Intuition for work and line integrals

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Intuition for work and line integrals

This might seem like an embarrassingly simple question.

It's a situation where I understand the underlying mathematics, but I want to develop the correct physical intuition.

Suppose I have a vector field F that represents a force, and I have a curve C along which a particle travels.

Then I know there's a certain line integral that represents the work done by the force F on a particle moving along the curve C. There may be different ways to write the integral, but roughly speaking, you integrate the dot product of F and T, where T is the tangent vector to C.

The curve C can be any curve -- it doesn't have to have any relationship to F. So roughly speaking, it's like the particle moves along C for some possibly unknown reason -- it's not necessarily the case that F is the only force acting on the particle.

I also know that, roughly speaking, if F points "mostly close to the same direction" as C, then the work is positive. If F points "mostly close to the opposite direction" as C, then the work is negative. (I know in general when the dot product of two vectors is positive and when it's negative.)

When I first thought about this naively, it seemed backwards. Naively, if C goes "mostly close to the same direction" as F, then the particle is going "with the flow", which is easier. In the opposite situation, the particle is going "against the flow", which is harder. However, this must be incorrect intuition on my part.

My textbooks use the wording "the work done by the force". I know that mathematically, the work is positive if the force is pushing mostly close to the same direction that the particle is traveling.

Maybe part of my confusion is that "work" in the sense of physics doesn't have to mean "trouble" or "suffering" or "exerting yourself". I guess it's more along the lines of: there is a force that's being applied in the direction of travel.

Do parts of the above seem reasonable? Can anyone provide other verbal explanations that might help me develop the "right" intuition for this topic?
skullturf

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Re: Intuition for work and line integrals

I think your confusion is with an incorrect intuitive understanding of work. When we say work done by a force, we don't mean the "amount it had to exert itself".. rather, we mean the amount of contribution that the particular force had in displacing the particle. So a force does positive work when it was partially responsible for the displacement that the particle underwent, and it does negative work when it was "fighting against" the overall displacement of the particle. Simply, I think it will make much more intuitive sense if you think of work not as exertion, but more as quantifying the contribution of a force to the final outcome. Positive work if the force was trying to accelerate it towards that outcome, negative work if it was trying to accelerate it away from the outcome, and zero work if it had no impact either way.
Ankit1010

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Re: Intuition for work and line integrals

Cool, thanks. "Work" = "contribution" seems like good intuition.
skullturf

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Re: Intuition for work and line integrals

What might also be contributing to the confusion is that "work" is often used to refer to "work done against a force" as well as "work done by a force" and, obviously these both have opposite signs.
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