Matching Lagrange and Newton's equations for rolling mass system

[rcummings89]

Homework Statement

Please see the attached picture for the problem description. Now, I have a solution using the Lagrange method, (it should coincide with Newton's second law, I believe?) I just have a hard time getting my equations of motion to match.

Homework Equations

∑F_ext - M dv/dt = 0

The Attempt at a Solution

For mass M_o I have
K(x₂ - x₁) = M_o*x₁'' x (where mass M_o moves a distance x₁ and mass M moves a distance x₂) which agrees with the Lagrange solution

I have trouble matching the remaining equations. For the rolling mass m, I have the external forces as -mg (y-dir) and F_N (y'-dir) where x' and y' are at an angle θ depending on the location of the rolling mass.

Then my equation of motion is -mg y + N y' = m[x'' x + (R-r)θ'' y' - (R-r)θ'² x']

Now I realize that I need to organize everything into the x-y or x'-y' plane but when I do the solution doesn't match; do you see anything that I'm missing?

Finally for mass M

∑F_ext = -mg y - K(x₂ - x₁) x
My professor typically neglects gravity and normal forces acting on carts, so I don't include them here. But I feel like I'm neglecting a force. Should I include the momentum of the rolling mass with the external forces acting on M? Because setting what I have = M*x₂'' x doesn't match, and I feel like I'm over-simplifying the problem.

Thanks in advance

 

[SteamKing]

No attached picture.