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∣=

3 answers

∣ω1∣= omega/(1+(R_2/R_1)^2)

∣ω2∣= omega*(R_1/R_2)/(1+(R_2/R_1)^2)

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