Editing Angular Momentum Chair (section)

===Full Explanation===

Momentum is when a mass has a constant velocity, meaning that it has zero acceleration. This type of momentum we can think of being linear, or traveling along a single dimensional line (typically the x-axis). When an object is changing the direction of its velocity, it will experience '''''Angular Momentum'''''. This introduces a second, distance variable, which changes in relation to the velocity variable as the object rotates around a fixed axial point. Angular Momentum can '''''also''''' be seen in a third dimensional view, where the momentum of the system can be viewed based on the rotation of the two-dimensional system as a whole along a third axis, which allows the momentum to be transferred from one plane of rotation to another. In other words:

{| class="wikitable" style="color:black; background-color:#ddd; text-align: center; margin: auto"
| Linear Momentum (One-Dimensional)
| Angular Momentum (Two-Dimensional)
| Angular Momentum (Three-Dimensional)
|-
| ''p'' = m*'''v'''
| '''&int;''' ''p'' d'''r''' = m*'''v'''*'''r'''
| '''&int;''' '''&int;''' ''p'' d'''r''' d'''&theta;''' = m*'''v'''*'''r'''*sin('''&theta;''') + m*'''v'''*'''r'''*cos('''&theta;''')
|-
| Changes in the '''x'''-axis
| Changes in the '''x- AND y'''-axis
| Changes in the '''x- AND y'''-axis, angled to the '''z'''-axis
|}

Where ''p'' = Momentum (constant), m = Mass (constant), '''v''' = Velocity (variable), '''r''' = Radius of Rotation (variable), and '''&theta;''' = Angle to the z-axis (variable)

When the volunteer is holding the weights, we are looking at the two-dimensional representation of Angular Momentum. The center of the chair is where the radius of rotation is based, so the starting distance of the weights will determine the initial rotational radius. When they move the weights to the new radius, the change in radius ('''r''') will cause a proportional change in velocity ('''v''') in order to keep the momentum of the system conserved, resulting in the ''p'' value being unchanged.

When we introduce the wheel, we are seeing a change happen now in three dimensions. The wheel, while spinning, has a set momentum, with the velocity, mass and radius all being constant. However, when the wheel is rotated along the '''z'''-axis, we see that the momentum value stays unchanged, since it transfers between sine and cosine. When the wheel is held straight by the volunteer, the momentum is completely within a two-dimensional plane. When we turn the wheel on its side, it is now on a different two-dimensional plane, and causes a transfer from the sin('''&theta;''') to the cos('''&theta;''') in the equation above. To create this transfer, you need to put momentum into the system, and since the momentum in the system must be conserved, that addition will come out of the system into you, causing you to spin.

Another way to think of why you spin is by using the '''''Right Hand Rule'''''. Place your thumb along the axis of rotation for the wheel, and your fingers will face outward with the wheel, and curl over in the direction of rotation. When you turn your hand so your thumb is now facing upwards, the rotation of the wheel will be along the horizontal plane, and the rotation of the stool will be the ''opposite'' of the rotation of the wheel. This is because of the Conservation of Angular Momentum: the momentum of the system must stay conserved, so you cannot add or subtract from it. Therefore, since you have to apply a force upon the wheel to make it rotate between the horizontal and vertical planes, the resulting momentum that would be placed on the wheel must be opposed, and is put back into you, and therefore the stool.