Is the given force equation conservative?

[Calpalned]

Homework Statement

A particular spring obeys the force law $\vec F= (-kx + ax^3 +bx^4)\vec i$. Is this force conservative? Explain why or why not.

Homework Equations

Conservative forces depend on the beginning and endpoints, but not the path taken.

The Attempt at a Solution

The given equation for force clearly depends on $x$, which is distance. Therefore, this force is non-conservative. Additionally, if $\vec F$ depends on $x$, then the graph of F vs dl will vary depending on what value of $x$ is plugged in. Work (which is change in energy) will not be constant, so once again force is not conserved. My answer is incorrect, according to my solutions guide. Where did I mess up?

 

[ShayanJ]

Here's where you messed up:

Conservative forces depend on the beginning and endpoints, but not the path taken.

This is wrong. Conservative forces are forces that their work on a particle in moving it from one point to another, doesn't depend on the path taken and only depends on the two points. This is equivalent to saying that a force is conservative if and only if $\oint \vec F \cdot \vec{dl}=0$ for all closed paths. It can be proved that this is equivalent to $\vec \nabla \times \vec F=0$. So the latter equation is a good way of finding out whether a force field is conservative or not.

 

[Calpalned]

Here's where you messed up:

This is wrong. Conservative forces are forces that their work on a particle in moving it from one point to another, doesn't depend on the path taken and only depends on the two points. This is equivalent to saying that a force is conservative if and only if $\oint \vec F \cdot \vec{dl}=0$ for all closed paths. It can be proved that this is equivalent to $\vec \nabla \times \vec F=0$. So the latter equation is a good way of finding out whether a force field is conservative or not.

What does the symbol integral with circle mean?

The second part of the question asks me to find the potential energy function, which is the integral of $\vec F$. If force is conservative, then the potential energy function is equal to zero right?

 

[ShayanJ]

What does the symbol integral with circle mean?

It means the line integral is taken over a closed path.

The second part of the question asks me to find the potential energy function, which is the integral of $\vec F$ . If force is conservative, then the potential energy function is equal to zero right?

No. To obtain the potential energy from the force field, you should use $\phi(\vec r)=-\int_{\vec{r}_{ref}}^{\vec r} \vec F \cdot \vec{dl}$ where $\phi(\vec r _{ref})$ is defined to be zero, i.e. the point where potential becomes zero is arbitrary.

 

[mattt]

In one dimension (as it is this case) a force $\vec{F}$ is conservative if and only if there is a function $\phi(x)$ such that $\vec{F} = -\frac{d}{dx}\phi(x) \vec{i}$

In your example the force is a continuous function of the position $x$ so the above is satisfied (i.e. the force is conservative).