A small wheel of 1 inch radius is put inside a circular track of 10 inch radius as shown. If the small wheel is rotated clockwise on the circular track without slipping until it comes back to its starting point P, how many turns will the small wheel make? Justify your answer.
[Problem submitted by Steve Lee, LACC Professor of Mathematics.]
Solution A for Problem 5:
Let the radius of the small wheel be r and the radius of the large circle R, and P a point on the small wheel. At the beginning, P coincides with P. Since the small wheel rotates without slipping, arc PQ and arc PQ have the same lengths. rB=RA. The angle C between the two arrowheads shown is the angle that the small wheel actually turned. C = B – A = (-) = () = 9. When the small wheel finished its rotation,. C=9. The small wheel made 9 turns clockwise.
Solution B for Problem 5:
If we rotate the small wheel clockwise on a straight line whose length is the same as that of the circumference of the large circle, then obviously the small circle made 10 turns clockwise.
Then we bend the line segment into the large circle with the small wheel at the right end. As we are bending, the arrowhead is turning counter clockwise (see figure below.) When we finished bending, the arrowhead made one complete turn counter clockwise. Therefore the total number of turns is 10 clockwise and 1 counter clockwise, that is 9 turns clockwise.