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 a vertical plane and rotates steadily about its vertical diameter with a period T. The position of the bead is described by the angle theta that the radial line from the center of the loop makes with the vertical.

R=.16m
T1=.44s
T2=.86s
g=9.8m/s^2

1) 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?

2) at T2?

Explanation / Answer

R = 0.16m
T1 = 0.44s
T2 = 0.86s
g = 9.8m/s2

1) 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 = 72.5o 2) at T2? cos = g(T2/2)2/R doesn't exist, so the bead can't be stationary relative to the loop

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