Gears are important components in many mechanical devices, from mechanical clock
ID: 2191899 • Letter: G
Question
Gears are important components in many mechanical devices, from mechanical clocks to bicycles. In fact, they are present whenever a motor produces rotational motion. An example of a simple gear system is shown in the figure. (Figure 1) The bigger wheel (wheel 1) has radius r1 , while the smaller one (wheel 2) has radius r2 . The two wheels have small teeth and are connected through a metal chain so that when wheel 1 rotates, the chain moves with it and causes wheel 2 to rotate as well. If the power needed to rotate wheel 1 is P1 , what is the ratio P1/P2 of the power of wheel 1 to the power of wheel 2? Express your answer in terms of any or all of the variables r1 and r2.
The answers i have tried that were wrong include: r1/r2, r2/r1
Explanation / Answer
You start by observing the tangential velocities of the two gears are equal where they touch each other. Why...because if they weren't equal, the gears would slide and strip the teeth. Therefore, we have v = Wr = wR = v; where v is the tangential velocity of one gear with radius r w, and of the other gear with R and w radius and angular velocity. R = r_1 and r = r_2 and w = omega_1 and W = omega_2 in your notation. a) Find w/W = r/R from noting the tangential velocities are equal. b) Let t = kmr^2 W = Fr be the torque on the r gear and T = kMR^2 w = FR be the torque on the R gear. We assume the torque forces F are applied at the point where the two gears touch. Thus T/t = FR/Fr = R/r since the two F's are equal but opposite forces where the gears mesh. c) Power = E/t = Fvt/t = FwRt/t = FwR = P for the R gear and e/t = FWrt/t = FWr = p for the r gear. Thus, P/p = FwR/FWr = Fv/Fv = 1.00; so that P = p, both gears expend the same power/energy.Related Questions
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