Rotational Mechanics

First, one should understand that any object in space has 6 degrees of freedom. 3 of these are cartesian translation (eg, the X, Y, and Z axes). The other 3 are rotations about those cartesian axes. The preceding laws of mechanics have applied (for the most part) to cartesian translation, but only partially apply to rotational motion of objects.

Rotational Mechanics are mechanics involving rotation of a body. This is different from centripetal forces are sometimes external forces applying to a body moving in a circular path, whereas rotational mechanics involve rotation of the bodies themselves.

Fundamentally, rotational mechanics uses polar coordinates to describe rotation. If you imagine a point in the XY plane, modeled in cartesian coordinates as X = 1, Y=1, or (1,1), that same point can also be modeled using polar coordinates as R=Sqrt(2), Theta = Pi/4 = 45 degrees (Sqrt(2), Pi/4). Using trig the R of the point can be derived using the pythagorean theorem as X^2+Y^2=R^2, which solves to R=Sqrt(2). Pi/4 is in radians, which is another way to describe angles, in that one circumference is 2*pi*R, so we let 360 degrees = 2*pi so that the integral works out to 2*pi*R (just trust me on this).

OK, you probably don't actually need to know all that, but it will help you understand where the following formulae come from:

Moment of Inertia: I = m * R^2. This is a scalar quantity, and is the rotational inertia of a point mass rotating around a stationary point one R away. m = mass.

Angular Displacement = theta = omega. In radians, this is the angle change of a point. It is a vector, using the right-hand rule to dictate direction of the vector.

Arc Displacement = R x theta. where theta is in radians. This is simply the distance drawn by an arc of radius R, over an angle theta. In general, all angular distances/velocities/accelerations become "linear" (arc) displacements/velocities/accelerations, when you cross them with the radius vector.

Angular velocity = theta' = nu (the greek letter). This is the derivative dtheta/dt, and is the rate of change of the angular displacement w.r.t. time.

Angular acceleration = theta'' = alpha (the greek letter). This is the second derivative, d^2theta/dt^2, and is the rate of change of the angular velocity w.r.t. time.

Angular momentum: L = R x P = I * omega. for an unchanging mass, I becomes a scalar quantity.

With these formulae, you can now extend all of the fundamental laws of physics from Chapter 1 into the angular/rotational (Polar coordinate) realm.

Newton's Law of Inertia (Rotational)

A body in rotation stays in rotation. In other words, a body rotating at a constant speed will continue to rotate at that speed until an external force is applied.

Newton's 2nd Law (Rotational) Defining Torque

T = dL/dt = I x alpha (when the moment of inertia is constant) = alpha dI/dt + I dnu/dt (if both moment of inertia and velocity are changing)

Torque/Moment is the derivative of Angular momentum w.r.t. time. Remember, because moment of inertia is related to the distance of the rotating body's center of mass to the center of rotation, and also the moment of inertia of the body spinning in one of its own "internal axes", the moment of inertia will often change over time.

Newton's 3rd Law (Rotational) the Law of Action-Reaction.

This is very much the same as the linear one, but just applying to rotational motion instead.

Conservation of Angular Momentum

The net angular momentum of a system will always stay constant.

Imagine an ice skater spinning with extended leg, then pulling the leg inward. Her moment of inertia becomes smaller, when her leg pulls in, but her angular momentum must remain constant, so she spins faster.

We use this same concept in shuaijiao, when we spin/coil before throws. We spin with leg extended and then pull the leg inward to turn faster. The intention is for your main torso to turn faster, so you do it this way.

The concept is used for kicks through the "chamber" in an opposite manner. You lift your leg with knee bent because the moment of inertia is smaller (easier to lift), so you can chamber your kick quicker. Then you snap your lower leg out as you near the target, slowing down your rotation, but also increasing your moment of inertia to increase the penetration power of your kick. This is also why a muay thai roundhouse is more powerful than a TKD roundhouse. The TKD roundhouse goes through the chamber, so has more speed because it is more efficient to get the leg out at full extension, but at the end of the acceleration, has less total angular momentum. However the Muay Thai roundhouse does not go through the chamber, so your hip motion continuously adds angular momentum to your kicking leg. This is also why their kicks are slower than TKD kicks.

Conservation of angular momentum is also responsible for precession and gyroscopic effects. (to be continued)

[WIP 2/29/12]