Serway & Jewett — Circular Motion and Other Applications of Newton's Laws: Problem 48

A single bead can slide with negligible friction on a stiff wire that has been bent into a circular loop of radius \(15.0\text{ cm}\) as shown in Figure P6.48. The circle is always in a vertical plane and rotates steadily about its vertical diameter with a period of \(0.450\text{ s}\). 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. (a) At what angle up from the bottom of the circle can the bead stay motionless relative to the turning circle? (b) What If? Repeat the problem, this time taking the period of the circle’s rotation as \(0.850\text{ s}\). (c) Describe how the solution to part (b) is different from the solution to part (a). (d) For any period or loop size, is there always an angle at which the bead can stand still relative to the loop? (e) Are there ever more than two angles? Arnold Arons suggested the idea for this problem.