Two disks are mounted (like a merry-go-round) on low-friction bearings on the same axle and can be brought together so that they couple and rotate as one unit. The first disk, with rotational inertia 1.46 kg·m2 about its central axis, is set spinning counterclockwise (which may be taken as the positive direction) at 412 rev/min. The second disk, with rotational inertia 7.37 kg·m2 about its central axis, is set spinning counterclockwise at 837 rev/min. They then couple together. (a) What is their angular speed (rev/min) after coupling? (b) If instead the second disk is set spinning clockwise at 837 rev/min, what is their angular velocity (in rev/min, using the correct sign for direction) after they couple together?

To solve this problem, we need to apply the principle of conservation of angular momentum. The total angular momentum before coupling should be equal to the total angular momentum after coupling.

The angular momentum of an object can be calculated using the formula:

L = Iω

Where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.

(a) First, let's find the initial angular momentum of each disk before coupling.

Disk 1:
Angular momentum of disk 1 before coupling = I1 * ω1
I1 = 1.46 kg·m^2 (moment of inertia)
ω1 = (412 rev/min) * (2π rad/rev) * (1 min/60 s) = 43.26 rad/s

L1 = 1.46 kg·m^2 * 43.26 rad/s = 63.16 kg·m^2/s

Disk 2:
Angular momentum of disk 2 before coupling = I2 * ω2
I2 = 7.37 kg·m^2 (moment of inertia)
ω2 = (837 rev/min) * (2π rad/rev) * (1 min/60 s) = 87.59 rad/s

L2 = 7.37 kg·m^2 * 87.59 rad/s = 644.23 kg·m^2/s

Now, let's calculate the total angular momentum before coupling:

L_total_before = L1 + L2 = 63.16 kg·m^2/s + 644.23 kg·m^2/s = 707.39 kg·m^2/s

After coupling, the two disks rotate as one unit with a common angular velocity ωf. The moment of inertia of the coupled system is the sum of the individual moments of inertia:

I_total = I1 + I2 = 1.46 kg·m^2 + 7.37 kg·m^2 = 8.83 kg·m^2

Now, let's find the final angular velocity (ωf) after coupling:

L_total_after = I_total * ωf

Since angular momentum is conserved, L_total_after = L_total_before

I_total * ωf = L_total_before

ωf = L_total_before / I_total

ωf = 707.39 kg·m^2/s / 8.83 kg·m^2 = 80.01 rad/s

Finally, let's convert the angular velocity to rev/min:

ωf = (80.01 rad/s) * (60 s/min) / (2π rad/rev) = 764.16 rev/min

So, the angular speed of the two disks after coupling is 764.16 rev/min.

(b) If the second disk is set spinning clockwise at 837 rev/min, then its angular velocity ω2 will be negative (-87.59 rad/s). In this case, the calculation will be the same, but with a negative value for ω2:

L2 = -7.37 kg·m^2 * 87.59 rad/s = -644.23 kg·m^2/s

L_total_before = 63.16 kg·m^2/s + (-644.23 kg·m^2/s) = -581.07 kg·m^2/s

ωf = -581.07 kg·m^2/s / 8.83 kg·m^2 = -65.76 rad/s

Converting ωf to rev/min:

ωf = (-65.76 rad/s) * (60 s/min) / (2π rad/rev) = -627.90 rev/min (clockwise)

So, if the second disk spins clockwise at 837 rev/min, the angular velocity of the two disks after coupling will be -627.90 rev/min.