Potential Energy Homework: Particle x Separation and Kinetic Energy E

[Parallel]

Homework Statement

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 separate 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)

The Attempt at a Solution

well actually I don't 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 don't know where to plug it,(the force,the potential energy)?

I really need some help

thanks

 

[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

 

[courtrigrad]

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

 

[Parallel]

why are you integrating the potential(I don't understand the physical meaning of it)?

any hints for (b)?

thanks!

 

[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.

 

[Parallel]

maybe I'm wrong,but if you integrate the potential you don't get the force!
if you differentiate the potential you get the force.

 

[AlephZero]

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

 

[Hootenanny]

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)?

 

[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

 

[Hootenanny]

well,the potential energy is zero.

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

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?

 

[radou]

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

 

[Parallel]

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?

yea it's clear now thanks :)

any suggestions for (b) ?

 

[radou]

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

 

[Parallel]

I tried to think about this since yesterday.

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?

thanks for your help guys

 

[Hootenanny]

I would agree with that in general. However, I would make one small change;you should label your r on the RHS something different, such as r'' since r is a variable in your equation.