# How to check and find potential functions given a force? [closed]

I was given a force, let's say $$F = 15xy$$ N to the right which moves a system from $$A(0,0)$$ to $$B(3,4)$$ and was to find the work done by the force, in Joules.

My approach was by using $$\vec{F} = -\vec{\nabla}U$$ to find the potential energy in $$A(0,0)$$ and the potential energy in $$B(3,4)$$, then using the relationship that $$W = -\Delta U$$ to find the work done.

Although, I have problems on how to find $$U$$ based on the equation $$\vec{F} = -\vec{\nabla}U$$. Based on the problem, I made $$\vec{F} = 15 xy \hat{i}$$ but I have no luck finding the $$U$$.

The choices to the problem (in Joules): $$540, 405, 1080, 1280, 2700$$. I tried another way but the answer was not in the multiple-choice.

• There is no basis vector attached to force? Or is $F = xy \vec{i}$...? Oct 19, 2020 at 7:33
• Do you know the direction of $\vec{F}$? I think you should try the integral formula for work. Oct 19, 2020 at 7:40
• @Buraian The problem only says the force is directed to the right, so I assumed it was in the $\hat{i}$ direction. There was no path given, so I assumed that it's conservative, but I just found out that there's no way for that force to be conservative?
– aco
Oct 19, 2020 at 7:40
• @JulianDeV Yeah, I assumed "right" to be in the direction of $\hat{i}$ but I've got no luck continuing it because the problem doesn't tell me the path the particle took. The previous problems given to me has it but this one doesn't.
– aco
Oct 19, 2020 at 7:44
• I wrote an answer, does it help? Oct 19, 2020 at 7:47

Suppose our force has expression as shown below:

$$F = F_x \vec{i} + F_y \vec{j} + F_z \vec{k}$$

If a potential function $$U$$ exists then,

$$F = -\nabla U = -[\frac{\partial U}{\partial x} \vec{i} + \frac{\partial U}{\partial y} \vec{j}+ \frac{\partial U}{\partial z} \vec{k}]$$

Now, you can compare any one of the coefficients, say we do $$x$$,

$$- \frac{ \partial U}{\partial x} = F_x$$

We can run a definite integral on both sides to get the potential:

$$U = - \int F_x dx + h(y,z)$$

We need to include the extraneous $$h(y,z)$$ term as all function of that form is differentiated to zero when derivated with $$x$$. Keeping it in their lets us fit the potential with other force components. So, we must find $$h(y,z)$$ such that the following system is satisfied:

$$\begin{bmatrix} \frac{ \partial U}{\partial y} \\ \frac{ \partial U}{\partial z} \end{bmatrix} = - \begin{bmatrix} F_y \\ F_z \end{bmatrix}$$

Also, to check that a potential function exists, just need to check that force field is irrational that is:

$$\nabla \times F = 0$$

For this case:

$$( \frac{ \partial}{\partial x} \vec{i} + \frac{ \partial}{\partial y} \vec{j} + \frac{ \partial}{\partial z} \vec{k}) \times (xy \vec{i})$$

Doesn't seem this one is conservative 😥 This problem seems to be unsolvable unless you specify a path. Without a lack of context, I'd assume they mean a straight-line path.

On further thought: The force can be made to be conservative if assume that 'right' means the $$\vec{k}$$ direction i.e:

$$F= xy \vec{k}$$