Calc Work: F<y,x,z^3> from P(1,1,1) to Q(2,3,4)

[PsychonautQQ]

Homework Statement

Calculate the work performed by the force field F = <y,x,z^3> from P = (1,1,1) to Q = (2,3,4).

Homework Equations

The Attempt at a Solution

So first I found that F must be a conservative field because ∇(xy + (z^4)/r) = <y,x,z^3>

Since it is a conservative field it is path independent... which means something that my teacher would be mad at me for not knowing (lol)

Can I parameterize the line integral in terms of t?

like
c(t) = <1+t,1+2t,1+3t>
c'(t) = <1,2,3>

∫F dot c'(t) dt = W? then what would be the bounds on my integral?

 

[haruspex]

Since you've found the potential function, why not just take the potential difference?

 

[zahbaz]

^what he or she said. Which is, coincidentally, a very useful result of path independence!

 

[LCKurtz]

c(t) = <1+t,1+2t,1+3t>
c'(t) = <1,2,3>

∫F dot c'(t) dt = W? then what would be the bounds on my integral?

What values of $t$ correspond to the two points?

 

[PsychonautQQ]

0 and 1?
so ∫F dot c'(t) dt = W between 0 and 1 will give me the correct answer?

 

[PsychonautQQ]

how do I put F into terms of t though? like x = cos(t) y = sin(t) and z=z? ;-( I'm confused

 

[LCKurtz]

What values of $t$ correspond to the two points?

0 and 1?

Why do you answer every hint with another question? Aren't you capable of plugging $t=0$ and $t=1$ into your equation of the line and checking whether you get the two points yourself?

so ∫F dot c'(t) dt = W between 0 and 1 will give me the correct answer?

Yes, if you insist on doing the problem the hard way. Had you heeded the advice in post #2 you would have been done with this thread long ago.

 

[PsychonautQQ]

I got the work = 68.75.

f(r(Q) - f(r(P)
f = xy + (z^4)/4
from <1,1,1> to <2,3,4>
so
(2)(3) + ((4)^4)/4 - ((1)(1) + ((1)^4)/4
70 - 1.25
68.75

Is this the correct method for doing this problem?

 

[haruspex]

I got the work = 68.75.

f(r(Q) - f(r(P)
f = xy + (z^4)/4
from <1,1,1> to <2,3,4>
so
(2)(3) + ((4)^4)/4 - ((1)(1) + ((1)^4)/4
70 - 1.25
68.75

Is this the correct method for doing this problem?

Yes. (Btw, you wrote /r instead of /4 in the OP. Adjacent keys, I see.)