sciencecapsules

If you go on increase the number of  sides of a polygon. the circle will be the limit.      1/2^1 + 1/2^2 + 1/2^3 + 1/2^4+........  =1                            or      Sum of all the terms of 1/2^n  = 1      When n goes from 1 to infinity, the sum of the above series approaches 'one'; but never touches 'one' or crosses 'one', however bigger the 'n' may be.  Hence we can say, the limit of the sum of the series is 1. (1+1/1)^1+ (1+1/2)^2 +(1+1/3)^3 + .....     =2.71... = e The limit of the above series is the famous Euler constant e.     The concept of limit is the corner stone of calculus.  Let us see an example. Let y= x^2 Let dx be the smallest change in x , let dy be the corresponding small change in y. I want to find dy/dx Let x be 3, let the small change be 0.1 (dx) The new x is 3.1 corresponding change in y is                            y=x^2       old x: 3        9      new x: 3.1    9.61 hence   dy = 9.61 - 9 = .61

       Today, we live in a world of digital images.  Ads, Media, movie, legal documents, CCTV or just picnic photos.  Everywhere, there are images.  We know, this images can be easily doctored and engineered by any body with the computer.  Further, when the photos are taken at different perspectives, the lengths, the angles and their relationships are distorted.  Then, how to find the truthiness of the image?  There is a mathematical way.  Let us go into detail.      Take a torch.  Switch it on.  Allow the cone of light to fall on the wall.  If you hold the torch straight, you see a circular patch of light.  If you tilt the torch slightly, you see an ellipse of light.      The circle is there in the light cone always. The circle is distorted into an ellipse, when the light patch hits the wall.  I want to prove, the ellipse is only the projection or an image of the circle.  Here we go.  Draw a diagram as given below.      Here, when the beam of light is sliced vertical t

    x = V - E + F Euler characteristic = Vertices - Edges + Faces. Draw triangle on a sheet of paper.  It has 3 vertices, 3 edges and 1 face. Hence the Euler char. = 3 -3 + 1 = 1 For a square Euler char. = 4 - 4 + 1 = 1 For a circle, imagine two points or vertices on the circle. So, Euler char. 2 - 2 + 1 = 1. The circle, square and triangle all have same Euler characteristic 'one'.  They are geometrically different, but topologically same.  That means, all the shapes can be converted into one another, if they are made of rubber sheet. " Topology is concerned with the properties of space that are preserved under continuous deformations; such as stretching, crumpling and bending, but not tearing or gluing" (Wikipedia).      Now we get the idea of topology.  Let us see one more example.     Take a 'tetrahedron' solid (prism).  It has 4 vertices, 6 edges, 4 faces.     Euler char. = 4 - 6 +4 = 2     for a cube     Euler char. = 8 - 12 +

         We know well about the circle, the distance between any point on the circle and the center is constant and it is called 'radius'.  Let us now consider an ellipse.  There are two centers (focus) in an ellipse.  The total distance between any point on the ellipse and two foci is constant.      Kepler discovered that all planets revolve around the sun in the elliptical orbits.  The comets appear to us at some point of time.  They reappear after a gap of long time(for example 76 years).  People were not able to predict the motions of comets for a long time.  Then it is found that the comet travel in long elliptical path with sun at one of the foci.  Hence the long time gap for every appearance.      There are two  famous whispering gallaries.  1. United states statuary capitol hall. 2. London's st. paul's cathedral.  They are designed in the shape of ellipse.  If a person whispers at one focus in the elliptical gallery, another person standing at the se

        First, we must know about relative velocity.  Say, you are travelling in a express train at the speed of 100km/hour.  In the next track another train is moving at the speed of 120km/hour.  Now subtract your speed (100) from 120 km/hour.  You get 20km/hour.  That means, another train is running at the speed of just 20km/hour with respect to you.  Hence it will apear to go slowly for you.      Say, rain is falling vertically at the speed of 4 kmph.  Draw a 4 cm arrow pointing downwards.  It is the vector representing rain's speed.      You are fast walking at the speed of 6 kmph towards east.  Draw 6 cm arrow from the head of rain vector representing your speed.  To find the relative velocity, we have to subtract.  Hence reverse your vector (-6cm) towards west side.  If we now add both the vectors, we will get the relative velocity.      To add, complete the triangle, the third side gives the result of addition.(triangular law  of vector addition).The length

        Water, a peculiar liquid.  It is the only liquid that can exist in all the three forms in room temperature.  That is, ice, steam, and water.            It is a life giving substance since it is one of the ingredient needed to make energy giving glucose by the life forms.  It also regulate the body temperature.  Water helps to transport nutrients, clear waste products in our body.  It goes on and on.  Yet, the science books misses to term it as a nutrient.      We have, spherical drops of water, slashing sheet of rain, interesting shapes of ice crystals, ice mountain called ice bergs, flood of water which sweeps away everything, giant waves called Tsunami, cloud forms pregnant with rain drops, a placid lake, the deep ocean, water dust raising due to water fall, high energy steam-so many avatars of water.     Men has made very high pressure jet of water which cuts even a metal.  Also man has built ice houses and castles.        Highest amount of water is required

        A cricket game in progress.  The batting team has scored 60 runs so far in 10 overs.  Dividing 60 by 10 we get 6.  6 is the average number of runs per over taken so far.  . That is , the team seems to have scored 6 runs in each over  (Practically, they might have scored 12 runs in a over and zero runs in another over).  'Average runs' is also called run rate.  If the rain bring the game to a halt, run rate is used to decide the winner.  Remember, as the game processes, the run rate changes.      Toss a coin 5 times  .  4 times you may get the head and 1 time tail may appear.  So the mean is 80% for head and 20% for tail.  Toss the coin 10 times.  6 times  may be head.  4 times may be tail.  Now the average is 60% and 40%.      Throw the coin for 1000 times.  (A computer simulation helps).  You may get the head 502 times and tail 498 times.  The average of getting head or tail per throw approaches 50% or 1/2.  When the volume of samples or trials go on increasin

            Head is a kind of spherical surface covered with hair.  Hair grows in all the directions.  When you comb the hair, you are trying to flatten the hairs smoothly all over the head.  In this process, there will be at least one point in the head which cannot be covered with hairs or at least one tuft will be there.  Or the hairs will stand straight at one point.      In short,"when you comb the hairs, there will be one null point."  This is called "hairy ball theorem".(HBT)      Wind flowing over the earth is similar to the hairs flowing over the head.  Suppose the wind flows from East to west on the globe, there will be two null points- North pole and South pole.      In a cyclone, the air whirls around.  At the centre of it, no air flows- a null point called eye of the cyclone.  Logically, the wind cannot cross over the 'eye'. If it does, it will disturb the whirling wind.  So there is a calm at the centre of the storm.      This top

         The space mission called "Mariner 9" had gone to mars, took close photographs and were sent to the earth without errors.  The photographs traveled millions of kilometers without contamination.      They have used what is called 'Hadamard code or Matrix'.  Let us first see the construction of that matrix.              A = 1  1                     1  0      In the above matrix, all the three entries are the same.  The bottom right corner entry is just reversed.  Hence, between the first and the second row, only one bit is different.  We will extend the idea to 4 X 4 matrix.       B  = 1  1  1  1               1  0  1  0               1  1  0  0               1  0  0   1      In the above matrix, the matrix A is repeated in all the three corners.  Again, the matrix A is reversed in the right bottom corner.  Here, between any two rows, two bits are different. For example: II row  :  1010 IV row:   1001 Here last 2 bits are different.  This i

      A body can move in three directions, x,y,and z.  Also the body  can rotate around three axes (x,y,z).  Hence we say, the body has "six degrees of freedom".  A helicopter or a bird has six degrees of freedom.  In aerodynamics language, the three rotations are called, pitch, roll and yaw.      Take a plank of wood.  In this three dimensional world, it has six degrees of freedom.  Fix the plank as a door in a building.  It can now do only one rotational motion about the joint.  The six degrees of freedom has come down to one.  We understand that the joints restrict the freedom.      There is a formula to calculate the degrees of freedom. M = 6*n-SUM (6-fi)                     for i = 1 to k M - Degrees of freedom n - Number of moving parts. k - number of joints. fi - number of degrees of freedom of each joint.     That is,  (6-fi) should be found for each joint and summed up.  Then the total sum is subtracted from 6n to get no. of degrees of freedom.  

         Physics says:  If a body is moved to some distance by a force, work is said to done.  The capacity to do work is energy.  In short, the amount of work that can be done gives the measure of energy.     A body at a height can fall down (does work).  So it said to posses "potential energy".      A fast moving body can hit another body and do some work.  So it has "kinetic energy".            In a chemical battery, oxidation takes place in the negative end and the reduction takes place at the positive end.  Hence the electrons are pulled from the negative to positive through a wire and the electric current is established.  Here chemical energy turns into "electrical energy".     If the electric current flows through a coil of wire, magnetic field is established.  This magnetic field can repel and attract another magnet.  So motions will be created.  Here electrical energy turns into mechanical energy.          In a chemical reaction, whe

          A jet of water shooting from a hose pipe will follow a parabolic path.  What is the so special about parabola.    Y= x^2 Draw a graph for the above equation.  It will result in a parabola.  This parabola is also called unit parabola.  Any equation involving square will yield a parabola. Example:  Y = 2x^2 +3x+3 (also called quadratic equation)    X= 2 and -2, both  satisfies the equation 4 = X^2.  Parabolic equations always have two solutions.     Any motion taking place freely under gravity follows parabolic path. Examples:   An object dropped from a moving train,   A bomb dropped from flying plane,  A ball kicked upwards.      If a beam of light rays fall on the parabolic shaped mirror, they will be reflected and brought to focus on a point.  This fact is made use of in Dish Antenna, Telescope mirrors, etc.      Inverted parabola shape is used in the construction of buildings and bridges.  Because the shape is able to bear more weight.      A plane

       In the universe, there are two entities, Life and non-Life.  In the physical world, there are two entities, particles and waves. 1. Electron, proton are particles.  You and me; table and chair; earth and water; all made of particles.      Sound is the pressure waves; light is the electromagnetic wave.  Water wave is the mechanical wave. 2. If force is exerted, particles transport themselves in a straight line.  A wave is a disturbance or the energy travelling through a medium or in empty space. 3. A particle cannot deviate from its path without a force.  But waves from a source can spread in spherical form in all directions. 4. The energy of the particle is determined by the mass and speed.   But the waves energy only depends on its frequency.     Energy = Planck constant *frequency 5. The waves can bend over the edges and obstacles and also it can spread through a hole.  But particles cannot do so. 6. Waves can be concentrated  and made like a solid object.  Examp

        Imagine a positive charge and a negative charge kept at some distance apart.  There is a force of attraction between them.  An invisible electrical line of force connects them.  Hold one charge and oscillate it up and down ( in imagination).  The electrical line oscillates like a rope.  Or we can say, electric field oscillates.  The oscillating electric field creates an oscillating magnetic field(Maxwell's law).  That is, electromagnetic waves are produced by the oscillation of charges.       Take two metal rods. Send alternating current through them. Opposite charges oscillate in the rods. Loops of waves are created and moves away.        This process is like the production of  soap bubbles.  For each blow of air, one bubble is created and it flies off.  For each oscillation of the charges, one wave is created and it moves away. Hence, the oscillating charges create the electromagnetic wave which propagates itself in the space.  If the charges vibrates 1000 t

                              NaOH + HCl = NaCl+ H 2 O     Here acid and base reacts to yield salt and water.  It is ordinary 'High school"  reaction.  Let us understand the dynamics behind it.      Why should sodium chloride salt form?      ( Note:  All the objects want to occupy lowest energy level.  That is why all the object fall and reach the ground, if opportunity is available.)     Sodium has a total of 11 electrons.  2 electrons in the first orbit, 8 electrons in the second orbit and one in the last shell.  The sodium is always ready to loose the last one negative electron for the energy advantage and become a positive ion.     Chlorine has 17 electrons.  In the last shell it has 7 electrons.  It will happily accommodate one more electron in the last layer to complete it's shell.  By acquiring one negative electron, chlorine will become a negative ion.      Hence Na will give one electron and the Cl will accept it.  Now the positive sodium and

        We are going to see some important phenomena which utilizes only  THREE entities.      Only using three particles namely proton, neutron and electron, all atoms, molecules, cells and matter are built.  Protons and neutrons are responsible for mass.  Proton and electron carry positive and negative charges.  Electron is responsible for electromagnetic property of the element and also interact with other atoms and molecules.  Hence these three particles are the required minimum to build the entire world .  Hence 20 particles does not exist in the atom.     Only four chemicals ( cytosine  [C],  guanine  [G],  adenine  [A] or  thymine  [T])  are used to write the entire bio-information of a species in the DNA molecules.  Here only three letter (chemicals) words are used to indicate the amino acids which are building blocks of life.   We live in three dimensional world.and we are three dimensional being.  Even we find it difficult to navigate in the two dimensional