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 determine the total angular momentum, we need to consider the individual angular momenta of the two metal disks and add them together. Angular momentum is defined as the product of the moment of inertia and the angular velocity.
1. Start by finding the angular momentum of each metal disk separately. The angular momentum (L) of a rotating object can be calculated using the formula: L = I * ω, where I is the moment of inertia and ω is the angular velocity.
2. The moment of inertia (I) of a circular disk rotating around its center axis is given by the equation: I = (1/2) * m * r^2, where m is the mass of the disk and r is the radius. Since the two disks are identical, they have the same mass and radius, so their moments of inertia will also be the same.
3. Consider the angular velocity (ω) of the first disk, denoted as ω1. The angular velocity of the second disk, ω2, is opposite in direction to ω1 (ω2 = -ω1) as you mentioned.
4. Calculate the angular momentum for each disk using the formulas. Let's denote the angular momentum for the first disk as L1 and the second disk as L2.
L1 = (1/2) * m * r^2 * ω1
L2 = (1/2) * m * r^2 * ω2
5. To find the total angular momentum (LT), simply add the angular momenta of the two disks:
LT = L1 + L2
Substituting the values, you have:
LT = (1/2) * m * r^2 * ω1 + (1/2) * m * r^2 * ω2
Since ω2 = -ω1, you get:
LT = (1/2) * m * r^2 * ω1 - (1/2) * m * r^2 * ω1
Simplifying further:
LT = 0
Hence, the total angular momentum of the combined system is zero.