sciencecapsules

  If a dog bites man, it is not a news.  If a man bites dog, it is a news.  Anything surprise is the news and an information.  If an event certainly is not going to occur then its probability of happening is 0 and it is not an information.  If an event is certainly going to happen then its probability is 1 and it is also not an information.      An event which is having probability between 0 and 1 carries some information.  For example:  In tossing of a coin getting a head or tail is having a 50% chance or 0.5 probability.  Out come of the toss is an information.  In the case of rolling of dice, each outcome has 1/6 chance or 0.167 probability.  Certainly the outcome of a  dice is  the surprise and an information.      Shannon gave a formula to calculate information content in an event based on probability.      I want to calculate the information content of coin toss and I also want to represent it in bits-binary digit format.  Here  probability p = 0.5 and to repr

      Consider this equation              p(n+1)=p(n)*r*(1-p(n))     This equation is taken from population modeling.  Here p represents population and r the rate of growth.  We need not go into the origin of the equation.  Here, the important point;  p(n) is calculated from the equation and the p(n) is fed back into the equation to get the next p(n).  The output to the input feed back calculation is repeated many times(iterated).  In computer language, the equation can also be written as                  p = p*r*(1-p)   Let initial p be 0.25 and let r be fixed at 1.5.  Let us calculate p repeatedly in a spreadsheet and draw a graph between p and number of iterations.  The screen shot is shown below.  The p initially increases and then stabilizes at 0.333.      Now let us change only r to 2.5.  As you can see in the next figure, the curve is slightly disturbed before settling down at 0.6.      When r = 2.9, the curve oscillates for a long time(more iterations)

                                                          N = n*e^(r*t)    The equation yields the natural growth or decay of any quantity.  Here, the initial amount of quantity   n becomes N after time lapse of  t at the rate of growth r .  When we apply the equation for decay, the r becomes negative. we know Euler's constant e =2.71...        The applet given below draws the curve for the equation, you can play with the rate r  and see it's effect on the curve. The initial quantity n = 1.  The curve gives the quantity N (y) as the time t (x) progresses.  The curve rapidly grows or decays and then it becomes slow.  But the curve tends to go to infinity.  To view the curve properly adjust the graph sheet in the applet.                                            1.    When a capacitor is connected to a battery, it charges up.  When disconnected, it discharges.  The discharging voltage is given by the following relationship.                                  

      Everybody knows Pythagoras theorem. The figure 1 states it clearly. If sides are equal to unity, what will happen to hypotenuse c^2=1^2+1^2 = 1+1 = 2 c = root of 2  = 1.414... (figure 2) Hypotenuse turns into root of 2 , a never ending irrational number. Suppose hypotenuse is 1, what will be the Pythagoras relationship? According to trigonometry, adjacent side is 1*cos x and opposite side is 1*sin x.  Applying Pythagoras theorem we get,  (1*cos x)^2+(1*sin x.)^2 = 1^2 (cosx)^2+(sinx)^2=1 So this is the trigonometric equivalent of Pythagoras equation.(figure 3) We can extend the Pythagoras principle to higher dimensions.  For example, we can calculate the diagonal inside the 3 D cube as shown in the figure 4. We can link number of triangles and apply the Pythagoras theorem as shown in the figure 5 .   Pythagoras equation is satisfied by some whole numbers or integers as given below. x^2+y^2 = z^2 3 ^2+ 4 ^2 = 5 ^2 5 ^2+ 12 ^2 = 13 ^2 7 ^2+ 2

   Newton law says, every object in the universe attract every other object by gravitational force.  Scientist Cavendish proved that two lead balls attract each other using torsion balance experiment.  He also measured the universal gravitational constant G using that experiment.  The gravitational force is the weakest one but a long range force.       Gravitational force is responsible for the formation of galaxies, stars and planets.  It keeps planets revolving around the sun and satellites around the planets.      On launching, a rocket must reach the speed of 11.2 km/sec.  Then only it can pierce out of earth's gravitational field(escape velocity).  Even if the speed is slightly less, it will fall back to the earth due to gravity.      Once out of the gravitational field, everything start to float.  Astronaut can sleep in any position since there is no gravity and consequently there is no up and down .  But they have to strap themselves to the walls of the space

  ELECTRONS AT WORK  Electron was discovered by J.J. Thomson  more than 100 years ago.  It is said to carry negative charge and negligible mass.  It is present in all the elements      When electron flows in a conductor, it is electric current.  We know, nothing serves like electricity.       Around the moving electrons magnetic field is formed and it gives rise to electromagnet.   When the electron oscillates electromagnetic waves like radio waves are radiated.  It is used for communication purpose.     While the electrons are moving in a conductor, they collide with the atoms and produce heat.  Many heating devices are made using this principle.  In atoms, when electron jumps from higher orbit to lower orbit, it loses energy in the form of light waves, X-rays etc,.   When the electrons moves in the magnetic field  at an angle, it experiences a force called Lorentz force.  This force is responsible for the working of motors.       The electrons in the outer

  Take a piece of salt harvested from ocean.  It  basically has cubical shape.  It is made up of sodium chloride molecules(NaCl).  The sodium and chlorine atoms are joined by ionic bond.  They are alternately arranged in rows, into planes and take the shape of cube.  We know sodium and chlorine are atoms that are made up of protons, neutrons and electrons.  Now they say, protons and neutrons are built from the particles called quarks.        Hence all non-life matters are made from fundamental particles like electron, proton etc,.  The particles constitute themselves into atoms.  Atoms are binded together to form molecules.  Molecules arranged themselves into some shapes or go shapeless like liquid and gas.        Life starts from non-life.  The atoms of some elements particularly carbon and hydrogen form complicated molecules.  The different set of molecules make plant or animal cells.  Collections of cells make tissues.  Then organs are constructed from cells and tissues

   In Cartesian coordinates, we marked two points P and P1.  The coordinates of P is x, y and that of P1 is x1, y1.  Both are connected to the origin by straight lines.  They make angles A and B with the x-axis respectively.  The length of the lines connecting both the points to the origin is equal and it is   ' r'.  That means, P1 is rotation of the point P by an angle B .   From the figure we can write x = r*cosA y = r*sinA similarly, x1 = r*cos (A+B) y1 = r*sin (A+B) using trigonometry formula, we can expand the equations. x1 = r*(cosAcosB - sinAsinB) y1 = r*(sinAcosB+cosAsinB) If we substitute x=r*cosA and y= r*sinA in the above equations we get,  x1 = xcosB -ysinB y1 = xsinB+ycosB writing in matrix form,   So R(B) is a matrix rotation operator .  If we multiply coordinates of any point by the matrix operator R(B), the point will be rotated by an angle B and we will get the new coordinates .  The image shown in the figure is letter T.

        What happens when just a cup of water is boiled.  Let us have a some deep look.      On the surface of water, atmospheric air gives downward pressure.  Water is heated up.  The water molecules slowly loose themselves from their bonds and gain momentum  using heat energy and moves up.  The loose molecules accumulates as water vapour above the surface of water.  Then the vapour pressure slowly builds up and acts upward against air pressure.  When the vapour pressure equals or slightly greater than atmospheric pressure, the water vapour pushes itself into the atmosphere and evaporates.  Next, the loose molecules(gas form) inside the bulk of water bubbles up rapidly and evaporates easily.   This moment is what we call boiling .  The temperature at this moment is called boiling point.      so boiling occurs when vapour pressure is equal to atmospheric pressure. Pressure cooking :      In pressure cooker, the steam (vapour) is trapped for sometime allowing the pressure to bui

  Definition:  "When a particle is subjected to two sine wave motion or two oscillatory motion at right angles, the particle describes lissajous figures".      We know sine wave motion and circular motion is basically same.  Hence we draw two circles A and B perpendicular to each other.  The circle B rotates twice faster than circle A.  That is, frequency of circle B is two times than that of A.        A particle at the intersection of two circles is subjected to two sine wave motion   A and B at 90 degree simultaneously.  The particle will describe figures depending on the frequency and phase of A and B .  In our case, the ratio of frequency is  1:2 and the two waves are in phase.        To draw lissajous figures :  A moving point in both the circles are chosen.   Here we should remember; during the time taken by the circle A to complete one rotation, circle B completes two.  Hence the points are marked on the circles according to their speed.  Then straight lines

     Let us take 60 samples of 10 students each(total 600 students).   For each sample, we calculate a average mark in a particular subject.  We get the following reports.          MARKS RANGE                                        NO.AVERAGES IN THE RANGE                 30.0--44.0                                                                        5         44.1--58.0                                                                        13         58.1--72.0                                                                        23         72.1--86.0                                                                        12         86.1--100                                                                          6                                                                                                 ----------         maximum  marks:150                                                        60        total no. averages or samples                              

  Imagine a square of unit side. Inscribe a quarter circle within the square.  The area of the square is  1 unit. The area of the quarter circle is pi/4 unit.      Let us randomly select N points within the square.  Out of N points, let C points fall within the circle.  Logically, the area of the square is proportional to N points and the area of the circle is proportional to C points .                                                   Circle:  pi/4  proportional  C                                                   Square: 1 proportional  N.                                    Dividing,                                                    ( pi/4)/1= C/N                                                           pi = 4*C/N   So we can estimate pi in this method.  As the number of random points increases, the accuracy of the pi increases.  A computer program is highly suitable to implement this technique. This method is named monte carlo technique because monte carlo is fa

      When a body vibrates- for example voice box in the throat- sound waves are produced.  They travel as a pressure wave in the air medium. In atoms, when an electron accelerates, it looses energy in the form of light waves.  They travel as electromagnetic wave without the help of the medium.        Since sound is produced by big objects, their wavelength is also big. (From 17 meter to 17 mm).  Since light is atomic phenomenon, their wavelength is smaller.  (From 400 nanometer to 700 nanometer).      We know , the speed of sound and light is vastly different(330 m/s and 3*10^8 m/s).  Hence we see the lightning first and hear thunder later.      Frequency of sound waves is responsible for the pitch and tone.  Frequency of light waves is responsible for the colour and energy.      Sound can be electronically amplified.  The amplification of light is LASER.      TWO eyes are responsible for the depth of a scene.  That is, you can roughly calculate the distance between you and

   To our knowledge, first scripts were sculpted in stone slabs and walls.  Then letters were engraved in leafs, leathers, and metal sheets using nails and knives.  Next paper, ink, quills came.  Followed by nib, fountain pen, etc,.The major step is printing technology.  It revolutionized the news and information industry.           With the development of science,  storage was also developed. After the invention of cine films, pages were photographed and stored as microfilms.  Audio and video information were stored in plastic tapes using magnetization.  Even today, ATM cards, hard disc uses magnetization.      Next came the big digital revolution .  Here all the info. is represented in binary form.  The 0 and 1 is called binary digit or a bit .  8 bits or 16 bits make one byte .  Today we talk about info in terms of mega bytes and giga bytes.  Digital info has one major advantage.  Errors or noise cannot creep into digital storage because they are strictly either 0 or 1 and not

  Triangle of numbers called pascal triangle has so many properties.  It touches all the branches of mathematics.  This one article is not enough to describe the glory of the triangle.  So we will meet this miracle maths object now and then.     Construction:  Start with "1".  Place numbers below it in triangular pattern based on this rule. " Each number is the sum of the pair of numbers residing above it." (figure 1) There are so many patterns in the triangle.  We will see some of them.(figure 1)  The first diagonal is just 1s.  The second diagonal has the counting numbers 1,2,3,etc,.  The third diagonal has the triangular numbers.  The triangle is symmetrical.  The left side of the triangle is the mirror image of the right side.  The sum of each row gives the powers of  2.      The rows in pascal's triangle can also show you the coefficients of binomial expansion as given figure 2.  The square of the number in the second diagonal is the sum

 Matter:  Electron is a particle and a book is a body made of particles. Waves:  Sound is a mechanical wave and light is a electromagnetic wave. Particles transport themselves and move in a straight line.  A wave is a disturbance or energy traveling through a medium or vacuum. A particle cannot deviate from its path without a force.  But waves from a source can spread in spherical form through all directions. A particle is characterized by its mass and speed .  Wave is characterized by frequency and wavelength . Waves can bend over the edges and spread through the hole.  But particles cannot do so. High frequency waves may behave like particles(light photons).  High speed particles may behave like waves(electron waves) . Matter can be converted into energy (waves) and vice versa, according to E=mc^2 Hence the border line between particle and wave blurs. Science update: US congress approves NASA manned mars mission by 2033.  Funds alloted from this year.