GPE and gravitational force exerted by a ring

[GwtBc]

Homework Statement

Consider a homogeneous thin ring of mass 2.5 x 1022 kg and outer radius 3.9 x 108 m (the figure). (a) What gravitational attraction does it exert on a particle of mass 69 kg located on the ring's central axis a distance 3.7 x 108 m from the ring center? (b) Suppose that, starting at that point, the particle falls from rest as a result of the attraction of the ring of matter. What is the speed with which it passes through the center of the ring?

Homework Equations

U = - GMm/r
F = GMm/r^2

The Attempt at a Solution

[/B]
Seeing as the ring is uniform and the mass is sitting on it's central axis, the first part is simple enough, I think, it's just:

F = GMm/r^2 cos(arcsin(R/r))

where r is the distance from the mass to each dm on the ring and R is the radius of the ring.
I'm not sure about the second part. I currently have:

$W = \int_{R}^{r_{i}} \frac{GMm}{r}\cos (\arcsin(\frac{R}{r})) dr$
$KE_{f} = W \Rightarrow \frac{1}{2}mv_{f}^2 = W$

Where $r_{i}$ is the initial distance of the mass from the ring (the ring itself, not it's centre of mass)

but using my numbers, this returns $v_{f} = 8.03568 * 10^5$ which seems a bit off.

any help is appreciated.

 

[gneill]

We'd need to see how you carried out the work integral. It looks like it could be nasty.

Perhaps it would be easier to find the gravitational potential energy of the system at the initial and final locations and take the difference?

 

[GwtBc]

We'd need to see how you carried out the work integral. It looks like it could be nasty.

Perhaps it would be easier to find the gravitational potential energy of the system at the initial and final locations and take the difference?

It is nasty, but my calculator doesn't mind that. Also the denominator 'r' terms is supposed to be 'r^2' in the integral.

Aren't I essentially finding the initial and final GPEs and then taking the difference? How else can I find the GPE?

 

[gneill]

Since PE is a scalar you don't have to worry about vector components or integrating the contributions. Instead, you know that all the dm's are the same distances from the given location in each case. So you should be in a position to write an expression for the GPE at each location by inspection. For example, at the ring's center all the ring's mass is located at the same distance r. So the GPE there must be GMm/r.