Rolling without sliding on an accelerated platform

• Below you may select the platform acceleration a, as well as its initial velocity v. It is also possible to select the initial position and velocity (X and V) of cylinderr's center of mass. X may also be set by dragging the cylinder center with the mouse.
• By pressing Frame you change the reference frame (laboratory or platform) used in the animation.
• 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

1. First of all, let us assume the platform is at rest in the laboratory, v = a = 0, and the cylinder center is moving with non-zero velocity V non-zero. Sketch the velocity field in eight periphery points, including the topmost nd the one resting on the platform.
2. Use the animation to check your drawing.
3. Now select a small value for the acceleration, say a = 0.04. Before your run the simulation try guessing (or, better, computing) what will happen.
4. What changes when the problem is analyzed from the platform frame?
5. Try different (positive, negative and null) values for the initial V.
6. Check that the angle of a cylinder radius with the vertical is
   φ = φ₀ + ω₀ t - a t²/3R,
   where a = a is the platform acceleration R the cylinder radius (0.5 in the animation units) and
   ω₀ = (V₀ - v₀)/R,
   V₀ = V and v₀ = v being the initial velocity of the cylinder center and the platform.
7. Check also that the position of the cylinder center is given by
   X = X₀ + V₀ t + a t²/6.
8. Which is the minimum value of the friction coefficient necessary to avoid sliding?