The Clockblock Problem - problem and solutions
I'm preparing myself for AMC 10 (which I'm sure a lot of other students would be doing too), but then I just don't know how to solve this problem (and the solutions there don't quite help me either - they're actually confusing).
The problem's basically like this:
The diagram below shows the circular face of a clock with radius $20$ cm and a circular disk with radius $10$ cm externally tangent to the clock face at $12$ o'clock. The disk has an arrow painted on it, initially pointing in the upward vertical direction. Let the disk roll clockwise around the clock face. At what point on the clock face will the disk be tangent when the arrow is next pointing in the upward vertical direction?
From what I understand, the clock itself has a circumference of $40\pi$ cm and the disk a circumference of $20\pi$ cm (half the previous one). So it'd take the disk two whole $360^\circ$ turns to start from where it's at now and come back - meaning it'd be at $6$ o'clock when the arrow's pointing upwards vertically. But the answer's different ($4$ o'clock), and I'm not quite sure where I went wrong, so can anyone help me understand it? (I think it could be that I misunderstood the problem, too.)
$\endgroup$ 02 Answers
$\begingroup$The flaw in your reasoning is that as the disk rotates along the edge of the clock face, there are two components to its orientation: the disk's own rotation, which you accounted for, but also a second rotational movement corresponding to its changing position relative to the clock.
To understand this, take two coins of equal size, and roll one around the other. You will find that even though their circumferences are equal, the coin that is rolling around the other coin actually goes through two full rotations as it travels around to its original position.
$\endgroup$ $\begingroup$Hint: You must look at the distance travelled by the center of the circle.
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