Rolling ball inside a shell problem

[Hamiltonian]

Homework Statement

A ball of mass M and radius R is placed inside a spherical shell of the same mass M and the inner radius 2R. The combination is at rest on a tabletop in the position shown in the figure. The ball is released, and it rolls back and forth inside the shell. Find the maximum displacement of the shell.

Relevant Equations

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I was able to solve this problem easily by using the fact that the center of mass of the system is stationary as $\sum F_{ext} = 0$ for the ball and shell system. since COM's of both objects can be replaced with point masses at there center, the shell will have maximum displacement when its COM is in the position of the ball and the ball's in the place of the shell's COM, by the given geometry, we can conclude when the COM's of both the bodies exchange positions the shell's COM would have been displaced by $R$

I wanted to know if there is a way to write the equation of motion of the shell and from that evaluate its maximum displacement.
also, it's given in the problem that the ball rolls inside the shell(i.e. no slipping condition applies) hence I feel that the question was expected to be solved in this manner?
I am unable to come up with any equations in terms of the $x$-coordinate of the shell.
a hint to get started with this problem will be amazing! :)

 

[kuruman]

The use of a Lagrangian to get the equation of motion is the only way I can trust myself to do it right. However, I doubt that the equation of motion can be solved analytically. If the outer shell were stationary, then the motion of the ball is that of a large-amplitude physical pendulum which already cannot be solved analytically. Allowing the outer shell to move would be equivalent to having a movable pendulum support and would make an already intractable solution even more so.

 

[jbriggs444]

The use of a Lagrangian to get the equation of motion is the only way I can trust myself to do it right. However, I doubt that the equation of motion can be solved analytically. If the outer shell were stationary, then the motion of the ball is that of a large-amplitude physical pendulum which already cannot be solved analytically. Allowing the outer shell to move would be equivalent to having a movable pendulum support and would make an already intractable solution even more so.

If I understand the OP's solution, it exploits the unstated fact that the surface is frictionless. This simplifies things nicely and would seem to make this exactly equivalent to a large-amplitude physical pendulum.

But that's just my intuition talking.

 

[Chestermiller]

I would start out by trying to quantify the kinematics of the motion.

 

[haruspex]

it exploits the unstated fact that the surface is frictionless

@Hamiltonian299792458 , is the table surface frictionless? That is not stated, but your easy solution assumes it.

 

[Hamiltonian]

@Hamiltonian299792458 , is the table surface frictionless? That is not stated, but your easy solution assumes it.

The question only mentions the ball to be in pure rolling and doesn't say much about the shell in particular.

 

[Hamiltonian]

I found a very similar problem which also states that the horizontal floor is frictionless.

 

[haruspex]

View attachment 276920
I found a very similar problem which also states that the horizontal floor is frictionless.

Yes, it's probably just an omission in the problem statement in post #1. If the shell doesn't slip either then it's just a kinematic question.
I note that the version in post #7 differs in two other ways: it does not specify rolling contact between the balls, and the ball comes to rest.