Nonuniform ball. In the figure, a ball of mass M and radius R rolls smoothly fro

Question

Nonuniform ball. In the figure, a ball of mass M and radius R rolls smoothly from rest down a ramp and onto a circular loop of radius 0.44 m. The initial height of the ball is h = 0.32 m. At the loop bottom, the magnitude of the normal force on the ball is 2.0 Mg. The ball consists of an outer spherical shell (of a certain uniform density) that is glued to a central sphere (of a different uniform density). The rotational inertia of the ball can be expressed in the general form I = MR2, but is not 0.4 as it is for a ball of uniform density. Determine .

Explanation / Answer

force at bottom of an arc

Fn - Mg = Mv^2/r

given Fn = 2Mg

center of mass speed v^2 = gr

r = arc radius

w^2 = v^2/R^2 = gr/R^2

R is the ball's radius

Icom = 2MhR^2/r - MR^2 = MR^2[2*h/r -1]

beta = b

b*MR^2 = MR^2[2h/r - 1]

b = 2*h/r -1

b = 0.4545

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