A disk is spinning at a rate of 10 rad/s. A second disk of the same mass and shape with no spin is placed on the top of the frist disk friction acts between the two of the disk until both are eventually travelling at the same speed. What is the final angular velocity of the two disks?
Assuming conservation of angular momentum:
Angular momentum of first disk before interaction:
L1 = I1 * w1
where I1 is moment of inertia and w1 is initial angular velocity
Angular momentum of second disk before interaction:
L2 = I2 * w2
where I2 is moment of inertia and w2 is initial angular velocity (which is 0)
Total angular momentum before interaction:
L = L1 + L2 = I1 * w1
After interaction, the two disks will have the same final angular velocity w:
Angular momentum of first disk after interaction:
L1' = I1 * (w1 - w)
Angular momentum of second disk after interaction:
L2' = I2 * w
Total angular momentum after interaction:
L' = L1' + L2' = I1 * (w1 - w) + I2 * w
Since angular momentum is conserved, L = L':
I1 * w1 = I1 * (w1 - w) + I2 * w
Simplifying:
I1 * w1 = (I1 + I2) * w - I1 * w
2 * I1 * w1 = (I1 + I2) * w
w = 2 * I1 * w1 / (I1 + I2)
Since the two disks have the same mass and shape, they have the same moment of inertia:
I1 = I2
Substituting:
w = 2 * w1 / 2 = w1
Therefore, the final angular velocity of the two disks is 10 rad/s.
To find the final angular velocity of the two disks, we can use the law of conservation of angular momentum. According to this law, the total angular momentum before and after the interaction remains constant.
The angular momentum (L) of an object is given by the product of its moment of inertia (I) and its angular velocity (ω): L = I * ω.
Since both disks have the same mass and shape, they also have the same moment of inertia.
Let's assume the moment of inertia of each disk is I.
The initial angular momentum of the first disk (disk 1) is given by:
L1 = I * ω1, where ω1 = 10 rad/s.
The initial angular momentum of the second disk (disk 2) is zero since it is not spinning initially:
L2 = I * ω2, where ω2 = 0.
The total initial angular momentum (L_initial) is:
L_initial = L1 + L2 = I * ω1 + I * ω2 = I * (ω1 + ω2).
When the two disks eventually travel at the same speed, their final angular velocity is the same. Let's assume the final angular velocity of both disks is ω.
The final angular momentum of disk 1 is:
L1_final = I * ω.
The final angular momentum of disk 2 is also:
L2_final = I * ω.
The total final angular momentum (L_final) is:
L_final = L1_final + L2_final = I * ω + I * ω = 2I * ω.
According to the law of conservation of angular momentum, L_initial = L_final.
Therefore, we have:
I * (ω1 + ω2) = 2I * ω.
Canceling out the moment of inertia (I) from both sides, we get:
ω1 + ω2 = 2ω.
Substituting the values of ω1 and ω2 into the equation, we have:
10 + 0 = 2ω.
Simplifying the equation, we get:
10 = 2ω.
Dividing both sides of the equation by 2, we find:
ω = 5 rad/s.
Hence, the final angular velocity of the two disks is 5 rad/s.