Solved: The differential of an automobile allows the two

Chapter 20, Problem 20-23

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The differential of an automobile allows the two rear wheels to rotate at different speeds when the automobile travels along a curve. For operation, the rear axles are attached to the wheels at one end and have beveled gears A and B on their other ends. The differential case D is placed over the left axle but can rotate about C independent of the axle. The case supports a pinion gear E on a shaft, which meshes with gears A and B. Finally, a ring gear G is fixed to the differential case so that the case rotates with the ring gear when the latter is driven by the drive pinion H. This gear, like the differential case, is free to rotate about the left wheel axle. If the drive pinion is turning at \(\omega_{H} = 100 \ rad/s\) and the pinion gear E is spinning about its shaft at \(\omega_{E} = 30 \ rad/s\), determine the angular velocity, \(\omega_{A}\) and \(\omega_{B}\), of each axle.