Two beads of mass m are initially at rest at the top of a frictionless hoop of mass M and radius R, which stands vertically on the ground. The beads, being strung on the hoop, are constrained to move along the hoop’s circular path. The beads are given tiny kicks, and they slide down the hoop, one to the right and one to the left. What is the largest value of m/M for which the hoop never rises up off the ground? (Given a sufficiently large ratio of m/M the normal force of the beads on the hoop can indeed cause it to jump off of the ground!) Some thoughts/hints about this problem: (a) It is important to think about the direction of the Normal force as the beads move around the hoop. The hoop is exerting a normal force on the beads which are constraining them to stay on the wire as they move around the hoop. The beads are also exerting a normal force on the hoop which is what will make the hoop jump. (b) What happens to the direction of the normal force exerted by the hoop on the beads as the beads go around? Does it always stay the same direction or does it flip? (c) Define the position of the beads in terms of θ where θ is the angle that the beads have moved. (as is shown in the figure). (d) What type of acceleration happens when you move in a circle? (i.e. What direction of force should you consider for the beads as they move around the circle?) (e) At some point you’ll get something of the form a ≤ f(θ). Think about how to find where the max force is applied just using that f(θ)
Two beads of mass m are initially at rest at the top of a frictionless hoop of mass M and radius R, which stands vertically on the ground. The beads, being strung on the hoop, are constrained to move along the hoop’s circular path. The beads are given tiny kicks, and they slide down the hoop, one to the right and one to the left. What is the largest value of m/M for which the hoop never rises up off the ground? (Given a sufficiently large ratio of m/M the normal force of the beads on the hoop can indeed cause it to jump off of the ground!) Some thoughts/hints about this problem:
(a) It is important to think about the direction of the Normal force as the beads move around the hoop. The hoop is exerting a normal force on the beads which are constraining them to stay on the wire as they move around the hoop. The beads are also exerting a normal force on the hoop which is what will make the hoop jump.
(b) What happens to the direction of the normal force exerted by the hoop on the beads as the beads go around? Does it always stay the same direction or does it flip?
(c) Define the position of the beads in terms of θ where θ is the angle that the beads have moved. (as is shown in the figure).
(d) What type of acceleration happens when you move in a circle? (i.e. What direction of force should you consider for the beads as they move around the circle?)
(e) At some point you’ll get something of the form a ≤ f(θ). Think about how to find where the max force is applied just using that f(θ)
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