Derivative of Distance r in DW=F*dr: Explained

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Discussion Overview

The discussion revolves around the equation dW = F * dr, specifically questioning why the derivative is taken with respect to distance r rather than force F. Participants explore the implications of this choice in the context of work done by a force, particularly in scenarios where the force may change as the distance changes.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants assert that the equation defines work in terms of the force applied over a distance, suggesting this is a useful formulation leading to the work/energy theorem.
  • Others question whether the derivative should also account for changes in force (dF) when the force varies with distance, particularly in the context of gravitational potential energy.
  • One participant proposes an alternative perspective, suggesting that the derivative could be taken with respect to force instead, raising questions about the origin of the relation and its presentation in educational contexts.
  • A later reply emphasizes the specific example of gravity, reiterating the focus on the work done by gravity when moving an object over a small distance.

Areas of Agreement / Disagreement

Participants express differing views on whether the derivative should involve changes in force, indicating that multiple competing perspectives remain without consensus on the matter.

Contextual Notes

Some participants highlight the importance of context in understanding the derivation of the equation, suggesting that assumptions about the constancy or variability of force may influence interpretations.

Miraj Kayastha
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dW = F * dr


in this equation why is the derivative of distance r taken and not of the force?
 
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Because the equation is answering the question: how much work is done by force F(x) in moving an object from x=r to x=r+dr.
 
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This equation is a definition. We define work that way because it is useful for us, specifically, it leads to the work/energy theorem.
 
Changing force

But in a case where F changes as r changes shouldn't there be dF. For example shouldn't there be dF in gravitational potential energy derivation?
 
I'm wondering about the context - i.e. why presume the relation was arrived at by taking the derivative of work with distance? One could start with the work and take the derivative with force as suggested, and still keep the same definition. Perhaps there has been a lesson that presented the relation as coming from the derivative?
 
Miraj Kayastha said:
But in a case where F changes as r changes shouldn't there be dF. For example shouldn't there be dF in gravitational potential energy derivation?
Please reread post #2 - that answers your question.

In the specific example of gravity, how much work is done by gravity when it moves an object from r to r+dr?
 

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