Nonuniform cylindrical object . In the figure, a cylindrical object of mass M an

Question

Nonuniform cylindrical object. In the figure, a cylindrical object of mass M and radius R rolls smoothly from rest down a ramp and onto a horizontal section. From there it rolls off the ramp and onto the floor, landing a horizontal distance d = 0.532 m from the end of the ramp. The initial height of the object is H = 0.81 m; the end of the ramp is at height h = 0.15 m. The object consists of an outer cylindrical shell (of a certain uniform density) that is glued to a central cylinder (of a different uniform density). The rotational inertia of the object can be expressed in the general form I = (Beta)MR2, but Beta is not 0.5 as it is for a cylinder of uniform density. Determine Beta.

Explanation / Answer

let moment of inertia of ball be I

conservation of energy:

M*g*.81 = .5*M*v^2 + .5*I*w^2 + M*g*.15

to get v:

.15 = .5g*t^2

t = .174 sec

v*.174 = .532

v = 3.04

put this in equation:

w = 3.04/R

g*.81 = .5*(3.04)^2 + .5*(beta) *(3.04)^2 + g*.15

beta = .399

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