If we have two gear wheels of the same size, one of which rotates once around the other, which is stationary, how often will the first one turn around its own axle?

Question

anonymous · Answer

I just tried it with coins, and it looks like it's twice, what do you guys think?

anonymous · Answer

Let's unroll the stationary middle gear. It will have a length equal to it's circumference. $$l = \pi d = 2 \pi r$$

Now, we need to find a relationship between rotational angle and distance travelled. From kinematics, the distance travelled by the CG of the rotating gear can be expressed as $$d = \theta r$$One rotation of the rotating gear corresponds to an angle of $2 \pi$, therefore, $$d= 2 \pi r N$$ where N is the number of rotations. Let's set $l =d$ to obtain$$2 \pi r N = 2 \pi r$$ For the equation to be satisfied, $N =1$

anonymous · Answer

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I thought gear (b) would turn twice around the stationary gear (a), because gear (b)'s motion is composite, consisting of its own rotation around its center and its rotation around
(a). I tried this on coins and it seemed so.