (approximately 41.8^{0} for Glass-Air interface), the relationship between them can be found by manipulating Snell’s Law of refraction. But after critical angle, the relationship suddenly changes as Total Internal Reflection (TIR) begins. The relationship becomes linear as per the law of reflection. This relationship holds until θ_{1} becomes a right angle after which it once again follows Snell’s law but with incident ray in the rarer medium.

__The Graph__

The graph of this can be plotted as below. I could have used excel to plot but I wanted to kill a little more time so I plotted manually.

And finally we have some physical example for the abstract mathematical concept. I don’t know whether, this concept was derived because of some physical observation or the mathematical concept came first, but I was really happy that I could relate it. If you find any other physical example please share with me. Till then, thanks for reading.

References:

- For Total Internal Reflection: https://en.wikipedia.org/wiki/Total_internal_reflection (Or any fundamental physics textbook)
- For the opening quote: https://www.reddit.com/r/AskPhysics/comments/44tol0/good_lectures_or_books_on_hydrostatics_without/

The quiz had four levels. First three levels were done on paper. Each level had math questions and also some general knowledge questions related to math and mathematicians. After each level, some of the teams were eliminated. There were also spot questions asked in between two levels and spot prizes were awarded to those who answered them correctly. At the end of three levels, five teams were selected for the final level.

As a Quiz master I was in charge of the last level of the quiz. This level consisted of four rounds. First round was the classic quiz, in which a question is asked to one team and if they get it right, they are awarded 5 points and if they could not answer it right, the question is passed on to other teams. The team who answered the question after it was passed to them was awarded 2 points. The second round was the buzzer round in which the team who knows the answer must press the buzzer. Whoever presses the buzzer first will get to answer. This round had little difficult questions and the team who could answer it correctly was awarded 10 points, but the wrong answer fetched them -5 points. Third round was relay. In this round, each team had two sets of questions and the answer to the first question was an input to the second question. Each team had 90 seconds to solve both questions. The catch was, only one member is allowed to tackle the first question. Only when the first question is solved the second member of the team can join them to solve the second question. Each question in that set carried 5 points. And the last round was a rapid fire round in which a team has to answer as much questions as they can within 90 seconds, each question worth 2 points.

At the end of four rounds, Anand Chitrao and Akshay Sant from Ruia College, Matunga were the winners by a huge margin. We congratulate them for their success.

The question and answers for the rapid fire round are as below…

- If A=B and B=C then A=C, what is his property called?

Ans: Transitive Property

- On whose life was the movie “A Beautiful Mind” starring Russell Crowe was made?

Ans: John Nash

- For a triangle of uniform density, where will its center of mass lie?

Ans: Centroid

- Set of numbers which can form the lengths of a right angled triangle is also known as what?

Ans: Pythagorean Triplet

- If toffee costs Rs. 100 less than a pastry and the total bill was 101, what is the price of that toffee?

Ans: Rs. 0.5

- Word radiation comes from which geometric term?

Ans: Radius

- If y is inversely proportional to x, which curve do you find on the y vs x graph?

Ans: Rectangular Hyperbola

- When will the equation x^n + y^n = C represent a square?

Ans: When n tends to infinity

- If I take 80 minutes each to run three of the sides of a square ground and 1 hr 20 min for the last side, what is the ratio of my speed in the last leg to that of the first three?

Ans: 1

- Hypothetically if iron is twice as dense as copper what will be heavier, a kilo of iron or a kilo of copper?

Ans: They both weigh the same

- What sentence is called the Liar’s Paradox?

Ans: I am lying

- If a bacterium doubles its population every minute, and takes 45 minutes to occupy a quarter of the bottle, after how many minutes will they fully occupy the entire bottle?

Ans: 2 minutes

- If frog jumps a slope which is 10 feet long, covers 3 feet in every jump and slips down 2 feet, how many jumps will it require to jump over the top?

Ans: 8

- Who is known as the man who knew infinity?

Ans: Srinivasa Ramanujan

- Which geometric shape has the minimum surface area for a given volume?

Ans: Sphere

- Which shape is used for making dish antenna and solar thermal collector?

Ans: Paraboloid

- Which curve represents the path which requires minimum time for a body to reach from point A to point B under gravity?

Ans: Cycloid

- What number is 10^100?

Ans: Googol

- Which conic section do you get when you cut the cone parallel to its axis?

Ans: Hyperbola

- A guy walks 10 km due north in a straight line and ends up 5 km further away from North pole than he originally was, how is that possible?

Ans: He was 2.5 km away from North Pole initially

- What number is known as the Golden Ratio?

Ans: 1.618

- In a lucky draw contest there are 100 people participating including me, I cheated and put my name in the box 5 times. What is the probability that my name will come up?

Ans: 5/104

- Who is believed to be the first to prove that irrational numbers exists?

Ans: Hippasus of Metapontum

- If two matrices are multiplied and the resultant matrix is n times the second matrix which was multiplied, what is n known as?

Ans: Eigen value

- In numerical integration which rule approximates the curve as a straight line?

Ans: Trapezoidal Rule

- If the derivative of a function is 0, it can either be maxima, minima or ______?

Ans: Point of Inflection

- Which geometric shape has the least perimeter for the given area?

Ans: Circle

- Which curve do you get if you cut a cone parallel to its edge?

Ans: Parabola

- Which point marks the center of the circle which touches all the three sides of the triangle?

Ans: Incenter

- In a 30-60-90 triangle, what is the ratio of side opposite to 60 degree to the side opposite to 30 degree?

Ans: Sqrt(3)

- Who is known as the father of geometry?

Ans: Euclid

- What geometrical shape do we get by connecting two opposite Mobius strips?

Ans: Kline bottle

- Which geometrical shape does the equation x + y = 5 represent in three dimensions?

Ans: Plane

- Which medal is often described as the “
**Nobel Prize**of**Mathematics**“?

Ans: Fields medal

- Which conic section do we get if we cut a cone through its vertex?

Ans: Pair of straight lines

- In numerical integration, which rules approximate a curve to parabola?

Ans: Simpson’s rules

- Which curve shows brachistochrone property?

Ans: Cycloid

- How is 1000 written in Roman numerals?

Ans: M

- What number is 10^Googol?

Ans: Googolplex

- Which matrix is its own inverse?

Ans: Identity matrix

- Which natural number is doubled when added to its reciprocal?

Ans: 1

- What is the sum of the first 100 odd natural numbers?

Ans: 10000

- What is the name of the point which is the center of a circle which passes through all the three vertices of the triangle?

Ans: Circumcircle

- What are the diagonal elements of a skew symmetric matrix?

Ans: 0

- For a cube of volume 27 units, what is the length of its longest diagonal?

Ans: 3 * sqrt(3)

- Who is known as the father of computer?

Ans: Charles Babbage

- For a radioactive element, what curve describes the quantity of the material left with time?

Ans: Logarithmic curve

- If set A is a collection of all the integers with absolute value less than 100, what is the sum of those integers?

Ans: 0

- A bookworm eats it way through a book at the rate 1 inch per hour, how long will it take to eat its way from the front of volume one of the book to the back of volume three, each volume being half inch long (Volumes are kept in order)?

Ans: 30 minutes

- Who invented Principle of calculation machine with progressive transmission of tens?

Ans: Pafnuty Chebyshev

]]>Consider the crystal structure of a metallic bar as shown in the figure. Of course, the real structures are more complicated, but just for understanding purpose let us consider they have this simple structure with circles representing the molecules and lines connecting them as the inter-molecular bonds.

Now, **let’s bend the metallic bar**. As you can see from the second figure, the molecules present in the inner side of the bend (concave side), come close to each other, while the molecules on the outer side (convex side) of the bend move away from each other. As a result, the molecules on the inner side push each other away while those on the outer side try to pull each other towards them. Think of the bonds like springs, if you stretch it, it will try to contract and if you compress it, it will try to expand. The lattice also undergoes permanent deformation, but let’s not talk about that. This push and pull action tries to bring this bar back to its initial shape thus giving rise to internal stresses.

Now, **let’s heat the bar**. What happens is that the old molecular bonds weaken and break; and molecules also re-position themselves into new lattice pattern, kind of like people trying to be comfortable in a crowded train. On cooling, new bonds are formed to suit new lattice arrangement. Now the structure acts as if the metallic bar originally was supposed to be bent.

And now you get the stronger finished component.

Based on the way and how long it is heated and cooled, different types of properties can be achieved.

Stay tuned for more updates.

Thank you

-Arnav

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