Two disks are rotating about the same axis. Disk A has a moment of inertia of 3.1 kg · m2 and an angular velocity of +8.0 rad/s. Disk B is rotating with an angular velocity of -10.2 rad/s. The two disks are then linked together without the aid of any external torques, so that they rotate as a single unit with an angular velocity of -2.6 rad/s. The axis of rotation for this unit is the same as that for the separate disks. What is the moment of inertia of disk B?

The negative signs are confusing me.

I w before = I w after by conservation of angular momentum

3.1 * 8 - 10.2 * I = -2.6 (3.1+I)

Thanks!

You are welcome.

How would one go about simplifying that equation down to an answer? I'm stumped by it for some reason.

The negative signs in this problem indicate the direction of rotation. Let's break it down step by step to understand what is happening.

Initially, disk A is rotating with an angular velocity of +8.0 rad/s. The "+" sign indicates that it is rotating counterclockwise when viewed from above. Disk B, on the other hand, is rotating with an angular velocity of -10.2 rad/s. The "-" sign indicates that it is rotating clockwise when viewed from above.

When the two disks are linked together without any external torques, they rotate as a single unit. The given angular velocity for this unit is -2.6 rad/s. Again, the "-" sign indicates that it is rotating clockwise when viewed from above.

Now, let's analyze the effect of linking the two disks together. When the two disks are linked, their angular velocities should add up to the angular velocity of the combined system, according to the principle of conservation of angular momentum. Mathematically, this can be expressed as:

Iₐ * ωₐ + Iₐ * ωₐ = (Iₐ + I_b) * ω_ab

Where:
Iₐ = moment of inertia of disk A
ωₐ = angular velocity of disk A
I_b = moment of inertia of disk B
ω_ab = angular velocity of the combined system

Plugging in the given values:

3.1 kg · m² * (+8.0 rad/s) + I_b * (-10.2 rad/s) = (3.1 kg · m² + I_b) * (-2.6 rad/s)

Now let's solve this equation for I_b, the moment of inertia of disk B:

24.8 kg · m² - 10.2 I_b = -8.06 kg · m² - 2.6 I_b

Rearranging the equation:

7.14 I_b = 32.86 kg · m²

Dividing both sides by 7.14:

I_b ≈ 4.6 kg · m²

Therefore, the moment of inertia of disk B is approximately 4.6 kg · m².