Elena G.
asked • 05/31/23Math need help. please help solve this question. thank you sir
Show that the force field F(x,y,z)=(y2z3cosx-4x3z)i+(2z3ysinx)j+(3y2z2sinx-x4)k is conservative and find its potential function. Hence find the work done by F(x,y,z) in moving an object from (π,1,3) to (3,4,1) and then to (0,2,0).
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1 Expert Answer
Andrew B. answered • 07/12/24
Tutor
New to Wyzant
Theoretical Physicist
If F is a conservative force then its curl is 0, ∇xF=0, and F=-∇U Where U is the potential function.
First to show that the curl is zero:
∇xF=(∂Fz/∂y - ∂Fy/∂z)i + (∂Fx/∂z - ∂Fz/∂x)j + (∂Fy/∂x - ∂Fx/∂y)k=0
(∂Fz/∂y - ∂Fy/∂z)=(∂[3y2z2sinx-x4]/∂y - ∂[2z3ysinx]/∂z)=(6yz2sinx-6yz2sinx)=0
(∂Fx/∂z - ∂Fz/∂x)=(∂[y2z3cosx-4x3z]/∂z - ∂[3y2z2sinx-x4]/∂x)=(3y2z2cosx-4x3-(3y2z2cosx-4x3))=0
(∂Fy/∂x - ∂Fx/∂y)=(∂[2z3ysinx]/∂x - ∂[y2z3cosx-4x3z]/∂y)=(2z3ycosx-2yz3cosx)=0
Thus ∇xF=0i+0j+0k=0
Now that we have shown that each component of the Curl is 0, we have proven that it is a conservative force and can move on to find the potential.
F=-∇U
We can solve this as a system of differential equations. First, we recognize that:
-∂U/∂x=Fx -∂U/∂y=Fy -∂U/∂z=Fz as that is simply what the gradient demands.
pt 1
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Andrew B.
pt 2 Then we can just integrate each component. U=-∫F (We are breaking up the line integral here, which turns out to be easier in the long run, and is only allowed because our force is conservative, meaning the path of integration is arbitrary, so integrating each component separately should give the same answer, which we will take advantage of). So for each component: U=-∫Fxdx=∫-y2z3cosx+4x3zdx=-y2z3sinx+x4z+f(y,z) U=-∫Fydy=∫-2z3ysinxdy=-z3y2sinx+g(x,z) U=-∫Fzdz=∫-3y2z2sinx+x4dz=-y2z3sinx+x4z+h(x,y) Because we are integrating functions of multiple variables but only integrating with respect to one, the constants of integration are no longer constants but are arbitrary functions of the other variables (for example, since -∂/∂x[f(y,z)]=0, differentiating the first antiderivative would give us back Fx without any complications). Now since these are all solutions for U, they should all be the same, so our goal is to eliminate the ambiguity by determining what f,g, and h must be. After quick examination, it is clear that f(y,z)=h(x,y)=0 and g(x,z)=x4z. With all 3 solutions in agreement, we find that U(x,y,z)=-y2z3sinx+x4z. It's always good to retake the -∇U to see if you get exactly F back (you do, but I'll leave it up to you for extra practice). To find the work done we can take advantage of the Fundamental Theorem of Gradients. W=∫F⋅dl=∫(-∇U)⋅dl=U(a)-U(b) So U(π,1,3)-U(0,2,0)=3π4 Again since work done is independent of the path, we only care about the initial and final positions. (Note: some may anticipate U(b)-U(a) from the FToG, but since work is a scalar, the difference between plus and minus is work done on versus by the system, so as long as you are keeping track of who's doing the work, it's up to your choice of orientation.07/12/24