A solid disk 1 with radius R1 is spinning freely about a frictionless horizontal axle l at an angular speed ω initially. The axle l is perpendicular to disk 1, and goes through the center S of disk 1.
The circumference of disk 1 is pushed against the circumference of another disk (disk 2) with identical mass. Disk 2 is in all respects identical to disk 1, except that its radius is R2, and it is initially at rest. Disk 2 can rotate freely about a horizontal axle m through its center P. Axles m and l are parallel. The friction coefficient between the two touching surfaces (disk circumferences) is μ.
We wait until an equilibrium situation is reached (i.e. the circumferences of the two disks are no longer slipping against each other). At that time, disk 1 is spinning with angular velocity ω1, and disk 2 is spinning with angular velocity ω2.
Calculate the magnitude of the angular velocities |ω1| and |ω2| in terms of R1, R2 and ω (enter R_1 for R1, R_2 for R2 and omega for ω).
It is quite remarkable that ω1 and ω2 are independent of μ, and it is also independent of the time it takes for the equilibrium to be reached (i.e independent of how hard one pushes the disks against each other).
∣ω1∣=
∣ω2∣=
∣ω1∣= omega/(1+(R_2/R_1)^2)
∣ω2∣= omega*(R_1/R_2)/(1+(R_2/R_1)^2)
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To calculate the magnitude of the angular velocities |ω1| and |ω2| in terms of R1, R2, and ω, we can use the principle of conservation of angular momentum. According to this principle, the total angular momentum of a system remains constant unless acted upon by external torques.
Initially, the angular momentum of disk 1 alone can be calculated as the product of its moment of inertia and angular speed:
Angular momentum of disk 1 = I1 * ω
Where I1 is the moment of inertia of disk 1. For a solid disk rotating about its axis perpendicular to its surface, the moment of inertia can be expressed as:
I1 = (1/2) * m * R1^2
Where m is the mass of the disk and R1 is the radius of disk 1.
Similarly, the angular momentum of disk 2 can be calculated as:
Angular momentum of disk 2 = I2 * ω2
Where I2 is the moment of inertia of disk 2, and ω2 is the angular velocity of disk 2.
Since there are no external torques acting on the system, the total initial angular momentum of the system will be conserved. Therefore, we can equate the initial angular momentum of disk 1 to the total angular momentum of the system after equilibrium:
I1 * ω = (I1 + I2) * ω1
Simplifying the equation:
(1/2) * m * R1^2 * ω = [(1/2) * m * R1^2 + (1/2) * m * R2^2] * ω1
Cancelling out the mass and rearranging the equation, we can solve for ω1:
ω1 = (R1 / R1^2 + R2^2) * ω
Similarly, using the conservation of angular momentum, we can equate the initial angular momentum of disk 2 to the total angular momentum of the system after equilibrium:
0 = (I1 + I2) * ω2
Simplifying the equation and solving for ω2:
ω2 = - (R1 / R1^2 + R2^2) * ω
Note: Since the question asks for the magnitude of ω1 and ω2, we can ignore the negative sign.
Therefore, the magnitude of the angular velocities |ω1| and |ω2| in terms of R1, R2, and ω are:
∣ω1∣ = (R1 / R1^2 + R2^2) * ω
∣ω2∣ = (R1 / R1^2 + R2^2) * ω