Suppose I have a circular metal disk spinning at a constant angular velocity. Now imagine near to, but not in contact with the first disk, I have an identical metal disk spinning at the same angular velocity but in the opposite direction (ie. W1 = -w2) and the surfaces are parallel to one another. What is the total angular momentum?

To find the total angular momentum in this scenario, we need to consider the angular momentum of each disk separately and then add them together.

Angular momentum is calculated by multiplying the moment of inertia (I) of an object by its angular velocity (ω). The moment of inertia for a disk rotating around its center axis is given by the formula: I = (1/2) * m * r^2, where m is the mass of the disk and r is the radius.

Let's assume the mass and radius of each disk are the same, denoted as m and r, respectively. Since the disks are identical, their moments of inertia will be equal: I1 = I2 = (1/2) * m * r^2.

Now, let's calculate the angular momentum of each disk. The angular momentum (L) is given by L = I * ω.

For the first disk (disk with angular velocity w1), the angular momentum is L1 = I1 * w1 = (1/2) * m * r^2 * w1.

For the second disk (disk with angular velocity -w2), the angular momentum is L2 = I2 * (-w2) = - (1/2) * m * r^2 * w2.

To get the total angular momentum, we add the angular momentum of each disk together:

Total Angular Momentum = L1 + L2 = (1/2) * m * r^2 * w1 - (1/2) * m * r^2 * w2.

Since w1 = -w2 in this scenario, the two terms cancel each other out:

Total Angular Momentum = (1/2) * m * r^2 * w1 - (1/2) * m * r^2 * (-w1)
= m * r^2 * w1 + m * r^2 * w1
= 2 * m * r^2 * w1.

Therefore, the total angular momentum in this system is 2 times the angular momentum of each individual disk, given by 2 * m * r^2 * w1.