Calculate the work done by a force

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The discussion revolves around calculating the work done by a force defined as F = b(1-x^2/a^2)j along a rectangular path. The user initially attempts to compute the work using the integral of F dot dr but faces confusion regarding the correct limits and the contributions from each segment of the path. Clarifications are provided on how to define dr for each segment, emphasizing that two segments can be eliminated due to the force's direction. The user questions whether the final answer should be zero, indicating a conservative force, but acknowledges uncertainty about the force's conservativeness. The conversation highlights the importance of correctly substituting values for x and understanding the implications of the path's closed nature on the work done.
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I am asked to calculate the work done by a force as it moves around a path. The force is F = b(1-x^2/a^2)j. The path is a rectangle with coordinates at (0,0); (0,L); (a,L); (a,0). The force moves clockwise around the path beginning at the origin. A diagram is attached.

I know work is the integral of F dot dr.
So for the first path I should have the the force F=b(1-x^2/a^2)j dotted with Lj (the path from the origin to point (0,L)). The integral is thus bL (1 - x^2/a^2) dy with limits from y=0 to y=L. Is this the right approach? If not, can someone please point me in the right direction??
 

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You have an extra factor of L that you don't need (look at the units). The work is the integral of \vec{F}\cdot d\vec{r} along the path. For example, the first segment has d\vec{r} = dy \hat{j}. You have to figure out what d\vec{r} is for each of the four segments.
 
So, for the first segment, dr = dyj. For the second segment, dr = dx i.
For the third segment, dr = -dyj. For the fourth segment, dr = -dx i. Is this correct? Are the limits on my integration correct as well?

Also, should the answer be 0 (closed path, conservative force...not sure if the force is conservative though)?
 
If the answer is zero then the force is conservative, but not all forces are conservative so you can't use that as a check here. (Your dr vectors are correct).

-Dale
 
I get an answer of 2bL (1- x^2/a^2). This does not seem correct to me, since it contains an x^2 term? Is this right? Is there a substitution I can make for x? x=a or x=L, for instance? This problem is driving me crazy...any help greatly appreciated!
 
You can eliminate two of the legs from your problem since the force is in the \hat{j} direction.

In segment 1, x=0. In segment 3, x=a.
 
Last edited:
Yes...I have the forces for the dx direction to be zero. I'm still doing something wrong though?
 
Did you substitute in the values for x that I just edited into my last post?
 

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