Archimedes' Screw: Moment Arm and Pocket Volume

Abstract

Archimedes’ screw is an ancient pumping device consisting of a tilted, helical structure that captures water in pockets and elevates the water as the screw is turned about its axis.  In the accompanying report, early attempts to find the amount of torque required to turn the screw, the first one in 1705, are discussed.  It is found that although the torque calculation could be easily accomplished using the principle of virtual velocities, it was not amenable to solution by moment equilibrium due to the complex geometry of the water pocket, which hampered finding the length of the moment arm of the water weight vector.  Determining the volume of water that a pocket could hold proved difficult as well, and an accurate numerical algorithm for this purpose was not developed until the year 2000.  However, using a result by Daniel Bernoulli in 1736 for a thin tube screw, a solution for the torque by moment equilibrium in the general case is easily derived for the first time.  This approach also leads to a simple way to compute the pocket volume.

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