Is F=-kx^2+ax^2+bx^4 Conservative?

In summary, the conversation discusses the concept of conservative forces and how to determine if a force is conservative or not mathematically and theoretically. Different methods such as checking the line integral, finding a potential function, and checking the curl of the force are discussed. The conversation also includes an example of finding the potential function and gradient for a given force equation and a discussion about the physical significance of the gradient. The conversation concludes with a discussion about using the curl to determine if a force is conservative or not, using Stokes theorem.
  • #1
anikmartin
8
0
Hello everyone, I have a question about conservative forces.
I am given a function F = -kx^2 + ax^2 + bx^4, where a, b, and k are constants. I am asked to determine if this force is conservative or not, I don't know know how to prove this mathematically or theoretically. Please help!

In this particular example the force is a spring force, spring forces aren't always conservative are they?
Thanks
 
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  • #2
You can use the fact that Work down by a conservative force in a closed path is zero.
I guess spring forces are always conservative if the spring is ideal.
 
  • #3
anikmartin said:
I am asked to determine if this force is conservative or not, I don't know know how to prove this mathematically or theoretically.
There are several ways to mathematically test that a force is conservative. Here are a few:
(1) You can check that the line integral is zero along any closed path (equivalent to what rhia suggested).
(2) Or you can see if you can find a potential function whose -gradient equals the force function (hint: integrate).
(3) Or you can see if the curl of the force is zero.​
The last two checks should be easy. :smile:
 
  • #4
Hi Doc AI,
Can you please explain the physical significance of the last two checks?
I don't understand things if I can't relate them to physical world.:(
Thanks a lot!
rhia
 
  • #5
Okay I though that if the force was conservative, then the gradient could be used, I didn't know that it proves the force is conservative.

A correction in the force equation

F = -kx + ax^3 + bx^4

After integrating a found the potential equation to be
U = (kx^2)/2 - (ax^4)/4 - (bx^5)/5

using F = -dU/dx.

I found the gradient of U to be

grad(U) = (kx - ax^3 - bx^4)i

Is there a reason for the sign difference? I just learned gradients, in my Calc III class that I am taking right now, two weeks ago so I am still not comfortable with all of its properties.

Okay what if I have a force function of the type:

F = -kx^2y +(y^2)(z^2) + xyz ( I made this one up)

So took three integrals, with respect to x, y, and z seperately, to get the potential function in each direction. Then I found the gradient of the potential function to be

grad(U) = (kyx^2 - (y^2)(z^2) - xyz)i + (k(x^2)y - (y^2)(x^2) - xyz)j + (k(x^2)y - (y^2)(z^2) - xyz)k

To find if the gradient equals the force function, do I find the absolute value of grad(U) , to make it a scalar function? I am not sure how to find if these two are equivalent.

Okay I took a guess to convert the force function into a vector function. First I found the partial derivatives of F with respect to x, y, and z. Then I took the integral with respect to x, y, and z. I found

F = -(k(x^2)y - xyz)i - (k(x^2)y + (y^2)(z^2) + xyz)j + ((z^2)(y^2) +xyz)k

When I took the integral with respect to each I didn't add the Integral constant, so maybe that could account for the difference. According to the scalar function F, when x = O, F = (y^2)(z^2), when z = 0, F = -k(x^2)y . .so these could be the missing constants in my vector function F??
Thanks for all your help.
 
  • #6
I think the simplest way, is the curl.

If [tex]\vec{F} = M(x,y,z)\vec{i} + N(x,y,z)\vec{j} + Q(x,y,z)\vec{k}[/tex], then

[tex] \vec{\nabla} \times \vec{F} = \left| \begin{array}{ccc}
\vec{i} & \vec{j} & \vec{k} \\
\partial/\partial x & \partial/\partial y & \partial/\partial z \\
M & N & Q \end{array} \right| = 0
[/tex]

The reason for this, is the Stokes theorem, which says:

Let be S a surface oriented with a normal vector [tex]\vec{n}[/tex], limited by a closed simple curve C. If [tex]\vec{F}[/tex] is a vectorial field and its partial derivatives are continuous in that region, then:

[tex] \oint_{C} \vec{F}d\vec{r} = \iint_{S} (\vec{\nabla} \times \vec{F})\vec{n}dS[/tex]

--

If the curl is equal to zero, then the right term is zero as well, and the definition of conservative field is that the left integral is equal to zero.
 

1. What does it mean for a force to be conservative?

A conservative force is one in which the work done by the force on an object is independent of the path taken by the object. In other words, the total energy of the system is conserved, meaning that any change in the kinetic energy of the object is balanced by an equal and opposite change in potential energy.

2. How is F=-kx^2+ax^2+bx^4 related to conservative forces?

F=-kx^2+ax^2+bx^4 is the mathematical form of a conservative force. This equation represents a combination of a spring force (F=-kx^2) and a quartic force (F=bx^4). Both of these forces are conservative, meaning that the total force is also conservative.

3. What is the significance of the "+" and "-" signs in the equation F=-kx^2+ax^2+bx^4?

The "+" and "-" signs represent the direction of the force. A negative sign indicates that the force is acting in the opposite direction of the displacement, while a positive sign represents a force in the same direction as the displacement.

4. Can a non-conservative force be represented by the equation F=-kx^2+ax^2+bx^4?

No, a non-conservative force cannot be represented by this equation. This equation only describes conservative forces, which follow the principle of energy conservation. Non-conservative forces, such as friction or air resistance, do not follow this principle and therefore cannot be represented by this equation.

5. How can we determine if F=-kx^2+ax^2+bx^4 is a conservative force?

We can determine if F=-kx^2+ax^2+bx^4 is a conservative force by calculating the work done by the force on an object along two different paths. If the work done is the same for both paths, then the force is conservative. Additionally, we can also check if the force satisfies the conditions for conservative forces, such as being path independent and conserving energy.

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