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Consider the point P where the gears meet. When the smaller gear rotates clockwise, the larger one will rotate counterclockwise.
Through one rotation of the smaller gear, P will have traveled the circumference of the smaller gear, which is in.
At the same time, a point P' on the larger gear traverses the same distance along the larger gear's circumference. This point traces out an arc that is subtended by some angle . The arc is as long as the smaller gear's circumference.
The measure of a circle's interior angle subtended by an arc is proportional to a complete revolution, i.e. an angular displacement of radians:
For part 2, we apply the same reasoning to the larger gear. In one full rotation of the larger gear, the point P' traverses the circumference in, and so does the point P on the smaller gear.
A full rotation is 360 degrees, so the smaller gear would have rotated times.
