Variable mass, uniform body, force -- pulling a massive rope

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The discussion revolves around the complexities of analyzing the forces involved when pulling a massive rope, particularly focusing on the relationship between velocity and mass elements. Participants express confusion about the application of conservation principles, noting discrepancies between momentum and work conservation results. The conversation highlights the importance of considering internal forces and the role of tension in the rope, with some suggesting that the average speed of a mass element is taken as v0/2 due to its acceleration from rest. Ultimately, the consensus leans towards recognizing that the assumptions made about force and motion can lead to different interpretations and outcomes, emphasizing the nuanced nature of these physics problems. The intricacies of the analysis suggest that further exploration and clarification are necessary to fully understand the dynamics at play.
  • #31
erobz said:
For some reason, I'm not getting that.

The differential work ##dW## done by the force as it moves a distance ##dx## should be the change in kinetic energy of the pulled rope:

$$ dW = F~dx = \frac{1}{2} \lambda ( x + dx) v^2 - \frac{1}{2} \lambda x v^2 = \frac{1}{2}\lambda v^2 dx $$

$$ \implies F = \frac{1}{2}\lambda v^2$$

?
Is that setting the differential work equal to the differential change in kinetic energy? I did not know the work energy theorem could be applied for differentials!

Many thanks!
 
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  • #32
Callumnc1 said:
Is that setting the differential work equal to the differential change in kinetic energy? I did not know the work energy theorem could be applied for differentials!

Many thanks!
Well... if it hasn't been shouted down by the pros around here yet it might be ok.
 
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