sciencecapsules

      VISIT: 'MATH IN ACTION' SITE   Consider an egg. You can balance it on one of its ends. It won't last long. The slightest disturbance will topple it. The two tips of the egg are known as unstable equilibrium points. "Equilibrium" because you can balance the egg on them and 'unstable' because the egg will fall over at the slightest nudge. The points around the egg's body on which it balances robustly are called 'stable equilibrium' points.      Now consider a flat square (a cardboard piece). It has four 'stable equilibrium' points at the center of the sides. You can balance that square on these points. It also has four unstable equilibrium points at its corners. Similarly, a triangle has three stable and unstable equilibrium points. All ellipse has two stable equilibrium points in the centers of its long side and two unstable ones at the ends.     Can you find a two or three-dimensional shape that is convex(that is like full coconut

      VISIT: 'MATH IN ACTION' SITE   Put a coin in a plastic tray. Consider both as a two-dimensional object. The coin can move within the tray but cannot escape from it. Suppose you lift the coin from the tray, it escapes from the boundaries of the tray. That is, you have moved the coin into the third dimension, that is why it escaped.      Now put a marble in a box. Both are 3D objects. You cannot move the marble outside the box without damaging the walls of the box. But, if you move the marble into the 4th dimension, the container walls cannot stop its movement(as in the coin and tray- case) and the marble will easily escape from the containment of the box. Similarly, one can escape from the jail, if be could move into the fourth dimension.      Einstein said, that besides three dimensions(length, breadth, depth) the time can be considered as the fourth dimension. We are able to move back and forth along length or depth. Hence, it may be possible to move back and forth in th

     John lives in a warm country. He has just 30 square meters of land. He can plant cocoa and pineapples. How much of pineapple and cocoa he has to plant in his plot to get maximum profit? This is a classical problem. It can be solved with graphs.     Let x stand for the amount of cocoa and y for the amount of pineapple. When john harvests the crops, he will get a unit return of 3 dollars on the cocoa and of 2 dollars on the pineapples. His total return p will be  p = 3x+2y       ----(1)     The profit p, we want to maximize. Planting comes with a cost. The unit cost of cocoa seed is 2 dollars and of pineapple, the seed is 1 dollar. Also, 4 dollars is required to cultivate both the crops in his land.     Hence, the total cost c of growing the two crops is       c = 2x+y+4   -----(2)      John has only 50 dollars in the bank. That is the first constraint. We write it mathematically as     c= 2x+y+4 < 50    ---(3)   constraint I     The size of the field also provides the second con

     ADVT:  SELL ON AMAZON      Think of Usain bolt. He created a world record in 100m dash. Could he repeat the record performance the next day after winning gold? Probably, he could not. Because many factors come into play on that particular day like his mood, the condition of the track, the passion of the crowd, etc. Things will not fall into paces again.      After an extreme performance, all of us may fall back to our average or around the average. That is why, a student who excelled in school final year, may not make it at the college level.     We have our own average values of BP, sugar, temperature, speed of writing, speed of running, response time, etc. When running, the BP value may increase, after a while, it will fall back to 'our average'. This is called"regression to mean" mathematically.      The above is the probability curve drawn between performance and its quality. Mostly, our performance cluster around means. Sometimes, we perform very poorly or e

      ADVT:  SELL ON AMAZON       A car is running. It can hit and damage an object. It can do some work because it posses energy. A body gets energy by virtue of its motion or its dynamics. We know, it is called kinetic energy. The formula is 1/2 mv^2, m is the mass of the body, or we can say m is the inherent property of the body. v is the velocity of the body. or we can say v represents the dynamics of the body. Almost all the kinetic energy formula in physics follows this pattern.      Consider an oscillating spring. It is KE is 1/2kx^2. Here k is constant (property) of the spring. It depends on the material, thickness, and length of the spring. x is the maximum displacement of the spring from its rest position.  We say x represents the main dynamics of the spring.     Consider a rotating body. Its K.E is 1/2IW^2. I is the moment of inertia or property of the rotating body, w its angular velocity (dynamics).     The energy stored in an inductor (coil of wire) is 1/2LI^2. L is the

      In a medical emergency, you always want to rush to the nearest hospital. A city has a number of hospitals. Suppose we have a map showing each hospital catchment area (for anyone in that region, that hospital is closer than any other) that will be highly helpful. How do you make that map?     It can be easily done. But doing it manually is difficult. With a computer algorithm, we can easily create a map.      Start with two hospitals at points A and B on the city map. Draw a line that connects them. Find the midpoint of that line and then draw a line that passes through that midpoint and is perpendicular to the line from A to B. That line divides the city into two regions. One region having A contains all the points closer to A than to B. The other contains all the points closer to B than to A.   ADVT:  TODAY'S DEALS IN AMAZON Now, look at the third hospital, at point C. Repeat the above process to the hospitals at A and C, you get a second line of separation. Repeat for B an

      Is it possible to create a formula to locate a serial killer? Yes, one man has done it. Mathematics looks for a pattern. Serial killers leave some pattern. From that pattern, we can make a formula.      Serial killers strike some distance away from their residence, but not too far away.   ADVT:  TODAY'S DEALS IN AMAZON     We select a large area where the killer is supposed to be hiding and map it. We superimpose the map on a square grid. Each square can be located using x, y coordinates as shown in the figure. We select a particular square(shaded one) and we will find the probability that "the killer will be residing there" using a formula. The formula is              The formula seems more threatening than a serial killer. Anyway, we will go step by step and understand the formula. Pij- - represents the probability finding the killer at the location i, j (x,y coordinates). The answer ranges from 0 to 1.  higher the probability,  higher the chance of finding him a

        We now know that masks have an effect on slowing the spread of COVID-19.  Masks protect the wearer and the people around them, a two -way protection.  We are going to explain in this article that widespread mask-wearing is a powerful way to extinguish an epidemic. (with some little math).      When a person exhales they spray out saliva particles mixed with viral particles. This virus-laden saliva spray is the main way that COVID-19 spreads.      When a contagious person breathes, they spray out roughly a thousand viral particles every minute. When they talk, they spray out roughly ten thousand viral particles every minute. When they cough, they spray out roughly a hundred thousand viral particles.  And when they sneeze, they spray out roughly a million viral particles. The more viral particles travel from person to person, the higher the chance of infection. MASKS ARE TWICE EFFECTIVE      Let us assume that a contagious person wears a 50 percent effective mask. That is, wearin

                   ADVT:  REFURBISHED MOBILES IN AMAZON              222,222,222,222,222,222^2 -222,222,222,222,222,222,221^2 =?      A calculator or a computer cannot solve this problem. But we can.     We know  1^2-0^2 =1                       2^2-1^2=3                       3^2-2^2=5                       4^2-3^2=7      There seems to be a pattern here. To find the difference between two adjacent squares, you just have to add the unsquared numbers together. We are not sure that this pattern goes on forever.      Let us draw some pictures.        each square is made from the previous one.  To go from 2^2 to 3^2, add 2 dots to the bottom and 3 dots to the side. The difference between 3^2 and 2^2 is 3+2=5.  It is clear from the pictures.     Similarly, to go from 3^2 to 4^2, add 3 dots to one side and 4 dots to another side.  The difference here is 3+4=7. The picture shows the same. Now we understand that the pattern will go on forever.     This is proof by induction. In the same lines

                          ADVT:  TODAY'S DEALS IN AMAZON     Can you believe that a handful of gas molecules carry and transports us?  yes, that is true.      We know the air is filled at high pressure in vehicle tires.  High pressure means; a very large number of molecules bang the walls of the Tyre with great force.  The force given by these molecules bear our weight; the vehicles' weight and carry us.      In the engine, a set of cylinders and pistons arrangement is present.  Inside the cylinder, the mixture of air (oxygen) and petrol vapor is ignited by an electrical spark.  It then explodes.  Explosion means Sudden expansion of gases; gas molecules gain energy; move faster.  These molecules push the piston.  A visible mechanical movement is created by invisible molecules.      These movements of pistons are converted into rotational motion which will then power and drive the wheels.  The rotations of wheels transport us.  Hence, just a collection of molecules

                         ADVT:  SMARTWATCHES IN AMAZON       Today, "going viral" is the phrase that is frequently used in social media.  How the gossip, virus, or video clip spreads rapidly.  Let us look in detail.     GOSSIP     Imagine that you hear a bit of a hot scandal. You pass it on to just a 'couple' of your friends with the condition, "Do not tell anybody". But, naturally they cannot keep it to themselves. They reveal the secret to a couple of confidants with the same condition. Anyway, the trusted friends will do the same and so it goes on, each disclosure leading on to 'just a couple' of friends.     Let us assume that the disclosure takes place every half-an-hour. How many people 'the secret' will reach in 12 hours that is. in  24 half hours.     Since it doubles every half -hour, we can write.    The total number of people who are in the know after 12 hours is 1+2+4+8 ........2^24 = 33,554,431.      Yes, so many people. In 12 ho

                             ADVT:   REFURBISHED MOBILES IN AMAZON      When a bell is struck, it vibrates with a certain frequency (Number of vibrations per second).  The vibrating bell pushes the air molecules around it.  Hence air pressure variations are created with the same frequency.  These pressure variations travel as a sound wave and reach our ears.  The ears sense it. The brain interprets as a certain sound.     Frequency is the main feature of sound and music. It is measured in Hertz-Hz. One Hertz is one vibration or one cycle per second.     How the stringed musical instruments like the violin, the guitar produces music-sound of various frequencies?  Let us find out.         Frequency of vibrating string = 1/2*length*(root of (tension/linear density)       According to the formula, one can change the frequency in three ways. 1. tightening the string -tension changes 2. Using a thicker string -linear density 3. using fingers on frets - length changes.     Mainly frequency is

                                ADVT:   REFURBISHED MOBILES IN AMAZON        In a stadium, 50000 strong spectators are watching a cricket game. There must be at least two people who were born on the same day of the same year. Are we certain? Can we prove it? Yes, there is a way.      Let us assume that everyone in the crowd is aged between 0 and 80. In each year there are 365 days, so, in 80 years there are no more than 29200 different birth dates. But the crowd is bigger than 29200, then there is certain to be a duplicate. We can be 100% sure.     This is called "pigeon hole proof". That is, there are 29200 pigeon holes (dates), but there are 50000 pigeons(people). You can not put all the pigeons in a given number of holes. We have to reuse some of the holes. This proves the statement, "there must be at least two people who were born on the same day of the same year".     1. How many people do you need before you can be certain that two of them have the same number

       We have over 300 different proofs for the Pythagoras theorem. Is proving easy? Can you prove anything?     Let us take one example. Consider the following map. ADVT:  REFURBISHED MOBILES IN AMAZON     How many colors, (minimum) you need to color this map. No adjacent state should have the same color. You can say with confidence "At least 4 different colors are required". Are 4 colors enough to color any map?      Is there any map where five colors will be needed. Nobody had the answer until 1976. But four crayons seemed to be sufficient for any map. Nobody felt the need for the fifth color. At the same time, no one was able to prove that only 4 colors are a minimum requirement with certainty.     It is thrown as a challenge to the math community. They were forced to answer the question "what is a proof?     After the computers came into being, In 1976 with the aid of hundreds of computer processing time, the mathematician Appel and Haken firmly turned the four-col

  Consider the y=x^2 curve ADVT:   REFURBISHED MOBILES IN AMAZON     What is the area under the curve? If it is a rectangle or circle, we can easily calculate the area. But it is an irregular shape.     Let us do it in the following way. Let us draw small rectangles under the curves and fill the entire curve with rectangles. Let us now calculate the area of each rectangle and total all the areas.  That will give the area under the curve. The smaller the width of the rectangles, the more accurate the result will be. We can do this process easier by writing a small computer program. The width of the rectangle is taken as 0.01 or even 0.000001. We can get accurate results.     What they have done before the arrival of computers. They invented calculus. The 'integration' operation in calculus gives an accurate answer.     The equation of the curve is y=x^2. The integration of x^2 is  2x^3/3. Suppose, you want to find the area between x=0 to x=3. Apply these limits to the formula. A