2 beads and spring going round a loop

  • Thread starter Gogsey
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  • #1
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4. Two identical beads of mass m slide on a thin frictionless wire bent into a vertical circular hoop of radius R in the
xz plane (with z vertical), centred on the origin. A light spring of spring constant k and unstretched length b is
threaded onto the wire between the two beads, with the beads attached to the ends, so that the spring is bent into an
arc subtending an angle b = b/R when unstretched. Describe the positions by the angles theta1 and theta2 from the origin,
relative to the vertically downwards direction. Write out the lagrangian, and find the differential equations of
motion.

I am seriously lost when it comes to this question.

I've tried writng the distance ftom the negative z axis, but I end up getting different combinations of beta, theta1 and theta2, at different points on the loop.

Are the 2 angles from the x = 0 line?
 

Answers and Replies

  • #2
Avodyne
Science Advisor
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Yes, the angles are from the x=0 line.

What is the kinetic energy of bead #1 in terms of [itex]d\theta_1/dt[/itex]?

What is the kinetic energy of bead #2 in terms of [itex]d\theta_2/dt[/itex]?

What is the gravitational potential energy of bead #1 in terms of [itex]\theta_1[/itex]?

What is the gravitational potential energy of bead #2 in terms of [itex]\theta_2[/itex]?

What is the potential energy of the spring is terms of [itex]\theta_1-\theta_2[/itex]?
 
  • #3
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I'm gussing these energies tkae into account that the spring is compressing and decompressing periodically.

Honestly I have no clue for any of the potential or kinetic energies. I'm guessing we need to use arclength somewhere, but I have no idea how to use or when to use it.
 
  • #4
160
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I can't think how to write then in the terms you listed. All I can think of is

m/2(xdot^2 + ydot^2), but as for using the angles I think you need to know arc length, which I don't, and really just don't know what to do
 

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