Potential energy of gravitational force

[diablo2121]

Homework Statement

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.

Homework Equations

$$U = -\int F dx$$

$$F_{grav} = \frac{-GMm}{r^2}$$

$$E = U + K$$

$$K = \frac{1}{2}mv^2$$

The Attempt at a Solution

Integrating the force of gravity, I find that the potential energy of the force is $$\frac{GMm}{r^2}*{R_{M}}^2*(\frac{1}{R_{M}} - \frac{1}{r})$$ where $$r$$ is the distance straight from $$m$$ to one of the spheres $$M$$ and $$R_M$$ is the radius of $$M$$. 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.

 

[diablo2121]

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.

 

[LowlyPion]

Isn't the r distance really given by

$$r = \sqrt{x^2 + \frac{d^2}{4}}$$