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Power needed to rotate wheels 1 and 2, torque, rotation

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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. (Intro 1 figure) 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.

A) Let wheel 1 rotate at a constant angular speed W. Find the ratio W1/W2 of the angular speed of wheel 1 to the angular speed of wheel 2. Express your answer in terms r1 and r2.

B) The rotation of wheel 1 is caused by a torque T1. Find the ratio r1/r2 of the torque acting on wheel 1 to the torque acting on wheel 2. Express your answer in terms of r1 and r2.

C) 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?

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a) Since the gears are connected through a chain they have the same circumferential linear ...

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The power needed to rotate wheels 1 and 2, torque and rotation.

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Park Pobedy station in the Moscow Metro

Park Pobedy station in the Moscow Metro contains one of the longest escalators in Europe. Each one is 120 m in length and has 740 steps, of which 350 are available (i.e. not on their way back inside the mechanism) and on the inclined section at any given moment. The height difference between the bottom and top steps is 55 m, and it takes three minutes to get from the bottom to the top.

a.Calculate the extra power needed to maintain the speed of the up escalator when it is full (assume the capacity of the escalator is calculated at two people plus luggage - a total of 200 kg - on each available step).

b.The steps of the escalator are all linked together by a chain on each side that drives them round a guide system. Each chain is driven by two sprocket wheels, each 300 mm in diameter, and each linked to an electric motor by a gearbox, i.e. four sprocket wheels in total. Calculate the torque on each sprocket wheel when the escalator is fully laden (ignore the effect of friction in the escalator mechanism).

c.The electric motor rotates at 3600 rpm. Calculate the gearbox ratio and the torque on the motor shaft.

d. If the up and down escalators are mechanically linked, it is possible to provide some of the power needed to raise passengers up one escalator by using the weight of those traveling down the other. Suggest what measures might need to be incorporated in this system to allow for its safe operation when there are many more people travelling down than up.

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