Problem:
The diagram below shows the circular face of a clock with radius and a circular disk with radius externally tangent to the clock face at 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?
Answer Choices:
A. o'clock
B. o'clock
C. o'clock
D. o'clock
E. o'clock
Solution:
The circumference of the disk is half the circumference of the clock face. As the disk rolls of the way around the circumference of the clock face (from o'clock to o'clock), the disk rolls through of its own circumference. At that point, the arrow of the disk is pointing at the point of tangency, so the arrow on the disk will have turned of one revolution. In general, as the disk rolls through an angle around the clock face, the arrow on the disk turns through an angle on the disk. The arrow will again be pointing in the upward vertical direction when the disk has turned through complete revolution, and the angle traversed on the clock face is of the way around the face. The point of tangency will be at o'clock.
The problems on this page are the property of the MAA's American Mathematics Competitions