Ropes and pulleys in special relativity (relativistic statics of threads)

Abstract

The relativistic statics of threads, also leaning on smooth surfaces, is treated by making reference to the following picturesque, apparent paradox. A rope is partially leaned on a 90° arc of a sheave without friction and is subject to terminal forcesF1 andF2 equal toF0 for the rest observerS. For a second observerS′ having a velocity—v with respect toS and directed asF2, the latter forceF′2 appears equal to the rest valueF0 whilstF′1=F0(1−v2/c2)1/2. The apparent paradox is solved both in the synchronous and in the asynchronous formulation. In the former the key to the solution lies in the fact that the tensionT is not tangent to the rope. In the asynchronous formulationT is tangent to the rope but the reaction of the sheave is no longer perpendicular to the rope, thus creating a progressive variation of |T| along the rope.