Gravitational potential energy bead of mass

[ttja]

Homework Statement

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 v_o 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.

Homework Equations

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

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

Please help (with part a)

 

[Andrew Mason]

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

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

 

[ttja]

Thank you Andrew