WAXING AND WANING MOON

The moon appears to grow and decay. What really happens? It is just an interplay of light and shadow.

Like earth, half of the moon's surface is illuminated by sun light always. The moon circles the earth every 28 days. when it shows fully illuminated face, we say it is full moon day. When it shows night side(dark side), it is new moon day. If it shows partly night side and day side, it must be either in waxing or waning phase. The figure illustrate the concept beautifully.

When we see exactly half moon, the earth, the sun and the moon must form a right angled triangle as shown in the diagram. The angle suspended at the earth is 89.85 degrees.
The trigonometry says :

COS(89.85) = distance between the moon and the earth / distance between sun and the earth = .002617
Reversing both sides,
distance between sun and the earth / distance between the moon and the earth = 1/0.002617 = 382

Hence sun's distance is 382 times the moon's distance from the earth. The moon is roughly at 400000 km from the earth. So the sun is at 382 x 400000 km from the earth. The trigonometry always plays the important role in astronomical calculations.

Moon's cycle is used as measurement of time till date (a month). Moon's phases gave us romance; threatened us; made us quiet and artist. There are many beliefs that surrounds the moon's phases.

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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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