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Take two gears: One is fixed, the other rotates around it. If the gears are the same size the rotating gear has to rotate around its center twice as fast as it rotates around the other gear. I'm looking for intuition or reasoning why this is the case.

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As you turn the gears, the point where the outer gear touches the inner gear will travel around the circumference of the inner gear, making one full rotation.

But the centre of the outer gear is twice as far (because the gears are of equal size), so it will travel twice as far too. (That is, twice the circumference of the gears.)

And because there's no slippage, this means the outer gear will have to rotate twice too.

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  • $\begingroup$ Why does the fact that the touching point travels half as far as the centre imply that it rotates twice? $\endgroup$ – Sarien Oct 26 '14 at 23:23
  • $\begingroup$ @Sarien Because a rolling gear will always rotate exactly as much as its centre moves. If the centre moves 1 unit, the gear will have to rotate 1 unit too. If it didn't, it would mean the gear either deform or slip. In this case, the centre moves (2rπ)*2 units, therefore the gear has to rotate (2rπ)*2 units, which is (2π)*2 radians = 2 full rotations. $\endgroup$ – biziclop Oct 26 '14 at 23:35

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