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Bead on wire with constant velocity

  1. Sep 22, 2011 #1
    1. The problem statement, all variables and given/known data
    A bead, under the influence of gravity, slides along a frictionless wire whose height is given by the function y(x). Assume that at position (x,y) = (0,0), the wire is horizontal and the bead passes this point with a given speed v0 to the right. What should the shape of the wire be (that is, what is y as a function of x) so that the horizontal speed remains v0 at all times? One solution is simply y = 0. Find the other.



    2. Relevant equations



    3. The attempt at a solution
    I started out this problem by considering the initial and final energies of the bead, [itex]E_{i} = \frac{1}{2}mv^{2}_{0}[/itex] and [itex]E_{f} = mgy + \frac{1}{2}mv^{2}[/itex]. From there, I can find the velocity of the bead at a point on the wire to be [itex]v = \sqrt{v^{2}_{0} - 2gy}[/itex]. This is the part where I start to get tripped up. I know that the horizontal component of the velocity (v0) would be equal to [itex]v\cos{\theta}[/itex]. I already have an expression for v, but I'm not quite sure how to represent the cosine in terms of variables I already have. Once I have that, I'm pretty sure I'd be in the clear for solving this one. Any help would be appreciated! Thanks.
     
  2. jcsd
  3. Sep 22, 2011 #2

    PeterO

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    Homework Helper

    If the wire ended at (0,0), the bead would become a projectile, and the horizontal component of the velocity would remain the same, so perhaps if the wire just followed the appropriate parabolic path down that the projectile would have followed???.
     
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