physics question?

Question

A single bead can slide with negligible friction on a wire that is bent into a circular loop of radius 15.0 cm. The circle is always in a vertical plane and rotates steadily about is vertical diameter with (a) a period of 0.450 s. The position of the bead is described by the angle theta that the radial line, from the center of the loop to the bead, makes with the vertical. At what angle up from the bottom of the circle can the bead stary motionless relative to the turning circle? (b) Repeat the problem if the period of the circle's rotation is 0.850 s.

Answer 1

The gravity acceleration Ag along the wire has to equal the centripetal acceleration Ac along the wire. Ag = g*sin(theta), and
Ac = w^2*r*cos(theta), where w is the angular rate of rotation in rad/s and r is the horizontal distance from the vertical axis of rotation to the bead.
w = 2*pi/T, where T is the period.
r = R*sin(theta), where R is the radius of the loop.
So we have
Ag = g*sin(theta)
Ac = w^2*R*sin(theta)*cos(theta)
Ac = (2*pi/T)^2 *R*sin(theta)*cos(theta)
Ag = Ac
g*sin(theta) = (2*pi/T)^2 *R*sin(theta)*cos(theta)
g = (2*pi/T)^2*R*cos(theta)
theta = arccos(g/((2*pi/T)^2*R))
Plug in your numbers for R and two values of T and you'll be there.
EDIT: What an interesting problem! It turns out that there is a critical value of T^2/R, 4.0257, above which bead displacement becomes zero. When w^2*R < g (i.e., T^2/R > (2*pi)^2/g) or T^2/R > 4.0257), the argument of the arccos becomes >1. (This happens for the .85 s value of T, for which T^2/R = 4.817.) Physically this means that the loop is turning too slowly to maintain any displacement of the bead from the center.
After some experimentation, I found that the ratio of Ag/Ac approaches (T^2/R)/4.0257 as theta approaches zero (since cos(theta) in the above equations approaches 1). When T^2/R > 4.0257, Ag > Ac for all values of theta, causing theta to converge to zero. When T^2/R < 4.0257, Ag > Ac for all theta > the solution (equilibrium) value theta_eq, and Ag < Ac for all theta < theta_eq, so theta converges to theta_eq.