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This question is from Introduction to Classical Mechanics, David Morin.

A mountain climber wishes to climb up a frictionless conical mountain. He wants to do this by throwing a lasso (a rope with a loop) over the top and climbing up along the rope. Assume that the mountain climber is of negligible height, so that the rope lies along the mountain, as shown.

enter image description here

At the bottom of the mountain are two stores. One sells “cheap” lassos (made of a segment of rope tied to loop of rope of fixed length). The other sells “deluxe” lassos (made of one piece of rope with a loop of variable length; the loop’s length may change without any friction of the rope with itself). When viewed from the side, the conical mountain has an angle α at its peak. For what angles α can the climber climb up along the mountain if he uses a “cheap” lasso?A“deluxe” lasso? (Hint: The answer in the “cheap” case isn’t α < 90◦.)

Solution: http://www.physics.harvard.edu/uploads/files/undergrad/probweek/sol7.pdf

In the fourth paragraph, I couldn't understand why the rope should take the shortest distance between two points when the cone is on the plane. Why no friction implies that?

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  • $\begingroup$ Ok. I agree that the constraint forces should be normal to the cone, but how that implies in minimum lenght to the rope between two points? Could you explain in more detail? $\endgroup$
    – Rafael Deiga
    Dec 28, 2016 at 23:44
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    $\begingroup$ For extra credit, consider how to climb a frictionless spherical cow ;) (Sorry, but the old jokes are still the best ones). $\endgroup$
    – alephzero
    Dec 29, 2016 at 0:37

1 Answer 1

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At equilibrium, the rope's tension equals the climber's weight. The rope's tension, which is tangential to the rope should be balanced by an opposite constraint force.

Since there is no friction between the rope and the cone, and the rope and the knot, any force of constraint along the rope is normal to the rope, and the only constraint generating a tangential force is at the knot's end. Therefore, the rope's tension is uniform along the whole rope. For this to happen, the rope's length should be minimum, because a loose rope does not generate any tension.

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  • $\begingroup$ Do you consider the weight of the rope negligible? $\endgroup$
    – Rafael Deiga
    Dec 29, 2016 at 13:29
  • $\begingroup$ @RafaelDeiga Yes, it's a reasonable approximation, not worse than assuming, as the problem does, a zero-height climber. $\endgroup$
    – Massimo Ortolano
    Dec 29, 2016 at 13:34
  • $\begingroup$ Shouldnt the rope's tension be radial to the rope as the tangential components would cancel off? $\endgroup$
    – Schwarz Kugelblitz
    Oct 24, 2020 at 22:45
  • $\begingroup$ @SchwarzKugelblitz Recall that the tension in a rope is the force that a part of the rope exerts on the other at each cross section. Thus, in the case of an ideally infinitely thin rope with no bending stiffness, the only tension the rope could support is the tangential (axial) one, because tension in a different direction would just bend the rope. $\endgroup$
    – Massimo Ortolano
    Oct 25, 2020 at 9:56
  • $\begingroup$ Oh you mean like after equilibrium had been achieved $\endgroup$
    – Schwarz Kugelblitz
    Oct 28, 2020 at 14:52

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