I think I'm not understanding some conceptual part of rotational kinematics because all the questions seem connected. I want to figure it out as best I can so please don't solve it but any hints in the right direction would be really appreciated, thanks! The Question: A stiff piece of wire is bent into a circle and mounted (vertically) to rotate as shown. A wooden bead with a hole through its center can slide frictionlessly on that wire. The radius of the wire hoop is 15.0 cm and it rotates steadily with a period of 0.450 s . a) At what angle θ will the bead rotate, along with the hoop, without sliding (up or down)? Show that there are two possible angles (solutions) for this hoop size and rotational speed. Include a clear free body diagram of the bead as part of your solution. b) At what period would the ball not leave the bottom position (θ =0)? c) At what period would θ =90 degrees? What would that angular speed then be? Explain, in terms of forces on the bead, why one can’t spin the hoop fast enough for the bead to reach θ =90 degrees. d) If the hoop were twice the radius, what would be the non-zero angle (at which the bead would rotate)? Relevant Equations: Fg=mg a=mv2/r v=wr w=(2*pi)/T Attempt at solving: a) If the bead is at θ then there is a normal force directed towards the center of the circle and the force of gravity directed downward. Sum of forces in the 'y' gives Ncosθ-Fg=0 and in 'x' is Nsinθ=ma I think the acceleration is rw2 so lots of substitution gives θ=tan-1 (rw2 /g) θ=71.47 degrees This answer seems reasonable but I'm not sure if I skipped something. Also I don't know how to find the second angle. b) I think Fg=N but if θ is zero it seems like acceleration would be zero so period would be zero which makes no sense. c) Normal force would be zero and Fg=mv2/r so g=rw2 and w=2pi/T, after substitution I got: w=8.08 rad/s2 T=0.777 s I think I'm missing the conceptual part of c because my answer seems like normal numbers. d) I imagine this would be solved the same as part a but the question seems to imply it would only have one angle it rotates at, what changed? Thanks again for your time and help!