Physics question (Challenge)?

Question

Consider a thin circular disk of radius R and mass M. Imbedded in the material of the disk, at a distance r from the center, is a mass m. Imagine that the disk is placed on a ramp inclined at angle THETA from the horizontal. What is the maximum distance, as measured along the ramp, the disk can roll UP the incline under gravity alone? Simplifications:
Consider only a disk allowed to move slowly without slipping.
Consider m>>M.
For definiteness, give the answer for these parameters:
THETA = 30°. R = 7.500, r = 6.800 (Three decimals ought to do it)
The first correct answer gets 10.

Answer 1

Let u be the angular position of the mass m. The height of the center of the disc is:

Ru*sin(theta) + constant

and the height of the mass m is:

Ru*sin(theta) + r*sin(u)

which makes the total potential energy:

V = g(M+m)Ru*sin(theta) + gmr*sin(u)

In other words, a constant times:

A*u + sin(u)

where:

A = ((M+m)/m)(R/r)sin(theta)

The disc can roll upward only when the force:

F = -dV/du = -(A+cos(u))

is positive. That is, only between the two values for which:

cos(u) = -A

which are:

arccos(-A) , 2pi-arccos(-A) , and which have an angular difference:

2(pi - arccos(-A))

This corresponds to a distance along the ramp of:

2R(pi - arccos(-A))

Using the values given, A = .5515 , so this appears to be:

14.8

Answer 2

if there are no other forces acting the wheel should go on spinning upwards forever because of the law of conservation of energy, the potential energy is getting converted into angular kinetic energy which again gets converted to potential energy.

Answer 3

(5/6) Pi * R

= 19.635