A hard question from the Oxford interview

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The discussion revolves around a physics problem involving two disks, where a small disk moves under gravity without friction on a fixed large disk. The key to solving the problem lies in understanding the relationship between centripetal force and gravitational force acting on the small disk. Participants emphasize the need to apply conservation of energy principles and circular motion equations to derive the conditions under which the small disk will no longer contact the large disk. The centripetal force required for the small disk's motion must be provided by the gravitational force, with careful consideration of direction. Ultimately, the problem requires a mathematical formulation of these forces to determine the distance from the small disk's center to the ground at the point of separation.
Negi Magi
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Homework Statement


See the attached picture.
The big disk is fixed on the ground, and the small disk move under the gravity. There is no friction between two disks. When will the small disk is no longer touch the big disk? Just measure the distance between the center of the small disk and the ground.


Homework Equations


The conservation of the energy
The equation of the circular motion


The Attempt at a Solution


I just consider that the centripetal force on the small disk, which is one of the component of the gravity on the small disk, may be the key to answer this question. But I really don't know how to write it into an equation and to solve it
 

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For a given velocity v, you can calculate the centripetal force required to keep the disk sliding. Gravity has to provide this force (but pay attention to its direction).
 
Since there is no rolling the potential energy is just converted into kinetic energy. The speed will be tangential to the line connecting it to centre of the big disc. One can simplify it by considering a point mass at 3r following a circular path. The component of g along the line to the centre of the big disc provides the centripetal acceleration.
 
Kindly see the attached pdf. My attempt to solve it, is in it. I'm wondering if my solution is right. My idea is this: At any point of time, the ball may be assumed to be at an incline which is at an angle of θ(kindly see both the pics in the pdf file). The value of θ will continuously change and so will the value of friction. I'm not able to figure out, why my solution is wrong, if it is wrong .
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