A uniform sphere of radius R, mass M, and moment of inertia (2/5)MR^2 is rolling

Question

A uniform sphere of radius R, mass M, and moment of inertia (2/5)MR^2 is rolling without sliding on a roller coaster. It starts at an unknown height h above the ground at rest. Its diameter may be considered negligibly small compared to the dimensions of the system. The sphere is to go about a vertical loop with diameter D, and with its base at ground level. What is the minimum value of h such that the sphere will go completely about the loop. Note that no credit will be given for simply copying down the solution to the problem of a block sliding without friction along the same roller coaster as we solved in class.

Explanation / Answer

Initial Potential Energy = m*g*h

At top of the loop,

N = 0
m*g = m*v^2/r
v2 = gr.
Minimum kinetic energy at the top = 1/2 m*gr + 1/2*I*w^2
1/2 * m*gr + 1/2* 2/5*mR^2 * V^2/R^2
1/2 * m*gr + 1/5 * mgr
0.7 mgr

Using Energy Conservation,

m*g*h = 0.7m*g*r
h = 0.7*r

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