Gravitational potential energy bead of mass

1. Oct 21, 2009

ttja

1. The problem statement, all variables and given/known data
A bead of mass m slides without friction on a smooth rod along the x-axis. The rod is equidistant between two spheres of mass M. The spheres are located at x=0 , y= $$\pm$$ a.

a. Find the potential energy of the bead.

b. The bead is released at x = 3a with an initial velocity vo toward the origin. Find the speed as it passes through the origin

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

2. Relevant equations

F= GMm/(r^2)
U = $$\int$$F(r)dr

3. The attempt at a solution

First i found the net gravitational force for the mass m at a point, d, which equals: 2*GMm/(r^2) cos theta.

I thought cos theta to be d/r. Therefore, i have the final equation for force: 2*GMm/(r^2)*(d/r).

d = sqrt(r^2 - a^2)

F = 2*GMm/(r^2)*sqrt(r^2 - a^2)/r

taking the derivative from r to a where the variable is the radius from m to M, i got:

2GMm int( sqrt((r^2 - a^2)/(r^3)), r, r, 0)

= 2GMm*[ arctan( a / sqrt( r^2 - a^2 ))/(2a) - sqrt( r^2 - a^2 )/(2r^2) ]

Now, i dont know if this is even remotely correct, since i cannot get a potential energy of 0 when r = a.

2. Oct 21, 2009

Andrew Mason

What is the net force in terms of x? (hint: $\cos \theta = \frac{x}{r} = \frac{x}{(x^2+a^2)^{1/2}}$)

Then calculate:

$$\int_{x=3a}^{x=0}Fdx$$

to find the work done by the gravitational force.

AM

3. Oct 21, 2009

ttja

Thank you Andrew