Setting up a centrifugal force question

[rn433]

Suppose we have a merry-go-round of radius 10 meters. There's a thin glass tube with cross-sectional density d going radially from the center of the merry-go-round to the edge. In the glass tube one meter away from the center, there's a marble of mass m. The merry-go-round completes one counter-clockwise rotation every 2pi seconds, and the glass tube is frictionless. At time t, the marble is at rest. I want to find the trajectory of the marble.

I'm taking the center of the merry-go-round to be at the origin and the marble's initial position to be at (0, 1); obviously its initial velocity is 0. Since the tube rotates counter-clockwise every 2pi seconds, I can define the marble's position at t as <r(t)cos(t), r(t)sin(t)>. What I'm having trouble with is finding the force acting on the marble. I know the normal force on the marble from the glass tube is in the direction <-sin(t), cos(t)>, perpendicular to the tube. However, I don't know how to determine its magnitude. I initially assumed it was constant, but this leads to an incorrect answer.

 

[RoyalCat]

It is not constant as the radial displacement of the marble is constantly changing. You have a long exercise in calculus ahead of you, since you can only tell what the tangential and radial components of the force are, and you'll have to work your way up from there to get a function of $$\theta (r)$$

I'm pretty sure you can't solve this problem without using the rotating frame of reference, which would mean you have to refer to both the fictitious centrifugal force (Directed radially outwards) and the fictitious Coriolis force (Directed tangentially in the direction of the spin) as well as the real normal force (Directed tangentially against the direction of the spin).

 

[rn433]

It is not constant as the radial displacement of the marble is constantly changing. You have a long exercise in calculus ahead of you, since you can only tell what the tangential and radial components of the force are, and you'll have to work your way up from there to get a function of $$\theta (r)$$

I'm pretty sure you can't solve this problem without using the rotating frame of reference, which would mean you have to refer to both the fictitious centrifugal force (Directed radially outwards) and the fictitious Coriolis force (Directed tangentially in the direction of the spin) as well as the real normal force (Directed tangentially against the direction of the spin).

How could I do this, though? I don't know how to determine the magnitude of those forces, either. :(