The angular momentum explained.
Remember the ice-skater doing his pirouettes faster and faster, because he pulled in his first fully stretched arms? Well, here is a simple, but somewhat more mathematical explanation.
Let's take an object rotating about some axis. We can devide the object in numerous small parts. When we take those volumes to be infinitessimaly small, it's just like we're talking about a cloud of particles rotating about the same (and for simplicity only) axis. Every particle is rotating at a certain position, which I will denote by a position "vector" r with a velocity "vector" v. A vector is just a mathematical tool, describing something in a solution space, having a direction in the coordinates of that space and a magnitude, or modulus or simply length. A velocity vector for instance, can have a "lenght"
20 mph and a direction say "west". Or (-1,0,0), where the first number stands for the x component of direction, the second for y and the last for z, where z is up.
Check the image above. The position vector has coordinates x', y', z', and length r.
Now, every particle in a system where the forces are "conservative", which is a difficult concept, but for our cloud in the formation of planets, the forces are, the outer product of the velocity vector and the position vector is conserved, i.e. a constant. This quantity is called the "angular momentum" (In fact it's the outer product of the position and the "momentum" vector, but with a constant mass, momentum is just the mass times the velocity.)
In a formula:
Here, C is the constant value and L is the so called angular momentum.. The X stands for the outer product. An outer product goes with the Sine of the angle between the two vectors. For particles in an circular orbit, the angle between the velocity and position is always 90 degrees. Where the origin of our coordinate system is the center of our cloud at the rotation axis. So, the Sine has the value one. And the angular momentum
becomes simply the product of the lengths of both the vectors.
Now we are ready to explain what happens. When I take the arms of the skater as a bunch of particles, the length of the position vector for all of the particles gets less when he retracts his arms. But, since the product of the velocity and the position is a constant, the velocity of the particles "orbiting" about the axis of rotation (through the skater's body, from between his legs through the head) must get bigger. So he starts to spin faster and faster!!!
The same happens with the particles of our gas cloud in the formation of planets. Neet huh?
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Back to planet-formation.