I would like to solve the following mechanics problem and would like to get few hints.

There is an inclined plane with angle $\alpha$. A ball of radius $r$ rolls on the inclined plane that has a spring connected to it at the bottom (first figure). The spring constant is $k_{1}$. Let the initial angular velocity of the ball be $\omega_{\circ}$. When the ball rolls down the inclined plane, it comes in contact with the spring (second figure) and it rebounds up the inclined plane (third figure). After it rebounds, it again rolls up the plane until the angular velocity is reduced to zero.

It starts rolling down the inclined plane again and gets rebounded by the spring up the inclined plane. This continues until the ball comes to rest in contact with the spring (last figure).

Problem: I would like to obtain the equation of motion for the rolling ball. After how many rebounds, let us say $n$, will the ball come to rest? Can this problem be solved using the Lagrangian formulation?

P.S. It may be possible that if the mass of the rolling ball is more than a certain limit, it may comes to rest with the spring compressed.

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  • $\begingroup$ Is there any friction involved? $\endgroup$ – Gert Apr 15 at 16:27
  • 1
    $\begingroup$ If you just want a few "hints", I'm not sure exactly what an "answer" would correspond to, but...the ball will never stop bouncing without some frictional losses (in the spring or in the ball-ramp contact), so the Lagrangian method will work as long as you know how to take into account the work that the ramp does on the spring. $\endgroup$ – levitopher Apr 15 at 16:30
  • $\begingroup$ The frictional coefficient is $\mu$. I am a physics enthusiast trying to invent problems and solving them. But I am not getting any starting point to the solution to this problem. $\endgroup$ – Suddhasattwa Ghosh Apr 15 at 16:41