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Last week a couple of students came to me with a related problem which
they were assigned in their mechanics course.
A rope* of length L is laid out perpendicular to the edge of a
frictionless tabletop. Initially the rope is at rest with a
portion of length x hanging over the edge. The rope is released
and falls to the floor. How much time t(x) is required for the
rope to leave the tabletop?
I believe the problem as stated cannot be solved. The simple
view that only the gravitational force can do work I'll grant,
but the motion of this rope is more complicated than the poser
of the problem recognized. One way to make the problem well-
formed is to specify, for example, the radius of curvature of
the edge of the table, since clearly the edge exerts a
horizontal force on the rope. having said that I'll confess
that I'm glad the radius wasn't specified, since the solution
would then perhaps have been determinate, but I was not up to
being the one to determine it.
Leigh
*inextensible, uniform, ideally limp, etc.