A question in the derivation of work done

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Discussion Overview

The discussion revolves around the derivation of work done by a force, specifically addressing the notation used in the integral formulation of work, ∫ F·dl. Participants explore the implications of using F instead of dF and the conditions under which integration is applied, particularly in the context of varying forces.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions why the force is represented as F rather than dF in the work integral, suggesting that integration implies summing infinitesimal parts.
  • Another participant asserts that the force remains finite while acting over an infinitesimal distance, countering the initial question.
  • Concerns are raised about whether the full force is applied over small segments of distance, with differing opinions on this point.
  • Participants discuss the scenario of varying forces, indicating that integration is necessary in such cases to account for the changes in force along the distance.
  • One participant emphasizes that at any point along the path, the force has a specific value, F(x), and relates this to the concept of differentiating speed with respect to time.

Areas of Agreement / Disagreement

Participants express differing views on the application of force in the work integral, particularly regarding the use of F versus dF and the implications of varying forces. The discussion remains unresolved with multiple competing perspectives.

Contextual Notes

There are assumptions about the constancy of force and the nature of integration that are not fully explored, particularly in the context of varying forces and the implications for the work done.

ehabmozart
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In our formulation of work done by a Force we say it is ∫ F.dl ... My knowledge of integration means adding Infinitesimal parts. We write dl because it is the smallest part of the distance moved. My question is why have we written F and not dF... Thanks for clarifying!
 
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ehabmozart said:
In our formulation of work done by a Force we say it is ∫ F.dl ... My knowledge of integration means adding Infinitesimal parts. We write dl because it is the smallest part of the distance moved. My question is why have we written F and not dF... Thanks for clarifying!
The force doesn't get infinitesimally small. It remains finite as it acts over an infinitesimal distance.
 
But not the full force is applied to this small part
 
ehabmozart said:
But not the full force is applied to this small part
Sure it is.

Say I exert a force of 100 N over a distance of 1 meter. I'm exerting the full 100 N over every bit of that distance.
 
What if the force is varying??
 
ehabmozart said:
What if the force is varying??

That's when you integrate. If the force is constant (and the object moves in a straight line), you simply have ##W = \vec F \cdot \vec {\Delta l}##.
 
ehabmozart said:
What if the force is varying??
As jtbell said, that's why you integrate.

But the force at any point is whatever it is, not some infinitesimal.

Say I push with some varying force for a distance of 1 meter. You can break that path up into as many small sections as you like. (When you integrate, you are making them infinitesimally small.) At any point in the path, the force has some value F(x) as it pushes through the tiny distance dx.

By analogy, think of calculating the distance traveled by something moving with some speed v:
X = V*t
Expressing it as a differential, you get dX = V*dt, not dX = dV*dt. Over the tiniest interval of time, the speed does not go to zero.
 
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