Elasticity and mechanical advantage in cables and ropes

摘要

The conditions under which one can gain mechanical advantage by pulling with a force F perpendicular to a cable ( or rope) that is fixed at both ends are examined. While this is a commonly discussed example in introductory physics classes, its solution in terms of fundamental properties of the cable requires one to model the elasticity of the cable. This solution has several complex and interesting features, e. g. a large increase in the tension in the cable may occur upon application of F if (i) F lies in a certain range and (ii) the initial tension T-1 in the cable (before F is applied) satisfies T-1 < 0.0340 kappa. Here, kappa is the spring constant of a unit length of the cable. For steel cables and cables composed of other materials such as carbon nanorods where the elasticity is low, significant increases in tension are possible. Examples involving walking a tightrope and attempting to increase the tension in a rope hauling a load are considered. Two programs to solve the equations of this work are available in the electronic version of this journal.