# Motion of an object spiraling outwards on a rotating disk with friction

I recently came across the following problem:

A spinning disk has an object initially located at radius $$R$$. The object is moving with the same angular velocity as the disk (which I will call $$ω$$). There is a string tying the object to something in the center of the disk. At time $$t = 0$$, the string is cut. The static friction between the object is less than the centripetal force required to keep the object in uniform circular motion (the disk's motion is constant throughout the problem), so the object begins to spiral outwards.

I am trying to find an equation for $$r$$ as a function of time and the initial conditions (we are given the initial radius $$R$$, and initial angular velocity $$ω$$).

In particular, I was wondering if the angular velocity of the object would be constant, what direction the friction would be directed in, and if this equation could then be applied to situations in which the disk is frictionless (by plugging in $$μ = 0$$ into my final equation).

Intuitively, I felt that this would somehow involve centrifugal force. Please let me know what you think.

• Is this a ball that rolls in two directions, a cylinder that rolls in one direction or a block that only slides? Dec 31 '18 at 18:32
• a block that only slides Jan 2 '19 at 2:47
• There isn't an analytical solution to this problem. If you run a simulation you will find that the general solution involves stick-slip phenomenon and a non-smooth acceleration curve. Jan 3 '19 at 14:00
• @ja72 I'm not sure what you mean by stick-slip phenomenon. I had intended to set up the question in such a way that we're only considering a slipping block, although perhaps I have erred in my phrasing... Jan 12 '19 at 4:04
• Object is located at periphery when you cut the spring it just falls outside as friction it is not sufficient, then what kind of equation you want to find. Jan 11 '20 at 14:31