Spring on a Hoop: A massless spring with spring constant k and relaxed length ze

Question

Spring on a Hoop: A massless spring with spring constant k and relaxed length zero has one end attached to a point on a fixed frictionless horizontal hoop of radius R. The other end is attached to a bead of mass m that is constrained to follow the hoop. The spring is initially stretched across the diameter with the bead at rest. The bead is given an infinitesimal nudge and the spring pulls it around the loop. What is the normal force in the horizontal plane (so ignoring gravity) that the hoop exerts on the bead at the moment the bead has gone 1/4 of the way around the circle?

Explanation / Answer

Force by the spring = 1.414kR

acts at 45 degree from the horizontal plane.

N = 1.414 kR cos(45) = kR (Ans)

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