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issisoccer10
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[SOLVED] Bead confined to a circular hoop
A bead with mass m is confined to traveling along a cirlcular hoop with radius r. The bead can slide without friction along this hoop. The hoop, which is vertical, rotates with an angular velocity (omega). What is the angle (theta), measured from the bottom of the hoop and with counterclockwise as positive, in which the bead will not move for this angular velocity?
angular momentum (L) = I * (omega)
torque = dL/d(theta)
Force = torque/R
I have been working on this problem for a solid day and a half and really haven't made too much progress with it. The bead, at this certain value of theta, will basically be rotating in a circle. Then I found the angular momentum (at least what I thought it was) by taking m*(R sin (theta))^2 * (omega). Then I took the derivative of L to get torque and then divided to torque by R, thinking I would be obtaining the force. The only other force that I knew to be at work on bead was the force of gravity. So I set the net force to equal zero and then solved for theta. Yet upon reflection, I realize that the angular momentum has a vertical and horizontal component going out from the center, but i don't know how to find it. But that could be entirely wrong as well.
I have also thought that this scenario seems as if it could be explained by a string attached to the bead and anchored at the origin. As the angular velocity increases, the angle would increase, and the particle would be moving along as if confined to circular hoop like in the original problem. However, I'm not able to figure out where this force in the vertical direction comes into play that cancels out the force of gravity...
I also considered the position of the bead could be described parametrically with x = sin(theta) and y = -cos (theta). Maybe I could differentiate this to get acceleration and then eventually force, but this course of action didn't seem all that logical, so I didn't really pursue it.
All in all I'm pretty lost, but I hope that I've got some bits and pieces of "correctness" in there...but in any case, I would certainly enjoy some guidance on this problem
thanks a lot
Homework Statement
A bead with mass m is confined to traveling along a cirlcular hoop with radius r. The bead can slide without friction along this hoop. The hoop, which is vertical, rotates with an angular velocity (omega). What is the angle (theta), measured from the bottom of the hoop and with counterclockwise as positive, in which the bead will not move for this angular velocity?
Homework Equations
angular momentum (L) = I * (omega)
torque = dL/d(theta)
Force = torque/R
The Attempt at a Solution
I have been working on this problem for a solid day and a half and really haven't made too much progress with it. The bead, at this certain value of theta, will basically be rotating in a circle. Then I found the angular momentum (at least what I thought it was) by taking m*(R sin (theta))^2 * (omega). Then I took the derivative of L to get torque and then divided to torque by R, thinking I would be obtaining the force. The only other force that I knew to be at work on bead was the force of gravity. So I set the net force to equal zero and then solved for theta. Yet upon reflection, I realize that the angular momentum has a vertical and horizontal component going out from the center, but i don't know how to find it. But that could be entirely wrong as well.
I have also thought that this scenario seems as if it could be explained by a string attached to the bead and anchored at the origin. As the angular velocity increases, the angle would increase, and the particle would be moving along as if confined to circular hoop like in the original problem. However, I'm not able to figure out where this force in the vertical direction comes into play that cancels out the force of gravity...
I also considered the position of the bead could be described parametrically with x = sin(theta) and y = -cos (theta). Maybe I could differentiate this to get acceleration and then eventually force, but this course of action didn't seem all that logical, so I didn't really pursue it.
All in all I'm pretty lost, but I hope that I've got some bits and pieces of "correctness" in there...but in any case, I would certainly enjoy some guidance on this problem
thanks a lot