Proving F is Conservative: Steps to Show

[mopar969]

F = a(r to the n)(r hat) Prove F is conservative. Please show steps.

 

[alphysicist]

Hi mopar969,

F = a(r to the n)(r hat) Prove F is conservative. Please show steps.

What have you tried so far?

 

[mopar969]

I know you have to do a substitution with r hat j hat and k hat but I don't know how. Please show steps.

 

[alphysicist]

I know you have to do a substitution with r hat j hat and k hat but I don't know how. Please show steps.

I'm sorry, but the rules of this forum are that you have to make an attempt and show your work; then somebody should be able to help show where you might be going wrong.

 

[mopar969]

Here is what I have done so far. Can you please show me how to finish this problem because it is due at 4 today. And thank you for all the help.

 

[alphysicist]

View attachment 29049
Here is what I have done so far. Can you please show me how to finish this problem because it is due at 4 today. And thank you for all the help.

Now what are the characteristics of a conservative force? If you look in your book, they probably have a list of several things that are true about conservative force fields; what you need to do is to prove that one of those is true. Which property would be easiest to prove is true for this problem?

 

[mopar969]

The teacher wants us to show mathematically that it is conservative so the partial fractions Fx/y must equal the partial fraction of Fy/x etc. Please help me finish this problem and thank you!

 

[alphysicist]

The teacher wants us to show mathematically that it is conservative so the partial fractions Fx/y must equal the partial fraction of Fy/x etc. Please help me finish this problem and thank you!

Okay, so you're almost done. You have written out the force F in terms of x, y, and z.

There are really only two steps left. What is the x component of that force that you wrote out?

And what is its partial derivative with respect to y, which is $$\frac{\partial F_x}{\partial y}$$ ?

That last answer might look a bit messy, but once you do those steps for all three components (you'll end up with a total six derivatives to take) you'll see that they'll all cancel in pairs.

 

[mopar969]

So I don't have to simplify the problem I can take the partial derivatives of what I have. (I am asking because I thought that I had to get one big fraction) Also if I am allowed to take the partial fractions now I can do this for all the requirements for the force to be a conservative force. Please help me finish the problem and thank you for the help.

 

[alphysicist]

So I don't have to simplify the problem I can take the partial derivatives of what I have.

I would simplify your force formula a bit, by combining the two square roots (the one in the numerator and the one in the denominator) into one square root.

(I am asking because I thought that I had to get one big fraction) Also if I am allowed to take the partial fractions now I can do this for all the requirements for the force to be a conservative force. Please help me finish the problem and thank you for the help.

I'm not quite sure what you're asking, but once you have the x component of the force, you can just take its (partial) derivative with respect to y; and once you have the y-component of the force you can take its partial derivative with respect to x. If the force is conservative, those should cancel.

You then check the other four possible derivatives the same way. What do you get for these steps?

 

[mopar969]

How do I combine the square roots because the one in the numerator is to the n.

 

[alphysicist]

How do I combine the square roots because the one in the numerator is to the n.

You have n in the numerator, and one in the denominator, so the total will be n-1. And you can also pull out the square root part into the exponent, so:

$$\frac{(\sqrt{x^2+y^2+z^2})^n}{\sqrt{x^2+y^2+z^2}}\Longrightarrow (x^2+y^2+z^2)^{(n-1)/2}$$

 

[mopar969]

can you explain how you got that. Also how do I do the partial fraction with the n-1 all over 2 as the power of the x^2 +y^2+z^2.

 

[alphysicist]

can you explain how you got that.

From algebra we know that

$$\frac{a^n}{a}=a^{n-1}$$

and also

$$\sqrt{a}=a^{1/2}$$

So we know that:

$$\frac{(\sqrt{x^2+y^2+z^2})^n}{\sqrt{x^2+y^2+z^2}}=(\sqrt{x^2+y^2+z^2})^{n-1}$$

right? This is just saying if we have n radicals on top and 1 in the bottom, together we have a total of n-1.

Then you can rewrite the radical as an exponent:

$$\left\{ \sqrt{x^2+y^2+z^2}\right\}^{n-1}=\left\{ (x^2+y^2+z^2)^{1/2} \right\}^{n-1}$$

and when you combine the exponents you will get the result in my post.

 

[mopar969]

Thank you that makes sense I just wasn't seeing it as being that simple. But my next question is how do I do the partial derivative of the (x^2+y^2+z^2) to the n-1 all over 2 power. Thanks for all the help.

 

[mopar969]

Also I wanted to ask what does this force formula represent or mean. In other words what is it used for?

 

[alphysicist]

Thank you that makes sense I just wasn't seeing it as being that simple. But my next question is how do I do the partial derivative of the (x^2+y^2+z^2) to the n-1 all over 2 power. Thanks for all the help.

Two points here:

1. What is the derivative

$$\frac{d}{dy} y^m$$

equal to?

2. We don't want just the derivative of that factor, we want to take the derivative of the entire x-component of the force. The components are found by matching it to the form:

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

So the square root part is part of the x-component, but what is the entire x-component of the force.

 

[mopar969]

I ended up getting the problem right and got 20 bonus points for my physics class. Thank you a lot and a lot for all the help. Thanks again Josh.

 

[alphysicist]

I ended up getting the problem right and got 20 bonus points for my physics class. Thank you a lot and a lot for all the help. Thanks again Josh.

Sure, glad to help!