What are the effects of different tensions in a rope?

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SUMMARY

The discussion focuses on the effects of varying tensions in a rope, particularly in the context of calculating wave speed using the equation v=sqrt(F/μ). The tension (F) at both ends of a vertical rope differs due to the weight tied to the bottom, complicating the application of the equation. David highlights the misconception that tension is uniform throughout the rope, while it actually varies along its length. The solution involves analyzing the tension for infinitesimally small segments of the rope to accurately determine the forces acting on each section.

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Hi,

I'm having some trouble with an equation: v=sqrt(F/μ). If I understand correctly, F corresponds to the tension in both ends of the rope. I'm calculating the wave speed for a vertical rope with mass as well as a weight tied to the bottom. It's also tied to the ceiling. This makes the tensions at both ends different. I don't know what I'm supposed to do. It doesn't appear I can use the beginning equation. If anybody could please help, I'd be very appreciative. Thank you.

David
 
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The trouble is that in easy situations, we get used to thinking of tension as the force on the ends of a rope. But that view assumes the tension is the same all the way along the rope (which, as you note, isn't the case here). So the tension will vary as we head up the rope in question.

While we can't calculate the tension for the whole rope, we can calculate it for each tiny piece of the rope. Take a tiny length the rope (with negligible mass), and since the force on top and bottom is the same (given that there is, again, negligible mass), we can figure out tension in the usual way. And you can do that for any individual spot on the rope.

I hope that helps get you on the right track.
 

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