WHY THE SATELLITES ARE NOT FALLING DOWN ON US?

A fast running bi-cycle hit by an object may not fall down. It gets stabilized due to gyroscopic effect. A boat or ship tilted by the strong wind will not turn turtle because it is so designed to stabilize itself. A space station orbiting around the earth may get hit by a rock. It may slow down. But it will not plummet to the earth. It automatically get stabilized. Let see how.

A satellite is taken by the rocket and pushed into the orbit. The satellite is given a preciously calculated velocity called 'orbital velocity'. If the velocity is slightly less than the orbital velocity, it will fall down on the earth pulled by gravity. If the velocity is more, it will fly away from the earth or earth's gravity.

Any orbiting or rotating body have what is called "angular momentum'. given by the formula
Angular momentum = mass * velocity* radius of orbit
This quantity is always remain constant and conserved. As the radius increases, the velocity has to increase to keep the momentum constant. Consider a spinning dancer. As the dancer brings the stretched arms close to her body, she spins even faster.(The distance decreases, speed increases).

If the space station or satellite is hit by a rock accidentally, its speed will decrease, and it will start falling down. But immediately, the principle of conservation of angular momentum comes into play. That is, the space station's speed will get increased as the distance decreases. The station will get back the required speed and will stay in the orbit. Hence "angular momentum" plays the main role in orbital stability.

The earth goes around the sun in slightly elliptical orbit. When the earth is close to the sun, it moves fast. When it farther from the sun, it slightly slows down. All because of angular momentum principle.
It seems that, nature has given laws to stabilize our vehicles. And the nature's intent is not to destabilize our life.

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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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