1 MASSACHUSETTS INSTITUTE OF TECHNOLOGYDepartment of Physics Physics 8.01 Fall Term 2010 Problem Set 11: Two Dimensional Rotational and Translational Motion Solution Problem 1: A drum Aof mass mand radius Ris suspended from a drumBalso of mass mand radius R, which is free to rotate about its axis. The suspension is in the form of a massless metal tape wound around the outside of each drum, and free to unwind. Gravity is directed downwards. Both drums are initially at rest. Find the initial acceleration of drum A, assuming that it moves straight down. Solution: The key to solving this problem is to determine the relation between the three kinematic quantities A!, B!and Aa, the angular accelerations of the two drums and the linear acceleration of drum A. One way to do this is to introduce the auxiliary variable zfor the length of the tape that is unwound from the upper drum. Then, 22Bd zRdt!=. The linear velocity Avmay then be expressed as the sum of two terms, the rate dzdtat which the tape is unwinding from the upper drum and the rate AR!at which the falling drum is moving relative to the lower end of the tape. Taking derivatives, we obtain 22AABAd zaRRRdt!!!=+=+. Denote the tension in the tape as (what else) T. The net torque on the upper drum about its center is then BTR!=, directed clockwise in the figure, and the net torque on the
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2 falling drum about its center is also ATR!=, also directed clockwise. Thus, /2/BTRITMR!==, /2/ATRITMR!==. Where we have assumed that the moment of inertia of the drum and unwinding tape is 2(1/ 2)IMR=. Newton’s Second Law, applied to the falling drum, with the positive direction downward, is AMgTMa!=. We now have five equations, 222222,,,,,BAABAAd zd zTTRaRMgTMadtdtMRMR!!!!==+=="=in the five unknowns A!, B!, Aa, 22d zdtand T. It’s easy to see that AB!!=. Therefore 2ABAAaRRR!!!=+=. The tension in the tape is then 244AAAMRa MRMaTR!===Newton’s Second Law then becomes 4AAMaMgMa!=. Therefore solving for the acceleration yields aA=45gThis result is certainly plausible. We expect Aag<, and we also expect that with both drums free to rotate, the acceleration will be almost but not quite g.