If a massless string is stretched by applying extensive forces of magnitude $x$ on both sides, a tension $T$ is developed at every point of the string. If the same massless string is placed over a frictionless pulley and the same extensive forces $x$ are applied on both ends, is the tension $T$ developed at every point (including the semi-circular part of the string over the pulley)? Furthermore, if that same string is placed over a fixed nail and stretched again with the same forces $x$, is the tension also $T$ at every point?
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$\begingroup$ This question is similar to: Can there be tension in an inextensible string?. If you believe it’s different, please edit the question, make it clear how it’s different and/or how the answers on that question are not helpful for your problem. $\endgroup$– Bob DCommented Aug 4 at 10:59
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2 Answers
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In the idealized world of undergraduate physics, the above statements and previous answer are correct. But in the real world there will be an unequal distribution of tension if the string is wrapped around an object with finite radius. Conceptually similar to beam bending except all tension, no compression. This is why there is a little bit of science behind knot tying - trying to make the tension distribution as constant as possible in the rope cross section. Frictional forces will also produce tension inequalities, especially in the case of the nail.
So the answer kind of depends on how deep you want to dig.
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If the same massless string is placed over a frictionless pulley and the same extensive forces $x$ are applied on both ends, is the tension $T$ developed at every point (including the semi-circular part of the string over the pulley)?
Yes it is the same, because the pulley is frictionless. There will be a normal force from the pully, but this is perpendicular to the tension at every point of contact so will not affect the outcome.
The tension has to be the same everywhere because the string is massless and at every point the net force on every infinitesimal segment of the string must remain zero. So the string will stretch at every point as needed to have a constant uniform tension to support the external weight $x$ added on each end.
Furthermore, if that same string is placed over a fixed nail and stretched again with the same forces $x$, is the tension also $T$ at every point?
Yes. Now the fixed nail will provide a reaction force in the opposite direction equal to $x$ to prevent the nail from moving. This is identical to the case of two people pulling in the string in opposite directions with a force $x$.
