ENERGY - IN A NEW ANGLE

A car is running. It can hit and damage an object. It can do some work because it posses energy. A body gets energy by virtue of its motion or its dynamics. We know, it is called kinetic energy. The formula is 1/2 mv^2, m is the mass of the body, or we can say m is the inherent property of the body. v is the velocity of the body. or we can say v represents the dynamics of the body. Almost all the kinetic energy formula in physics follows this pattern.

Consider an oscillating spring. It is KE is 1/2kx^2. Here k is constant (property) of the spring. It depends on the material, thickness, and length of the spring. x is the maximum displacement of the spring from its rest position. We say x represents the main dynamics of the spring.

Consider a rotating body. Its K.E is 1/2IW^2. I is the moment of inertia or property of the rotating body, w its angular velocity (dynamics).

The energy stored in an inductor (coil of wire) is 1/2LI^2. L is the inductance or the magnetic property of the coil. I is the current flowing through it (dynamics).

The energy of the capacitor is 1/2cv^2. c is the capacitance of the capacitor (property). v is the voltage applied across it (electrical dynamics - in a capacitor voltage plays, more important role than a current).

The kinetic energy of the running fluid is 1/2dv^2. Here d is the density of the fluid and it is the main property of fluid and not the mass.

Einstein said the energy of a body is mc^2. m is property of the body, we understand. But c is the velocity of light, why it appears here? When the body explodes and destroys itself, all the particles in it are converted into electromagnetic waves and they move with the velocity light. Next, why 1/2 is not present here? The energy of a body is the sum of rest energy and the energy when it is moving. Hence 1/2+1/2 =1. We have looked at the famous Einstein formula from a new angle.

Finally, when a body is active, its energy will be equal to = 1/2 * inherent property*(main dynamics)

This formula will help us understand the nature of energy and make new formulas in new situations.

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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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