Tension of a “Girdle of the Earth”

Abstract

IT is perfectly true, as Prof. Lodge has asserted, that a cord or chain running on its own track as an endless band in a frictionless groove of any form will not require the sides of the groove to keep it in that form. But whatever velocity it moves with, such a tension will exist all round it as to resist the centrifugal forces of its windings; and to preserve them by virtue of the curvature and constant tension, invariable in shape however the speed of coursing of the belt may be increased, without any external guidance and assistance. If w is the cord's mass per unit of its length, and v the velocity with which it pursues its course, WV² is the tension in dynes which will be set up all round it. The speed may of course be so increased as to tear the cord or chain to pieces; and this will occur in steel tires of railway wheels, for example, if the train's velocity on which the wheels are carried is much more than 120 miles an hour. Mr. Bourne long ago pointed out, in his works on the steam-engine, that a very low limit of speed in railway trains is enforced for safety in view of this dynamical condition so as not to approach and exceed working and proof-stresses at least in the material of which steel wheel-tires are formed.