**Problem 5**

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 P_{}Q 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.