What Happens to X When Force and Total Energy Are Defined by U(x)?

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Homework Help Overview

The problem involves a potential energy function U(x) = a/x^2 + bx, with a focus on determining the force when it is zero and analyzing the conditions under which an object with a given total energy is limited in its position.

Discussion Character

  • Exploratory, Conceptual clarification, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the method of finding where the force is zero by taking the derivative of the potential energy function. Questions arise regarding the relationship between total energy and potential energy, particularly in the context of setting equations equal to each other. Some participants express confusion about their attempts and the correctness of their solutions.

Discussion Status

There is ongoing exploration of the problem with participants sharing their attempts and questioning their reasoning. Some guidance has been offered regarding the interpretation of energy equations, but no consensus has been reached on the correct approach or solutions.

Contextual Notes

Participants are grappling with the definitions of potential and total energy, and there is uncertainty about the correct application of mathematical methods, such as derivatives and solving polynomial equations. The specific values for a and b are provided, but there is a lack of clarity on how to integrate these into the problem-solving process.

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Homework Statement



A potential energy function is given by U (x) = a/x^2 + bx.
a- Find where the force described is 0, in terms of ab and b.
b- Suppose a = 10.0 J/m^2 and b = 2.00 J/m. if an object has a total energy of 20.0 J, for what values of x would it be limited?

Homework Equations





The Attempt at a Solution



I don't understand this. Should I just take the derivative and set it to 0? Then, for part b, set the entire function equal to 20?
 
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(a) Yes.

(b) You're close, and would likely receive most of the available credit by doing that.
It says total energy is 20 J, whereas the U(x) expression is for potential energy. They are not the same thing, so setting them equal is not quite right.
 
Thanks for your help, but I ams till having trouble. For part a, I came up with
x = - ( 2a/b) ^ (1/3).

I think that is right, but I have been working on part b for almost half an hour, and I cannot solve it. I have 10mx^2 - x^3 -mx^2K = 0 where K is the unkown kinetic energy. I don't know how to solve this, or if it is only so difficult because I made a mistake somewhere.
 
This may help for part a:

dU=-f*ds=0

For part b You have the right idea:

U(x)=Umax and solve for x.
 
I'm sorry, I don't understand... my answer for part a is wrong? I thought I should just take the derivative, set it to 0, and solve for x. And for part b, how do I solve it? Do I need to use the quadratic equation or something? I feel like I did it all wrong.
 
-2*x^3-20*x^2+10=0
x^3+10*x^2-5=0
solve for x
hint divide by (x-1)
 
Last edited:

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