Work Done from (1,0,0) to (1,0,1) in Conservative Force Vector F

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In summary, the conversation discusses finding the work done from (1,0,0) to (1,0,1) using a force vector F = (x^2 + y)i + (y^2 + x)j + ze^z k). The question gives multiple paths to integrate, but the speaker uses the curl to determine that the force is conservative and thus the work done is path independent. They create a function of x,y,z based on F and continue to build it by taking partial derivatives with respect to x,y, and z. However, there is a discrepancy in the final result and the speaker is unsure of where they went wrong. Another speaker suggests integrating along a straight line to find the work done, which
  • #1
Mindscrape
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I have a force vector that is [tex]F = (x^2 + y)i + (y^2 + x)j +ze^z k)[/tex] and I am supposed to find the work done from (1,0,0) to (1,0,1). The question gives a bunch of paths to integrate, but I used the curl and found that the force was conservative (hence path independent), so I was going to make a function of x,y,z based on F.

[tex]\frac{\partial f}{\partial x} = x^2 + y[/tex]

[tex]f(x,y,z) = (1/3)x^3 +xy+g(y,z)[/tex]

[tex]\frac{\partial f}{\partial y}=x+\frac{\partial g}{\partial y} = y^2 +x[/tex]

[tex]\frac{\partial g}{\partial y} = y^2[/tex]

[tex]g(y,z)= (1/3)y^3 +h(z)[/tex]

so now continue to build the function

[tex]f(x,y,z)=(1/3)x^3 +xy+(1/3)y^3 + h(z)[/tex]

[tex]\frac{\partial f}{\partial z} = 0 + \frac{\partial h}{\partial z} = ze^z[/tex]

solve by parts to get

[tex]h(z)=ze^z - \int e^z dz = e^z(z-1)[/tex]

[tex]f(x,y,z)=(1/3)x^3 + xy+(1/3)y^3 + e^z(z-1)[/tex]

Now I know that the last part is the part that I screwed up because [tex]f_z[/tex] is supposed to equal ze^z, and it doesn't. What I can't figure out is what I screwed up. Anyone see it?
 
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  • #2
No I get the same answer as you, dx=dy=0, so the only term left in [tex]\int_{1,0,0}^{1,0,1}{F.dl}[/tex] is [tex]\int_{0}^{1}{z e^{z} dz}[/tex], which gets you [tex][e^z(z-1)]_{0}^{1} = -1[/tex]. Why should it be ze^z? df is an element of work done, not dF.
 
  • #3
Don't you have to regain F with its respective partials? Taking the partial of the function with respect to x gives [tex]x^2 + y[/tex], which is the x (or M) term; with respect to y gives [tex]y^2 + x[/tex], which is the y (or N) term; yet, with repsect to z it is [tex]e^z+ze^z-e^z[/tex] (errr... whoops I just realized it did work out - for some reason I assumed it didn't).

It would give +1 though, since evaluating gives 0--1=1.
 
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  • #4
Since you know the force is conservative, you know the work done is independent of the path. Integrate F along the straight line from (1, 0, 0) to (0, 0, 1): x= 1, y= 0, z= t. dx= 0, dy= 0, dz= dt so the integral is
[tex]\int_0^1 te^t dt[/tex].
 

Related to Work Done from (1,0,0) to (1,0,1) in Conservative Force Vector F

1. What is work done?

The work done is the measure of energy transfer that occurs when a force is applied to an object and causes it to move a certain distance in the direction of the force. It is calculated by multiplying the magnitude of the force by the displacement of the object.

2. What is a conservative force?

A conservative force is a type of force that does not depend on the path taken by an object, but only on its initial and final positions. This means that the work done by a conservative force is independent of the path taken by the object.

3. How is work done calculated for a conservative force?

For a conservative force, the work done is equal to the negative change in potential energy. Mathematically, it is represented as W = -ΔU, where W is the work done and ΔU is the change in potential energy.

4. What is the significance of the points (1,0,0) and (1,0,1) in the given scenario?

The points (1,0,0) and (1,0,1) represent the initial and final positions of the object in the x, y, and z coordinates, respectively. These points are used to calculate the change in potential energy and, therefore, the work done by the conservative force.

5. How is work done from (1,0,0) to (1,0,1) calculated in a conservative force vector F?

The work done from (1,0,0) to (1,0,1) in a conservative force vector F can be calculated by finding the potential energy at each point and then taking the difference between the final and initial potential energies. This difference represents the work done by the conservative force.

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