A thin disk, mounted on a frictionless vertical shaft of negligible rotational inertia, is rotating at 495 revolutions per minute. An identical disk (that is not initially rotating) is dropped onto the first disk. The frictional force between the disks causes them to rotate at a common angular velocity. Find this common angular velocity.
To find the common angular velocity, we need to apply the principle of conservation of angular momentum.
First, we need to calculate the initial angular momentum of the first disk. The angular momentum of a rotating object can be calculated using the formula:
L = I * ω,
where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
Given that the first disk is rotating at 495 revolutions per minute, we need to convert this to radians per second. There are 2π radians in one revolution, and 60 seconds in one minute, so we can convert the angular velocity as follows:
ω1 = (495 rev/min) * (2π rad/rev) * (1 min/60 s) = 51.7 rad/s.
Since the first disk is rotating on a frictionless shaft, it has a negligible rotational inertia. Therefore, its moment of inertia, I1, can be considered to be zero.
Now, when the second disk is dropped onto the first one, the frictional force between the disks causes them to rotate at a common angular velocity.
Assuming that the two disks stick together and rotate as one object, the total angular momentum is conserved. Therefore, we can write:
L_total = L1 + L2,
where L_total is the total angular momentum of the system, L1 is the initial angular momentum of the first disk, and L2 is the angular momentum of the second disk.
Since the second disk is initially at rest, its initial angular momentum, L2, is zero.
Therefore, the total angular momentum after the second disk is dropped is equal to the initial angular momentum of the first disk:
L_total = L1.
Thus, the common angular velocity, ω_common, can be found by rearranging the formula for angular momentum:
L_total = I_total * ω_common.
Since the two disks are identical, they have the same moment of inertia, I_total = I1 + I2 = I + I = 2I.
Substituting the values, the equation becomes:
L1 = 2I * ω_common.
Simplifying further, we have:
ω_common = L1 / (2I).
Since I1 is negligible (I1 ≈ 0), we can ignore it in the calculation. Thus, the common angular velocity can be simplified as:
ω_common = L1 / (2 * 0 + I) = L1 / I.
So, to find the common angular velocity, we need to know the moment of inertia of the disks. If the moment of inertia (I) is provided, we can calculate ω_common using the formula:
ω_common = L1 / I.