Solving Gravitational Force: F_g between Ring & Mass m at Distance x

[carlosbgois]

Homework Statement

What is the intensity of the force F_{g} between the ring and a mass 'm', which is at a distance 'x' from the center of the ring?

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The Attempt at a Solution

I have got to my own answer, but it is different from the given one. Here's what I did:

There's a potential energy between 'm' and a dM from the ring, which is given by dU=-G*m*dM/d and d=\sqrt{r^{2}+x^{2}}.

From this, I can find the total potential energy by integrating dU from 0 to M, which gives me U=-\int^{M}_{0}\frac{G*m*dM}{d}=\frac{-G*m*M}{\sqrt{r^{2}+x^{2}}}

As the variation in the potential energy is equal to the negative of the work done, I did -\int^{0}_{d}F(d)*dd=\frac{-G*m*M}{\sqrt{r^{2}+x^{2}}}\RightarrowF(d)=\frac{2*G*m*M}{(r^{2}+x^{2})^{3/2}}

So, what's wrong? Thanks

 

[carlosbgois]

I think -\int^{0}_{d}F(d)*dd should be -\int^{0}_{d}F(d)*cos\theta*dd, but even this way I didn't get the right answer.

 

[carlosbgois]

Sorry, forgot to post the given answer for the exercise: F = \frac{GMmx}{(r^2+x^2)^{3/2}}