Beads on a hoop that cause elevation

[michelle15g]

Homework Statement

There are two beads at the top of a hoop that is tied by a string. The beads start moving downward on the hoop on each side. M is the hoop's mass and m represents each bead's mass.
What is that mass ratio that will cause the hoop to elevate.

Homework Equations

T>= (M +2m)g

The Attempt at a Solution

I tried to go about this question but I couldn't finish the equations.
I know that the momentum on the x-axis isn't relevant since the beads cancel each other out.
I'm not sure how to use the hoop as a constraint. I can find the difference in potential energy but I can't seem to connect it to the force - only to the momentum.
I would really appreciate the help.

 

[tiny-tim]

welcome to pf!

hi michelle! welcome to pf!

I'm not sure how to use the hoop as a constraint. I can find the difference in potential energy but I can't seem to connect it to the force - only to the momentum.

assume the hoop is fixed, and use conservation of energy to find the bead-speed at a general angle θ

(i expect you've already done that )

then find the acceleration of the beads,

then the reaction force on the beads …

what do you get? ​

 

[michelle15g]

Thank you very much. My problem was that I kept skipping to the bottom instead of finding the speed at every angle in which case the integral was irrelevant.