What is the most important thing that an undergraduate needs to know?

A must learn lesson for all undergraduates.

I would like to start this story by using reference from the movie "$3$ idiots". In the movie, there is a scene where Joy Lobo came running to the director with a very happy mood, excited with the idea that he will be the first engineer from his village. The conversation between the two goes like this,

• Joy : Sir. Excuse me, Sir.
• Director: Mister Joy Lobo!
• Joy : Sir, I would like to know the dates of convocation.
• Director: Why?
• Joy : Actually, my dad wants to make a reservation. I am the first engineer from my village, Sir. Every relatives of mine would like to come for the convocation.
• Director : In that case, please call your Dad. (hands over his mobile phone to Joy) Please Please . .  Don't waste my time.
• Joy : (Calls his Dad.) Dad, director Sir would like to speak to you.
• Joy's Dad : (In excitement) Joy! 
• Director : Mr. Lobo, your son will not graduate this year. 
• Joy's Dad : What happened, Sir?
• Director : He has violated all the deadlines, with some unrealistic project. He is making some unrealistic Helicopter. So I recommend that, you don't need to make a reservation for the train. I am so sorry.
• Joy :  Sir, I am this close, Sir.
• Director : Is your project ready?
• Joy : Arrggh. . . 
• Director : (in a deeper voice) Is your project ready? 
• Joy : Sir, will you at least, please, take a look at my project. Please!
• Director : Submit your project and I will consider it.
• Joy : Sir, If you give me a little extension . . . 
• Director : (cutting in between) Why? Why should I give you an extension?
• Joy : After my Dad's stroke, I couldn't concentrate for $2$ months, Sir. Please?
• Director : Did you leave your food and drinks during those $2$ months?
• Joy : No.
• Director: Did you leave taking bath?
• Joy : No.
• Director : So, why did you leave studies?
• Joy : Sir, I'm very close. Please take a look at my project. Please.
• Director : Mr. Lobo, In a Sunday afternoon, my son died after falling off of the train. But, I still went to give a lecture to a college the next Monday morning. So, don't give me this non-sense.
• Joy : (Remains silent with an unwilling nod)
• Director : I can give you my sympathies but not an extension.
• Joy : (Cries!)

Finally, the story ends up with Joy committing suicide for having frustrated with his degree and life.

This is a typical story of a college boy who have struggled so much but nothing to show for. This is a really nice scene in the movie which teaches us a lot of lesson. From this we can learn not to take pressure in any form while living. There is also a very famous quote told by a very wise man.

Source : www.friendship-quotes.info/life-quotes/dying-to-live/

So my dear undergraduate friends, brothers and sisters; Never take pressure in your life. Instead, live and enjoy life as it is. Live in the present. Never dwell in the past nor in the future.

Make your choice and live free!

Make today be the day you'll always cherish in your life!

How to calculate Bending Moment

Lets consider the following beam under the action of the force $P$ at the middle of the length. The beam is supported by two fixed supports.

Now we can divide the beam into two parts based on the point where the force $P$ is applied, namely:-

1. Part A - where $x$ varies from $0$ to $\frac{L}{2}$ and
2. Part B - where $x$ varies from $\frac{L}{2}$ to $L$.

Now, let's calculate the bending moment in these two parts.

Part A

Assume a shear force $v$ acting along the direction shown.

Then on equating the forces and moments in equilibrium we have, $$v = -\frac{P}{2}$$ $$M_b = -v.x = \frac{P}{2}x$$

Part B

Similarly for this part the shear force and the bending moment obtained will be,

$$v = -\frac{P}{2}$$ $$ M_b = (L - x)\frac{P}{2}$$

Thus the bending moment is given as,

$$M_b = \frac{P}{2}x \hspace{1in} x \in (0, L)$$

$$M_b = (L - x)\frac{P}{2} \hspace{0.6in} x \in \left(\frac{L}{2}, L\right)$$

The bending moment diagram for the system will be,

Leaching of iron from Bauxite (experiment)

Aim

Leaching kinetics for removal of iron from low grade bauxite (Gibbsite) with HCl acid.

Apparatus required

1. Beakers 
2. Conical flask
3. Ultrasonic resonance spectrophotometer

Chemicals required

1. Gibbsite ore
2. $1$N HCl acid 
3. distilled water

Theory and principle of the experiment

Bauxite ores include minerals such as Gibbsite (Al$_2$O$_3$.3H$_2$O), Boehmite ($\gamma$ - Al$_2$O$_3$.3H$_2$O), Diasporite ($\alpha$ - Al$_2$O$_3$.H$_2$O) and Alumina (Al$_2$O$_3$) and higher concentration of impurities such as iron oxides and silica. Bauxite ore having high iron content is also used as starting material for the production of aluminium, once the removal of iron is achieved. Favorable bauxite ore will not have high contents of impurities such as iron oxides and silica. The removal of iron is important for two reasons. Low iron bauxite  having less than about $2\%$ Fe$_2$O$_3$ finds application in the production of refractory and removal of iron from bauxite prior to the Bayer's process reduces the disposal amount of red mud and provides high alumina content. 

In this experiment we use Hydrochloric acid to leach out the iron completely, given by the reaction 

Fe$_2$O$_3$ $+$ 6HCl $\longrightarrow$ 2FeCl$_3$ $+$ 3H$_2$O

Procedure and Observation.

1. A calculated amount of the ore was taken in the beaker.
2. $10$ ml of HCl and $90$ ml of distilled water was added in the beaker.
3. Different concentration ratios of HCl and water were added to different beakers. They were added to different beakers. They are left to react for some time.
4. This series of solutions were taken to a spectrophotometer and the absorbance of each solution was measured.

Results

HCl leached Fe at a higher concentration more effectively.

Precautions

Standard safety protocols apply for handling acids and use of spectrophotometer.

Cementation experiment (for Copper)

Aim

Cemenatation of Copper from an aqueous solution of Copper Sulfate using iron metal.

Chemicals and apparatus required for the experiment.

1. Copper Sulfate (CuSO$_4$)
2. Water (H$_2$O)
3. $1$ L beaker
4. Iron metal sheet
5. Magnetic stirrer

Principle and Theory behind the experiment

Precipitation of copper from an aqueous solution of copper sulfate (CuSO$_4$) by scrap iron is known as Cementation. The precipitated metal is usually cemented on the added metal's surface. In galvanic series, the metal having the higher emf series is more reactive  which is anodic metal. The metal present lower in the series is less reactive which is the cathodic metal.

In the galvanic series, iron is present above Copper, hence iron is more reactive comparatively. So, when iron gets dissolved in the CuSO$_4$ solution, iron substitutes the copper in the solution. Thus, Iron gets oxidized while copper is reduced. Thus copper is deposited on the iron metal sheet, hence undergoing precipitation.  

The pH (i.e. the negative log of the hydrogen ion concentration) of the solution is very important factor in cementation. H$_2$SO$_4$ is added in the solution to maintain acidic pH of the solution. It is maintained to about ~2 to prevent the hydrolysis of Fe$^{+2}$ and Fe$^{+3}$ ions. Appreciable amounts of iron is dissolved in the acid present with generation of hydrogen. Ferric iron is also formed and it's presence contributes to the dissolution of more scrap iron. 

The following reactions take place at the metal-solution interface.

Cu$^{+2}$ $+$ Fe $\longrightarrow$ Fe$^{+2}$ $+$ Cu$^{0}$

Fe $+$ 2Fe$^{+3}$ $\longrightarrow$ 3Fe$^{+2}$

The presence of trace amounts of arsenic and phosphorus as impurities in scrap iron favors the formation of highly poisonous gases Arsine (AsH$_3$) and Phosphine (PH$_3$), through the following reactions

Fe$_3$As$_2$ $+$ 6H$^+$ $\longrightarrow$ 3Fe$^{+2}$ $+$ 2AsH$_3$

Fe$_3$P$_2$ $+$ 6H$^+$ $\longrightarrow$ 3Fe$^{+2}$ $+$ 2PH$_3$

The product is usually contaminated by clay, iron oxide and metallic iron. A linear powder is obtained at high Cu$^{+2}$ ion concentration.

Results and Conclusions

The reaction involved is a zero order kinetics, i.e. the rate of reaction is constant.

Precautions to be measured while performing the experiment

1. Iron sheet should be polished properly to remove metal surface before reaction.
2. H$_2$SO$_4$ should be added before the reaction.
3. Time should be noted and proper time interval of reaction should be maintained while the iron is dipped in the solution.
4. The iron metal should be completely immersed into the copper sulfate solution.

Sneak peek into maths.

Q. What is the slope of the function $$y = x + \frac{1}{x}$$ at $x = 0.5$?

Soln. Simply differentiate $y$ with respect to $x$, to get $$\frac{dy}{dx} = 1 - \frac{1}{x^2}$$
          We also know that the slope of $y$ at $x_0$ is given by $\left(\frac{dy}{dx}\right)_{y_0}$

           $$\Rightarrow slope = \left(\frac{dy}{dx}\right)_{x_0 = 0.5}$$ $$\Rightarrow \left(\frac{dy}{dx}\right)_{x_0 = 0.5} = 1 - \frac{1}{0.5^2}$$ $$\Rightarrow slope = 1 - \frac{1}{0.25}$$ $$\Rightarrow slope = -3$$

          Thus the required slope of the function $y = x + \frac{1}{x}$ at $x = 0.5$ is $-3$.

Are you afraid of the dark?

It all starts when the sun enters the horizon in the west. The magic begins when the magnificent celestial objects becomes visible in the night's sky. Some nights are darker than others while some others are unbelievably bright and beautiful with a lot of wonders waiting to be seen. If you have ever seen a Nebula, then you might actually wonder about the creator and his imaginations and skills of craft and art. The beauty is beyond human proportions, it makes you think if you could ever do something as magnificent as nature itself.

This is a portion of the Carina Nebula, located 7500 Light
Years away in the southern constellation of Carina. This
composite image was made from filters that isolate emission
from iron, magnesium, oxygen, hydrogen and sulphur.
Source: hubblesite.org

Of course, the nature is great and it has been the source of inspiration for all major scientific discoveries and inventions. Starting from Galileo, Newton to Einstein, you name it. But do you know that the night sky and the celestial bodies are not just an inspirational source for beauty and science but astrology too. I have no idea how Astrologers define someone's life paths and tracks them using the stars in the night sky. But sometimes they are really convincing and some intelligent astrologer makes a well informed choice of what to deliver as advice for your future and they intimidate you with scientific proofs which are beyond a layman's reach. 

The night sky may also appear fearsome sometime, depending on the moon's cycle. Let's consider a new moon night and that the dark clouds are hovering some miles over your head. Now imagine the more likely situation of lightening striking down the dark sky and lights up the night for just a second or two. You may now be getting a picture of how it would appear to you seconds after the lightening strike. The fading white light, which just helped you decide where to go. It is really a terrifying picture to realize how dark it is just after the strike ends. There could be a Lion standing just behind you and you don't even realize it. You are now afraid of the lightening now, because you don't want to know what's out there and since every time it shines with the strike, you'll keep on doubting yourself more on how dark the night actually is. Let's make it even worse, consider speedy wind blowing in the neighborhood. Crackling, rustling and sounds of breaking stuffs due to collision due to the wind keeps bugging you. 

A dark stormy night.
Source: www.bhslaughter.com

You keep wanting everything to end. The only thing that you can think about at the moment is how badly you wanna go home or how badly you want to see the sun rise. Here's the best part about being in the dark all alone. You not only get to feel life at it's fullest but you are taken to great lengths of patience and being in the dark alone will teach you how to control your mind in stressful situations. 

The reason you feel life at it's fullest when you are in the dark is the fact that adrenaline levels in your body rises due to the situation being "Fight or Flight". Your heart keeps beating faster and it accelerates even more when the lightening flashes it's gates open to give you a hint of heaven's brilliance. This situation brings a thought in your mind and start making you doubt if you will make it through the night. It is when this thought strikes your mind, you want to make the best use of your little time left. Well secondly, you start developing stress and thoughts of whiling the night at the spot where you are stuck due to the rain, let it be an abandoned mansion or a creepy hut in the forest, or whatever creepy that comes in your mind right now. Now this stress is not going away until the dark of the night subsides in the light of the sun. But you have to bear it anyway, and now compare this with the stress you have while standing in a long queue. Yeah, now you got the picture of how you can learn to control your mind by being out in the dark, alone.

I would like to summarize this by the simple approximated inverse relation between $0$ and $\infty$, i.e. $$ \infty = \frac{1}{0}$$ 

You can see the difference between the values of these two numbers; while one of it is nothing, the other is everything. You just need to choose which side of the fraction you want to be.

Sneak peek into the Integrals of Complex functions

An integration of a function $f(z)$ over $z$ is defined as, 

$$I = \int^{b}_{a} f(z)dz$$

But we can convert this integral into a summation and write,

$$I = \sum^{n}_{k=1} f(\xi_k)(z_k - z_{k-1})$$

where, $z_0 = a$ and $z_n = b$. So the above expression becomes,

$$ I = \sum^{n}_{k=1}f(\xi_k)\Delta z_k$$

Now, let's talk about complex functions. So let's define a complex function $f(z)$, such that

$$f(z) \equiv u(x,y) + i v(x,y)$$

where, $z = x + i y$, So the equation follows, 

$$f(z)dz  = (u + i v)(dx + i dy)$$

$$f(z)dz = (udx - vdy) + i (vdx + udy)$$

Now, let's substitute some terms to simplify the equation. The following substitutions are made,

$\overrightarrow{A}(\overrightarrow{r}) = (udx - vdy) $ and $\overrightarrow{B}(\overrightarrow{r}) =  (vdx + udy)$

Now, having this equations and substitution in hand, we now integrate the above relation,

$$\int^{b}_{a}f(z)dz = \int^{b}_{a}(udx - vdy) + i \int^{b}_{a}(vdx + udy)$$

$$\int^{b}_{a}f(z)dz = \int^{b}_{a}\overrightarrow{A}(\overrightarrow{r}) + i \int^{b}_{a}\overrightarrow{B}(\overrightarrow{r})$$

Now, that being defined let's look into some theorems.

Jordan Curve Theorem Every non-intersecting curve divides the complex plane into an interior and an exterior. 

[Note: Interior and Exterior are just the concept of being inside or outside of a region defined by the curve or the function defined.]

Cauchy-Gowsal Theorem If $f(z)$ is analytic in a region $R$ and if it's boundary is $C$, then 

$$\oint_C f(z) dz = 0$$

Monera's Theorem If $f(z)$ is continuous in $R$ and $$\oint_C f(z) dz = 0$$ for every closed curve in $R$, then $f(z)$ is analytic in $R$.

Ok, so that's a little sneak peek into the basic of integrating a complex function. So, stay tune for more exciting materials.

4 obvious reasons why you should know your materials and the Science behind it.

(I guess the last one is not so obvious)         

If the first thing that comes to your mind when I say "Material Science" is the material that your T-shirt is made of, then you are probably in the right track (depending on how you are thinking about it). Let me tell you seven reasons why you should know 'How materials work?', 'What are their properties?' and 'What's the science behind it?'.

         1) Everything you see in your room is made in association with Materials Science and Engineering somehow, starting from the computer you are using right now to the pen you used to check the calendar or the wires carrying current or the bulbs on the wall or your smartphone and you name it... For instance, the wire carrying current in your room was once upon a time locked down deep in the earth, when one day a gang of engineers came over to dig it up and process it in many ways before it ended up in your room.

         2) You can save your life. If you are thinking how, I can give you a lot of examples on people getting into misery because they were not aware of the property of the materials present in their stuff or car or door or the house or the metal they used for constructing the bridge or ship, etc. I can assure you that knowing the science behind how the wire carrying current in your room works, you might actually end up preventing a fire. (PS. I have added some links to show you the harsh truth. If you are not well informed about the material you are using, you may also land up in an accident). Also, knowing which rubber was used to make the tires of your automobile, might mean the difference suffering a fatal accident or avoiding one.

a) Engineer Suffers Scary Electrical Accident.
b) Crane mechanics failure

***Long story short, a defected screw holds the key to the gate upstairs.***

         3) You'll be one step ahead in the game. Knowing about materials and their properties give you an extra edge on many day to day things starting from the question of whether you get bullied at school or you get a date at the end of the day or whether you ride the most economical yet best functioning car among your peers, etc. Crazy right! I know, and I thought about the same thing too. Do you know that if you wear a scent carrying the aroma of orange fruit mixed with lavender flower scent than you will appear younger than you actually. You can see where I'm going with this.

         4) You can become invisible. Have you ever wondered what it would be like if you were given the power to become invisible for a day or a week. Well, you won't be needing any super powers to do that from now on. Guess what, science and engineering are just one step away from perfecting the technology to make your fantasy come alive. You can follow the link below to see the news update on research related to this. The Invisibility cloak

So, what are you waiting for? Get going and start learning the properties of your materials.

A quick quiz on Maths.

If you are given that $x^{x^{x^{...}}} = 2$, then find the value of x?

Soln. Since, $x^{x^{x^{...}}} = 2$, (an infinite exponential in x)
we can write, $x^{2} = 2$
i.e. $x = \sqrt{2}$ (Q.E.D.)