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Circular Motion and Gravitation: bead can slide without friction on a circular hoop

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A bead can slide without friction on a circular hoop with a radius of .100 m in a vertical plane. The hoop rotates at a constant rate of 3.00 rev/s about a vertical diameter.

a) find the angle &#952; at which the bead is a vertical equilibrium.

b) is it possible for the bead to "ride" at the same elevation as the center of the Hoop?

c) what will happen if the hoop rotates at 1 rev/s?

https://brainmass.com/physics/rotation/circular-motion-gravitation-friction-circular-hoop-8588

Solution Preview

a.)
Radius of hoop (R) = 1.00 m (I think .100m is typo, but any way! I have solved for both values.)
angular rotation of hoop (w)= 2*pi*f

pi = 22/7, f = rev/sec = 3.00, w == omega (angular frequency)
therefore,
w = 2*pi*3 = 18.85 rad/sec
see attached file.