A mass on a rolling disc, the friction falls

[mrsobhan]

Homework Statement

We are familiar with the problems like this:
A rolling disc with a mass on it at the radius of r from the center of the disc. In this type of problems we know in advance that the mass turns with the disc and so has a circular motion. Then since this is a circular motion with constant angular velocity we must have an acceleration a equal to rw^2 which w is the angular velocity. Then we draw the free diagram of forces and we find that the friction provides the required force for the acceleration towards the center.
Now consider this disc is rolling and after a while the friction reduces. We know that the mass cannot stay at the same distance from the center. Is it possible to describe the motion of the mass?

Homework Equations

We know that the friction is:
Ff = μN
This is the magnitude. and we don't know the direction. We simply determine the direction knowing the motion. The friction opposes to the motion but in this case we don't know anything about the motion.
If we draw free diagram for the problem we don't know how to draw the friction, on the other hand we don't know about acceleration neither the direction nor the magnitude.

The Attempt at a Solution

First I considered the friction along with the radius and set the magnitude in formulas. But as I said above It does not seem reasonable.
Then I tried to solve the problem by dynamics of the rotation For example conservation of energy or momentum. but again no result

 

[jbsteele]

.Now I'm stuck and I don't know how to proceed or what to do.A:The motion of the mass can be described using the equations of motion for a particle in two dimensions. The force on the mass is composed of two components, the centripetal force due to the circular motion of the mass around the disc, and the frictional force between the disc and the mass. The equation of motion in the x-direction is$$m \frac{d^2x}{dt^2} = F_{fric} \cos \theta - m r \omega^2$$where $\theta$ is the angle the line joining the center of the disc to the mass makes with the x-axis. Similarly, the equation of motion in the y-direction is$$m \frac{d^2y}{dt^2} = F_{fric} \sin \theta.$$The acceleration of the mass is given by$$a = \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}.$$These equations can be solved numerically using an appropriate numerical technique such as Euler's method.