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). Disk 2 has the same thickness and density as disk 1, but has a radius 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∣=
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To calculate the magnitudes of angular velocities |ω1| and |ω2| in terms of R1, R2, and ω, we need to apply the conservation of angular momentum.
Angular momentum (L) is equal to the product of moment of inertia (I) and angular velocity (ω). It is conserved when there is no external torque acting on the system.
For disk 1:
Moment of inertia (I1) of a solid disk rotating about its center is given by:
I1 = (1/2) * M1 * R1^2, where M1 is the mass of disk 1.
Angular momentum (L1) of disk 1 initially is:
L1_initial = I1 * ω
For disk 2:
Moment of inertia (I2) of a solid disk rotating about its center is given by:
I2 = (1/2) * M2 * R2^2, where M2 is the mass of disk 2.
Angular momentum (L2) of disk 2 initially is:
L2_initial = I2 * 0 (as it is initially at rest)
In the equilibrium situation, we have:
L1_final + L2_final = L1_initial + L2_initial
Since disk 2 is free to rotate about axle m, the only torque acting on the system is the friction torque between the disks. The torque due to friction can be calculated as the product of the friction force and the radius of the disks (R1 or R2).
The friction force acting between the disks can be calculated using the equation:
friction force = μ * normal force
The normal force (N) can be calculated as the product of the mass (M2) of disk 2 and acceleration due to gravity (g).
Once we have the friction force, we can calculate the torque due to friction as:
friction torque = friction force * R2
Since the torque due to friction causes the angular acceleration of disk 1 and the angular deceleration of disk 2, the change in angular momentum can be written as:
change in L1 = -change in L2 = friction torque
Finally, we can calculate the final angular velocities ω1 and ω2 using the final angular momenta (L1_final and L2_final) and their respective moments of inertia (I1 and I2).
|ω1| = |L1_final| / I1
|ω2| = |L2_final| / I2
Therefore, the magnitudes of the angular velocities |ω1| and |ω2| in terms of R1, R2, and ω can be calculated using the above equations.