# Potential Energy

1. Dec 2, 2006

### Parallel

1. The problem statement, all variables and given/known data

Particle x is bounded to another particle P by force which is dependent on their distance 'r',the potential derived from this force is:

U(r) = a/r^3 - b/r^2

a,b>0

(a)how much energy is required to seperate the particles?.
(b)suppose particle x has kinetic energy E,how close/far can he get relative to P?(no need to solve the equation,just write it)

3. The attempt at a solution

well actually I dont even have a clue on where to start.
for (a) my only thought is that I need to see what happens when r-->infinity.
but I dont know where to plug it,(the force,the potential energy)?

I really need some help

thanks

2. Dec 2, 2006

### FunkyDwarf

well your on the right track except its not simply the function as it tends to infinity, as that would be zero, but rather the sum of all of em as it tends to infinity. which calculus function does this for you?

-G

3. Dec 2, 2006

You should take $$\lim_{r\rightarrow \infty} \int \frac{a}{r^{3}} - \frac{b}{r^{2}} \; dr$$

4. Dec 2, 2006

### Parallel

why are you integrating the potential(I dont understand the physical meaning of it)?

any hints for (b)?

thanks!

Last edited: Dec 2, 2006
5. Dec 2, 2006

### AlephZero

EDIT - the original version of this was garbage. Apologies!!!

From the definition of potential U, the force on the particle at any point = -dU/dr

Work = force times distance.

Work done in moving from radius r1 to r2 = integral (force.dr) = U(r1) - U(r2).

In both parts of the question, I think you need to know the initial distance between the particles.

Last edited: Dec 2, 2006
6. Dec 2, 2006

### Parallel

maybe I'm wrong,but if you integrate the potential you dont get the force!!
if you differentiate the potential you get the force.

7. Dec 2, 2006

### AlephZero

Yeah, sorry for complete brain failure. See my edited post #5.

8. Dec 2, 2006

### Hootenanny

Staff Emeritus
Let us try an take a less mathematical approach. By definition, what is the potential energy of the particles when they are at an infinite distance apart (i.e. when $r\to\infty$)?

Last edited: Dec 2, 2006
9. Dec 2, 2006

### Parallel

well,the potential energy is zero.

as you get them closer(i.e doing work),you increase their potential energy.

AlephZero:
the intial distance is not given in the problem

10. Dec 2, 2006

### Hootenanny

Staff Emeritus
Correct, so the work you do on the particle would be the change in potential energy, i.e;

$$W = U_{f}-U_{i} = U_{r\to\infty} - U_{r'}$$

Make sense?

11. Dec 2, 2006

The problem text should contain the fact that the initial distance between the particle is r' .

12. Dec 2, 2006

### Parallel

yea it's clear now thanks :)

any suggestions for (b) ?

13. Dec 2, 2006

Hint: the total energy is conserved, see how you could use that.

14. Dec 3, 2006

### Parallel

o.k so the energy is conserved,let's assume that the distance between the particles is R.so becuase energy is conserved:

E + a/R^3 - b/R^2 = a/r^3 - b/r^2
left side is 'Ei',and the right side is Ef(no kinteic energy just potential energy)

is this o.k?