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## Play It By Gear

Gears have been used for thousands of years to transfer and transform mechanical energy. The teeth of two gears fit together like puzzle pieces so that when you rotate either one, the other rotates as well.

The operation of a system of gears follows only one rule: teeth in contact always move in the same direction with the same speed.

In the gear system above, the bronze gear on the left has $10$ teeth and rotates at a rate of $\num{10}$ rotations per minute or $\si{rpm}.$ (The number of teeth is written at the center of the gear.) The bronze gear turns the gold gear, which has $50$ teeth.

It follows from the rule that the gold gear rotates at exactly $\SI{2}{rpm}$ — let's see how...

The rule says the teeth have equal speeds at their point of contact, and the gold gear has $5$ times as many teeth. Therefore, the bronze gear rotates $5$ times for every $1$ rotation of the gold gear.

Therefore, the rate of the gold gear is smaller by a factor of $5:$

$(\text{rate of gold gear}) =\frac{\SI{10}{rpm}}{5}=\SI{2}{rpm}.$

It follows from the rule that their rotation directions are also related. The teeth at the point of contact need to move in the same direction. This means if the small bronze gear rotates in a clockwise direction, the gold gear rotates in the opposite direction, counterclockwise.

These two observations about the relative rotation rates and directions can be extended to trains of more than two gears as well.

(The number of teeth on each gear is displayed at its center.)

The purpose of a gear system is to generate mechanical advantage for the operator of a device, amplifying the force they apply on the driven gear, although out of context they make fun brain-teasers to untangle.

# Today's Challenge

In this system of two gears, the silver gear is driven to rotate at $\SI{10}{rpm},$ which turns a large, gold gear.

You add a second silver gear to the system, and its teeth make contact with the other two.

If the teeth remain in contact when the power is turned on, at what rate does the newly-added, silver gear rotate?

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