A single bead can slide with negligible friction on a stiff wire that has been b

Question

A single bead can slide with negligible friction on a stiff wire that has been bent into a circular loop of radius R. The circle is always in the vertical plane and rotates steadily around its vertical diameter with period T (the period is the time to make one complete rotation). The position of the bead is described by the angle that the radial line, from the center of the loop to the bead, makes with the vertical. (a) At what angle up from the bottom of the circle can the bead stay motionless relative to the turning circle? (b) Is there always an angle where the bead is stable? If so, why? If not, why not? (c) Is there ever more than one angle where the bead is stable? If so, describe all stable positions.

Explanation / Answer

at what angle up from the bottom of the circular loop can the bead stay motionless relative to the turning circle when the period of the loops is T1?
normal force = N

Nsin = m(2/T1)2(Rsin), so N = m(2/T1)2R

Ncos = mg

cos = g(T1/2)2/R

cos = g(T2/2)2/R

doesn't exist, so the bead can't be stationary relative to the loop

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