A Hula-hoop problem with a paradox?

Classical Mechanics Level 4

A Particle of mass \(m\) is attached to a light, rigid circular hula-hoop of radius \(R\). The hoop is placed on a rough surface with the particle initially at the highest point. When the hoop is displaced slightly, find the speed of the particle (in \(m/s\)) when the radius to the particle makes an angle \(\theta\) with the vertical.

A Question to ponder about : Suppose the particle reaches the bottom of the hoop, i.e \(\displaystyle\theta = 180^o\). Then, there is a clear loss in Potential Energy of the particle. But, if the hoop is rolling without slipping on the surface, then the velocity at the contact point is zero. Where has the energy gone?

Details and Assumptions:
\(\bullet R = 30cm\)
\(\bullet \theta = 60^o\)
\(\bullet g = 9.8 m/s^2\)


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