WHISPERING GALLERY, COMET, KIDNEY STONE AND ELLIPSE

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 second focus can hear it loud and clear. The two foci are seperated by hundreds of feet. Here the sound waves starting from one focus, are reflected by the elliptically shaped walls and converge at another focus. That is why, even a pin drop can be heard.

Lithotripsy is the procedure of destroying kidney stone by sending high energy shock waves. Here again, the concept used in whispering gallery is adopted. The patients lie down in an elliptical tub. Shockwave source is at one focus and the patient's kidney stone is at the another focus. The stone is broken into minute pieces by the shock waves and then they pass out of the body without requiring surgery.
Like circle, the ellipse also has its own uses and status.

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LISSAJOUS FIGURES

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

THE PARABOLA

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

CASINO'S GAME

Let us find out how the casino survives with mathematics. Say, your friend invite you for a game of dice. You must bet (wager) 2 dollars. If you roll 'six' you will get back 8 dollars. The game will go on for 30 rounds. All sounds good. The probability of rolling 'six' is 1/6. Since the game will be played for 30 times, the 'expected win' is 30*1/6 = 5. That is, you are expected to win 5 rounds out of 30. Hence your gain will be 5 * 8 =40 dollars. ok. This also implies that you will loose 25 rounds. Hence your loss will be 25*2 =50 dollars. Your net gain will be gain-less = 40-50 = -10 dollars. For 30 rounds, the loss is -10 dollars, Hence, for one round =-10/30 = -1/3 dollars. There will be a loss of -1/3 or 0.33 dollars per round. It is not a fair game. Let us make a simple formula to calculate 'Pay out per round\. The probability for a win = p The pay-out in case of win = V No. of rounds = n The expect

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