Is he rolling forever?
Above are two tangent circles, a bigger one with radius
and a smaller one with perimeter
. Imagine you can roll the smaller circle counterclockwise around the bigger circle so that they stay tangent. Only whole revolutions are allowed. You now start rolling the small circle.
After how many revolutions will the midpoint of the small circle be back exactly at his red marked starting position as shown above? If you think the small circle will never be back at his starting point answer
The answer is 0.
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1 solution
The small circle will never be back at his starting position. The covered distance of the small circle after n revolutions is n ∗ 2 . When the small circle is back at his starting position, his covered distance has to ba a multiple of the perimeter of the bigger circle. Therefore it would exist an integer q with n ∗ 2 = q ∗ 2 ∗ π . But that obviously cant be true, because the right hand side will always be an integer and since π is irrational, the left hand side will never be an integer. Therefore the small circle will never be back at his starting point.