Solving Potential Energy: Force & Potential Exam Ques.

In summary, the conversation discusses how to show that the given force F is conservative and how to find the potential energy function V(x,y). It is mentioned that using the curl of F can easily show its conservativeness, but finding the potential function from the force is not as straightforward. It is suggested to take the negative gradient, although this may not be correct for the specific potential given in the question. It is also mentioned that a conservative field has the property that the work done along a path equals the change in potential, making it possible to find the potential function if the work is known.
  • #1
thenewbosco
187
0
Just studying for an exam and the following question appeared on the sample exam:
Given the force: [tex]F=-c(x-y)^2(\hat{i}-\hat{j})[/tex] where i and j are the unit vectors.
a) Show the force is conservative
b) Show the potential energy is given by [tex]V(x,y) = \frac{c}{3(x-y)^3)}[/tex] assuming V(0,0) =0.


So for a) it is simple to show using the curl of F. but for b i am not sure how to get the potential energy function given the force. [tex]F=-\nabla V[/tex] will give the force easily if i have the potential function, but I am not sure how to go the other direction. perhaps for the exam to show it i could just take the negative gradient, (which appears to be wrong for the potential given in this question) but i would like to just know how to go the other way for knowledge. thanks
 
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  • #2
if a field is conservative, then the work done against the field on a object from point a to point b is conserved and equals to the change in potential. Well, basically, if you know the work done alone a path, you know the potential function.
 
  • #3


a) To show that the given force F is conservative, we need to show that the curl of F is equal to zero. We can do this by taking the partial derivatives of each component of F with respect to the other variable:

∂F_x/∂y = -2c(x-y)

∂F_y/∂x = 2c(x-y)

Since these two partial derivatives are equal, we can conclude that the curl of F is equal to zero, and therefore F is a conservative force.

b) To find the potential energy function V(x,y) from the given force F, we can use the formula V(x,y) = -∫F⋅dr, where r is the position vector (x,y). Integrating the given force F along a path from the origin (0,0) to a point (x,y) will give us the potential energy at that point.

∫F⋅dr = ∫-c(x-y)^2dx + ∫c(x-y)^2dy

= -c∫(x-y)^2dx + c∫(x-y)^2dy

= -c∫(x^2-2xy+y^2)dx + c∫(x^2-2xy+y^2)dy

= -c(x^3/3 - xy^2 + x) + c(x^2y/2 - y^3/3 + y)

= -cx^3/3 + cxy^2 - cx + cx^2y/2 - cy^3/3 + cy

= -cx^3/3 + cxy^2 + cx^2y/2 - cy^3/3

= c(-x^3/3 + x^2y/2 + xy^2/2 - y^3/3)

= c(x^2y - xy^2)/2 - c(x^3 + y^3)/3

= c(x^2y - xy^2)/2 - c(x^3 + y^3)/3 + c(x^3 + y^3)/3

= c(x^2y - xy^2)/2 + c(x^3 + y^3)/3

= c(x^2y - xy^2)/2 + c(x^3 + y^3)/3 + C

= V(x,y
 

Related to Solving Potential Energy: Force & Potential Exam Ques.

1. What is potential energy?

Potential energy is the stored energy an object possesses due to its position or state. It is often associated with the force of gravity and can be converted into kinetic energy when the object is in motion.

2. How is potential energy related to force?

Potential energy and force are directly related. The force acting on an object is equal to the negative of the derivative of its potential energy with respect to its position. In simpler terms, the greater the potential energy, the greater the force acting on the object.

3. How is potential energy calculated?

Potential energy can be calculated using the formula PE = mgh, where m is the mass of the object, g is the acceleration due to gravity, and h is the height or distance from the reference point.

4. Can potential energy be negative?

Yes, potential energy can be negative. This occurs when the object has a lower potential energy at a certain position compared to a reference point. For example, if an object is dropped from a higher height, its potential energy will decrease and become negative as it falls towards the ground.

5. How is potential energy different from kinetic energy?

Potential energy is the energy stored in an object, while kinetic energy is the energy an object possesses due to its motion. Potential energy can be converted into kinetic energy and vice versa, but they are not the same type of energy.

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