Classical Mechanics - Potential Energy Function

  • #1
teme92
185
2

Homework Statement


The potential energy function of a particle of mass m is V(x) = cx/(x2+a2), where c and a are positive constants.

Qualitatively sketch V as a function of x. Find two equilibrium points: identify which is a position of stable equilibrium, and find the period of small oscillations about it.

Homework Equations


PE=0.5mv2=-0.5kx2

The Attempt at a Solution


I think I'm supposed to differentiate and let it equal to zero which gave me:

(c(x2+a2) - 2cx2) / (x2+a2)2 = 0
cx2 + ca2 - 2cx2 = 0
x = a

Putting that into the function gave me: V(x) = cx2/2x2 = c/2

I don't know if any of this is necessary but it doesn't answer the question. Also when it says sketch do I just sub value of x in? Any help would be greatly appreciated.
 

Answers and Replies

  • #2
Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
2021 Award
19,863
10,398
It answers part of the question, i.e., one of the equilibrium points. In order to find the other you must find the second solution to your equation (it exists as it is a second order equation in x).

To know which is stable you must look at the second derivative to deduce which is a min and which is a max.
 
  • #3
teme92
185
2
Hey Orodruin thanks for the reply. How do I find the second solution?
 
  • #4
Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
2021 Award
19,863
10,398
You start from the equation you obtained and solve it for the general case. You have x^2 = a^2. This has two possible solutions.
 
  • #5
teme92
185
2
Ah yes sorry, so the other value for x is -a. But this gives the same solution as it gets squared?
 
  • #6
teme92
185
2
I just realized I should be subbing a in for x so I get V(x) = ±c/2a. So from this how do I tell where the stable equilbrium is? Also how do I find the period?
 

Suggested for: Classical Mechanics - Potential Energy Function

Replies
15
Views
481
Replies
13
Views
804
  • Last Post
Replies
6
Views
314
Replies
2
Views
321
  • Last Post
Replies
14
Views
479
Replies
14
Views
219
Replies
1
Views
323
Replies
13
Views
509
  • Last Post
Replies
1
Views
296
Top