WHY JOY-STICK IN THE AIR CRAFT?

Imagine car moving in a plane surface. It is moving on a 2-dimensional surface. Hence it can move in two directions(x and y). In other words, the car can have 2 transnational motions. How about rotations? It can only turn left or right on the surface. It has only one rotational freedom. Hence one steering wheel is enough to turn the car.

Imagine an air-craft flying in a 3-dimensional air space. It can have 3 transnational motions (x,y,and z). It can also rotate about all the three x,y and z axes. Hence the air-craft also has three rotational freedom. ( In 3 dimensions only, the number of rotational freedom and the number of transnational freedom equals).

So, an air-craft needs a three steering wheels. Three wheels are not practically possible. Hence the joy-stick was invented. The mechanism of joy-stick is the same as that used in computer games. The stick can move up and down; can move side ways, can rotate. All the three rotational motions can be controlled by one joy-stick. The stick translates the physical motions of the hand into digital mathematics and there by controls the turning of the air-plane. In the same way, in a computer game, a character or an object is controlled by the joy-stick.

The following formula gives the number of rotational motions or steering wheels required in a given number of dimensions.

Number of steering wheels required = 1/2n(n-1)
n - number of dimensions.
putting n=3, for a air-plane
= 1/2*3*(3-1)
=1/2*3*2
=3. So it is.
putting n=2, for a car, we get 1.
say, putting n=1 for a train, we get zero. That means, no steering wheel is required for a train. The train simply follows the track. Track changing is controlled by the Rail-stations.
The formula says, in a four dimensional world six rotational motions are possible.
Thank god. We do not have to encounter four dimensional world.

The earth spins and revolves. The electron spins and orbits. Water molecules rotates. This three dimensional world with three rotational freedom is highly suitable for them and us.

Comments

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

sciencecapsules

Search This Blog