If the curve a particle moves in when calculating work is dependent on the force itself, why don't we find this path before doing work calculations?

Question

I am having some confusion getting my head around work done by a general force on a curve. Lets say I had some mass in space so we aren't considering gravitational potential, if I apply some force that makes the particle move in a curve how would I calculate the work done?

Would I need to find out the curve that the force creates? And then take the line integral of the force with that curve, also does this mean no other curve makes physical sense with this force since the force couldn't cause the particle to move in multiple different curves?

When in a force field I think I understand these questions since I am applying a force against the field to make it move, or the field is making the particle move. But in the case of no field and an applied force I don't understand how there can be a generic path surely the path is determined by the force or is it dependent on initial conditions i.e the velocity and position of the particle initially?

In summary I think I am asking isn't the curve a particle moves in when calculating work dependent on the force itself? Why don't we ever find this path before doing work calculations?

Answer 1

if I apply some force that makes the particle move in a curve how would I calculute the work done?

The work $W$ done by any force $\mathbf F$ along some path $C$ is, by definition, the line integral of that force along the path:

$$W=\int_C\mathbf F\cdot\text d\mathbf x$$

Would I need to find out the curve that the force creates? And then take the line integral of the force with that curve, also does this mean no other curve makes physical sense with this force since the force couldn't cause the particle to move in multiple different curves?

You need to find the path the particle moves along, yes. If the force you are finding the work of is the only force acting on the particle, then the path the particle moves along is the curve (trajectory) the force creates. Of course there could be other forces in general, and so you would need to know the trajectory of the particle due to all of the forces before you can find the work done by any particular force along this path.

In summary I think I am asking isn't the curve a particle moves in when calculating work dependent on the force itself?

Yes, the trajectory is dependent on all of the forces (the net force) acting on the particle. This is just Newton's second law.

Why don't we ever find this path before doing work calculations?

What are you referring to, specifically? You need the specified path $C$ to find the work done. You might be getting confused with introductory physics problems where the path is already given to you ahead of time; this is just to make the problems easier for students to grasp the concept of work and how to calculate it rather than also put onto the problem for them to find the specific trajectory of the particle themselves. This is also easier for the asker of the questions as well, as then they don't need to contrive the specific forces required to cause the particle to move on the path they want to with the specific force in the question.

Answer 2

Maybe in your problem you are dealing with conservative forces; the works done depends on the extremal points of the trahectory, not on the trajectory itself. In such a case you can choose the trajectory as you want to make the calculations simpler