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Potential energy of gravitational force

  1. Oct 16, 2008 #1
    1. The problem statement, all variables and given/known data
    A bead of mass m slides along the x-axis between two spheres of mass M equidistant from the x-axis (distance d) and attract the bead gravitationally.

    a. Find the potential energy of the bead.

    b. The bead is released at x = 3d with an initial velocity toward the origin. Find the speed at the origin.

    c. Find the frequency of small oscillations of the bead about the origin.

    2. Relevant equations
    [tex]U = -\int F dx[/tex]

    [tex]F_{grav} = \frac{-GMm}{r^2}[/tex]

    [tex]E = U + K[/tex]

    [tex]K = \frac{1}{2}mv^2[/tex]

    3. The attempt at a solution
    Integrating the force of gravity, I find that the potential energy of the force is [tex]\frac{GMm}{r^2}*{R_{M}}^2*(\frac{1}{R_{M}} - \frac{1}{r})[/tex] where [tex]r[/tex] is the distance straight from [tex]m[/tex] to one of the spheres [tex]M[/tex] and [tex]R_M[/tex] is the radius of [tex]M[/tex]. This is fine, but I'm stuck on how to set the potential in terms of d, which I will need to find the velocity in the next part.

    Conceptually, I know the y component of the gravitational forces cancel out since the spheres are equal mass and distance away. Also, the potential I get for the x component will need to be doubled since there are two masses.
     
    Last edited: Oct 16, 2008
  2. jcsd
  3. Oct 16, 2008 #2
    Wow, once you learn Latex, it's a lot easier to format your equations. I hope that helps the equations a little better to understand.
     
  4. Oct 16, 2008 #3

    LowlyPion

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    Homework Helper

    Isn't the r distance really given by

    [tex] r = \sqrt{x^2 + \frac{d^2}{4}} [/tex]
     
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