HOW MUCH FREEDOM YOU HAVE JUST TO MOVE ABOUT?

A body can move in three directions, x,y,and z. Also the body can rotate around three axes (x,y,z). Hence we say, the body has "six degrees of freedom". A helicopter or a bird has six degrees of freedom. In aerodynamics language, the three rotations are called, pitch, roll and yaw.

Take a plank of wood. In this three dimensional world, it has six degrees of freedom. Fix the plank as a door in a building. It can now do only one rotational motion about the joint. The six degrees of freedom has come down to one. We understand that the joints restrict the freedom.

There is a formula to calculate the degrees of freedom.

M = 6*n-SUM (6-fi)
for i = 1 to k

M - Degrees of freedom
n - Number of moving parts.
k - number of joints.
fi - number of degrees of freedom of each joint.

That is, (6-fi) should be found for each joint and summed up. Then the total sum is subtracted from 6n to get no. of degrees of freedom.

For example, take our arm. At the shoulder joint, we have 3 DOF (degrees of freedom). At the elbow, only 2 DOF. At the wrist 3. The fingers have only one DOF. Then, what is the total number of freedom for the entire arm. Explore using the above formula.

The DOF of a robotic arm is calculated and designed using the above formula.

Note: A gas molecule(example O2) is free to move (No joints). Hence it has all the six degrees of freedom and vibration freedom also. The molecule executes all the motions due to thermal energy. The energy is equally shared by all the degrees of freedom.

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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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