Show that the following force is conservative

[nbram87]

Homework Statement

Fx = K(2x + y), Fy = K(x + 2y)

Homework Equations

The Attempt at a Solution

I think what is confusing me is that it is two different forces (Fx and Fy). I know that the curl has to be zero for it to be conservative, and I am assuming I will have to figure out a value for the constant K for that too happen.

 

[gneill]

Fx and Fy represent two components of a vector, so they describe a vector field. It might be written as:

$\vec{F} = Fx\;\hat{i} + Fy\;\hat{j}$

How would you form the curl of that?

 

[nbram87]

I think that is one thing that is confusing me. How else could you determine that the force is conservative? Would you have to determine the work done by both Fx and Fy are equal to 0?

 

[gneill]

I think that is one thing that is confusing me. How else could you determine that the force is conservative? Would you have to determine the work done by both Fx and Fy are equal to 0?

You could show that the work done in moving a particle along any closed path is zero (start at point P, traverse all possible paths (!) ending again at point P). The curl looks like the easiest approach.

 

[nbram87]

When you do the curl of Fx and Fy, I think the constant K becomes useless because it equals to zero. What is the meaning of K in the problem then?

 

[gneill]

When you do the curl of Fx and Fy, I think the constant K becomes useless because it equals to zero. What is the meaning of K in the problem then?

I don't understand your meaning. How does K become zero? Can you show your curl calculation?

 

[nbram87]

Curl = d/dx(Fy) i - d/dy (Fx) j
= d/dx [K(x + 2y)] + d/dy [K(2x + y)]
= K(1+ 0) - K (0+1)
= 0
So K - K = 0?

 

[gneill]

Curl = d/dx(Fy) i - d/dy (Fx) j
= d/dx [K(x + 2y)] + d/dy [K(2x + y)]
= K(1+ 0) - K (0+1)
= 0
So K - K = 0?

That tells you that the curl is zero no matter what value K has.

 

[nbram87]

Is it correct? Is my calculation of the curl and the value of K being meaningless correct?

 

[gneill]

Is it correct? Is my calculation of the curl and the value of K being meaningless correct?

The curl calculation result is correct. K is not "meaningless" (it's a scaling constant for the magnitude of the force, and likely makes the force equation units balance). It simply turns out to be irrelevant to the question of conservation.

 

[nbram87]

Ok thank you.