Rolling with and without sliding

A spinning disk is gently placed on a rough horizontal surface. Initially there will be sliding, but gradually the center of mass accelerates, while the angular velocity and the velocity of the point on the floor decrease until the latter and sliding dissapear.

The unit length is the disk radius R and the unit time (R/g)^1/2, so that in these units g = R = 1.
Below you may select the initial angular velocity ω₀ = ω, as well as the friction coefficient between disk and floor: μ. It is also possible to select the disk initial position by dragging its center with the mouse.
If Graph is pressed a new window displays the time evolution of the angular velocity ω and the velocities of the center (V) and the contact point (v).
N is the number (from 0 to 32) of periphery points that will have their velocity shown as a vector. Checking V you will also see the velocity of the center of mass. All velocities in display (including that of the platform or the laboratory) are relative to the currently selected reference frame. If the velocity vectors are too long or too short, change scale.
To get information on an element, put over it the mouse pointer to see the corresponding tooltip.

Activities

Make sure you understand the relationship among angular velocity, the velocities of the center and the contact point and the disk radius R:
V = v + ω R.
By using the friction force and its momentum around the disk center, write and solve the evolution equations for the center-of-mass and the angular velocity.
How will change with time the angular velocity and the velocities of the center and the contact point?
When will start rolling without sliding?
Use the simulation to check your calculations.

This is an English translation of the Basque original for a course on mechanics, oscillations and waves.
It requires Java 1.5 or newer and was created by Juan M. Aguirregabiria with Easy Java Simulations (Ejs) by Francisco Esquembre. I thank Wolfgang Christian and Francisco Esquembre for their help.