# 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

The angular momentum of the object will change if there is a lateral force on it. Without friction the object would move in a straight line after the string is cut - so it would move outward and thus encounter disk "going faster". This means the angular momentum of the object will increase.

• Without friction there would be no change in the angular momentum unless the mass was in a frictionless groove so that the mass could interact with the disc. Then angular momentum of the mass would increase because of the torque exerted on the mass due to the groove. Apr 30 '17 at 5:42
• The problem without friction only interests me in that it should provide a method to test my final equation in the case where the coefficient of kinetic friction is zero. Also since it's radius would increase while it's velocity would remain constant (in the frictionless string being cut situation with no horizontal disk interaction) this would decrease angular velocity would it not? Initially I found what I thought was a solution involving hyperbolic sines and cosines, it did not become evident that there was a problem until I plugged in zero for μ. Apr 30 '17 at 9:47
• Also since angular momentum is the cross product of radius with momentum, this should be constant in the string being cut situation with no horizontal disk interaction (frictionless). Finally, since there is indeed friction in the problem I am concerned with, I don't think I can assume that anything is conserved. Apr 30 '17 at 9:51

Angular velocity of the object around the center of the disk will not be constant, but will decrease with time. You can analyze the system in the reference frame of the rotating disk, in which the object slides over a motionless surface. The frictional force is then very simple: it is always directed opposite to the motion of the object. There are two pseudoforces: the centrifugal force, which causes the object to move outward, and the Coriolis force, which acts perpendicular to the velocity of the object, curving it in a direction opposite to the disk rotation.

• Yes, both centrifugal and coriolis forces act on the object in additon to friction, and to solve for the equation of motion probably requires a numerical approach with a computer code. Jan 27 at 1:07