Energy, Momentum and Stress in a Belt Drive1

[2] showed the figure below with the caption “Transfer of mass without net transfer of particles or radiation”. A battery drives a motor that in turn drives a paddlewheel via a belt drive. The paddle-wheel turns in a water bath that heats up as a consequence. Hence, energy E is transferred from the battery to the water bath, and as a result the mass of the battery decreases by E/c2 while the mass of the water increases by the same amount, where c is the speed of light. How does this energy get from the battery to the water? Do observers in difference frames of reference agree as to the path of the energy flow? Discuss the momentum in this system, supposing that the apparatus rolls without friction on a horizontal table. The distance between the pulleys is L. The velocity of the belt relative to the baseplate of the apparatus is v and the tension in the tight side of the belt is T. The pulleys and the belt have teeth such that the belt cannot slip, and the tension of the belt is zero on the loose side. The motor is mounted on vibration dampers such that no oscillations of the frequency of the motor or higher are transmitted to the baseplate. To avoid possible ambiguities as to the role of the loose side of the belt, consider also a configuration in which a string of mass m and length l L is initially wrapped around the pulley attached to the paddlewheel. The string is also tied to the motor pulley, and the motor is operated for a time t = l/v, after which time the string is wrapped around the motor pulley. To support the tension T, both the string and belt must be elastic media, for which the spring constant of the stretched portion is k. 1This note is largely superceded by [1]. 1