Need help with graphs of potential energy and force

[Spyridoula]

Hi everyone!

Homework Statement

The potential energy U of a point mass is U(x)= U_ox_o x/(x² + x_o²) where U_o and x_o are positive contants, U_o in joule and x,x_o in meters. Draw the graphs of U(x) and F(x).

Homework Equations

F(x) = -dU/dx

The Attempt at a Solution

I found the F(x) function and made the U(x) graph but I'm stuck with the F(x) graph and I'm not totally sure for anything that I found so far.

Thank you!

 

[haruspex]

Please post what you got for the F(x) function. For the graphs, either post your attempts or at least a verbal description.

 

[Spyridoula]

For the F(x) I found that it is V₀x₀ (x+x₀)(x-x₀) / (x² + x₀²)²

I can't make a graph out of it. As for the U(x) the graph is in the attachment below.
U(x)=0 when x=0 and Vmax=V₀/2 when x=x₀

 

[haruspex]

When sketching a graph, you should take it far enough to illustrate asymptotic behaviour. What is that for U(x)?
I don't understand why cannot get a graph for F. What tool did you use to get the graph for U? You ought to be able to sketch graphs from the formula without using a plotting tool. That's how you know which sections are interesting to plot.
Can you at least describe in words what F(x) would look like by inspection of the U(x) graph? Say what the asymptotic values are, over what ranges it would be positive, where the zeroes are, and all extrema.

 

[Nickie]

Hi there!

To draw the graph of F(x), we can use the equation F(x) = -dU/dx. This means that the force at any point x is equal to the negative derivative of the potential energy function U(x).

To find the derivative, we can use the power rule. The derivative of U(x) is:

dU/dx = Uo(x2 - xo2)/(x2 + xo2)2

We can simplify this to:

dU/dx = Uo(1 - (xo2)/(x2 + xo2)2)

Now, to plot the graph of F(x), we need to plug in different values of x and find the corresponding values of F(x). We can use a table to do this, or we can use a graphing calculator.

Once we have the values, we can plot them on a graph with F(x) on the y-axis and x on the x-axis. This will give us the graph of F(x) as a function of x.

Keep in mind that the graph of F(x) will have the same shape as the graph of U(x), but the values will be different. Also, remember that the units for F(x) will be newtons (N) since force is measured in newtons.

I hope this helps! Let me know if you have any further questions. Good luck with your graphs!