A disk of mass m is spinning freely at 6.22 rad/s when a second disk of identical mass, initially not spinning, is dropped on it so that their axes coincide. In a short time the two disks are corotating. (The two disks have the same radius.)
(a) What is the angular speed of the new system?
rad/s
(b) If a third such disk is dropped on the first two, find the final angular speed of the system.
rad/s
a) 2.34 rad/s
b) .94 rad/s
Note: I might not be correct, but I did put more work into this answer by just making up answers than you did in the post.
To solve this problem, we need to apply the principle of conservation of angular momentum. The angular momentum of a system remains constant if no external torques act on it.
(a) Angular Speed of the New System:
The initial angular momentum of the first spinning disk can be calculated using the formula:
L1 = Iω1
where L1 is the initial angular momentum, I is the moment of inertia of the first disk, and ω1 is its initial angular speed.
After the second disk is dropped and the two disks are corotating, their combined moment of inertia can be approximated as the sum of their individual moments of inertia:
I_combined ≈ 2I
The final angular momentum of the new system can be calculated using the same formula:
L_final = I_combined * ω_final
Since angular momentum is conserved, we can equate the initial and final angular momentum:
L1 = L_final
Iω1 = I_combined * ω_final
Substituting the value of I_combined, we get:
Iω1 = (2I) * ω_final
ω_final = ω1 / 2
Therefore, the angular speed of the new system is half of the initial angular speed:
ω_final = 6.22 rad/s / 2 = 3.11 rad/s
(b) Final Angular Speed of the System with Three Disks:
To find the final angular speed of the system with three disks, we follow the same approach as above, considering the moment of inertia of the three disks combined:
I_combined ≈ 3I
Using the conservation of angular momentum:
L1 = L_final
Iω1 = I_combined * ω_final
Substituting the value of I_combined, we get:
Iω1 = (3I) * ω_final
ω_final = ω1 / 3
Therefore, the final angular speed of the system with three disks is one-third of the initial angular speed:
ω_final = 6.22 rad/s / 3 = 2.07 rad/s