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Bead on a wire, potential energy

  1. Oct 7, 2013 #1
    1. The problem statement, all variables and given/known data
    A particle is constrained to move without friction on a circular wire rotating with constant speed ω about a vertical diameter. Find the equilibrium position of the particle, and calculate the frequency of small oscillations around this position. Find and interpret physically a critical angular velocity, ω = ωc, that divides the particle’s motion into two distinct types.


    2. Relevant equations
    U=-∫Fdr


    3. The attempt at a solution
    I set it up in spherical coordinates. The wire is [itex]\rho^2+z^2=R^2[/itex] and it is being rotated about the z axis (rho lies in x-y plane). I set [itex]\theta[/itex] to be the angle measured around the circle counterclockwise from the rho axis. The forces acting on the particle are gravity (mg) and centripetal force ([itex]-m\rho\omega^2[/itex]). The potential due to gravity is then mgy and the potential due to centripetal force is:

    [itex]\int_{0}^{\rho}m\rho\omega^2 d\rho=\frac{1}{2}m\omega^2\rho^2=\frac{1}{2}m\omega^2R^2cos^2(\theta)[/itex]
    Gravitational potential becomes mgy=mgRsin(θ). Now my total potential energy is:
    [itex]U=mgRsin(\theta)+\frac{1}{2}m\omega^2R^2cos^2(\theta)[/itex]

    I differentiate with respect to θ and set equal to zero:
    [itex]U_\theta=mgRcos(\theta)-m\omega^2R^2cos(\theta)sin(\theta)=0[/itex]
    [itex]gcos(\theta)=\omega^2Rcos(\theta)sin(\theta)[/itex]
    [itex]sin(\theta)=\frac{g}{R\omega^2}[/itex]
    [itex]\theta=arcsin(\frac{g}{R\omega^2})[/itex]

    I'm concerned about this answer. As ω→0, θ diverges. I should get θ → -π/2

    Where did I screw this up? Thank you.
     
  2. jcsd
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