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From some MIT resources: Because there is no more rope below the bottom, there is no mass below this point, and therefore no weight pointing down. Therefore tension at the bottom end of the rope is zero - there is no force exerted at the bottom end

Why the mass of that point is being discarded as if the point itself wouldn't count? The argument "no mass below this point" seems sketchy as when we draw FBD we look at the forces on the object/point. Assuming that point has mass $ dM $, wouldn't the tension at the bottom be $ T - dM g = dM a $ and because the rope has no acceleration: $ T = dM $g? i.e. very small but not zero.

Qmechanic
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Jon
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2 Answers2

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The point being made is that as you approach the bottom of the rope, tension approaches 0. So we say that the tension at the very bottom of the rope is 0.

In other words, the differential mass $\mathrm{d}M$ is infinitesimal, so for the purposes of computing tension, we treat $\mathrm{d}M\to0$.

DanDan面
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Why is the mass of that point being discarded as if the point itself wouldn't count?

This is because we are working with infinitesimals. Essentially the mass of an infinitesimal slice of rope is vanishingly small. When we integrate these slices then they count infinitesimally and we get in toto the total mass of the rope.

Mozibur Ullah
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