Classical Mechanics - Potential Energy Function

In summary, the potential energy function of a particle of mass m is V(x) = cx/(x2+a2), where c and a are positive constants. To find the two equilibrium points, we differentiate the function and set it equal to zero, resulting in x=a and x=-a. Substituting these values into the function, we get V(x)=c/2a. To determine the stable equilibrium, we must look at the second derivative of the function. The period of small oscillations can be found by solving the second order equation in x.
  • #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.
 
Physics news on Phys.org
  • #2
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
Hey Orodruin thanks for the reply. How do I find the second solution?
 
  • #4
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
Ah yes sorry, so the other value for x is -a. But this gives the same solution as it gets squared?
 
  • #6
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?
 

What is potential energy in classical mechanics?

Potential energy is a type of energy that is associated with the position or configuration of an object in a physical system. It is the energy that an object has due to its position relative to other objects or forces in the system.

What is the potential energy function in classical mechanics?

The potential energy function is a mathematical representation of the potential energy of a system. It is a function that takes into account the position, mass, and other properties of the objects in the system in order to calculate the total potential energy of the system.

How is potential energy different from kinetic energy?

Kinetic energy is the energy that an object has due to its motion, while potential energy is the energy that an object has due to its position. They are two different forms of energy, but they are related to each other and can be converted from one form to the other.

What is the formula for calculating potential energy?

The formula for calculating potential energy in classical mechanics is PE = mgh, where m is the mass of the object, g is the acceleration due to gravity, and h is the height of the object. This formula can be used for objects near the Earth’s surface, but for more complex systems, a different potential energy function may be used.

How is potential energy used in real-world applications?

Potential energy is a fundamental concept in physics and is used in many real-world applications. It is used in fields such as engineering, mechanics, and thermodynamics to analyze and design various systems. For example, potential energy is used in the design of roller coasters, dams, and other structures that rely on gravitational potential energy. It is also used in the study of celestial bodies, such as planets and satellites, and their orbits.

Similar threads

  • Introductory Physics Homework Help
Replies
10
Views
616
  • Introductory Physics Homework Help
Replies
15
Views
300
  • Introductory Physics Homework Help
Replies
13
Views
2K
  • Introductory Physics Homework Help
Replies
1
Views
682
  • Introductory Physics Homework Help
Replies
2
Views
1K
  • Introductory Physics Homework Help
Replies
1
Views
808
  • Introductory Physics Homework Help
Replies
23
Views
286
  • Introductory Physics Homework Help
Replies
6
Views
990
  • Introductory Physics Homework Help
Replies
6
Views
656
  • Introductory Physics Homework Help
Replies
4
Views
2K
Back
Top