A circle with radius 1 rolls without slipping once around a circle with radius 3. How many revolutions does the smaller circle make?

Does it matter if the smaller circle rolls on the inside or outside of the larger circle?

Charlie, Half-Mad and rixar are all correct; 4 if the little circle is outside and 2 if the little circle is inside the bigger circle.

I find that with certain problems, "seeing" the solution can be incredibly difficult or surprisingly simple depending on how you think about it.  Sometimes a change of perspective makes all the difference.  So here was the "mind exercise" I did that made the answer simple.

Picture holding 2 rings, the little one outside the big one between your fingers; big ring above, little ring below.  Start pinching your fingers to the left, and continue so the little ring rotates clockwise, and the big ring counterclockwise.  The tangent point is stationary between your fingers.  When the little ring has rotated 360 degrees clockwise, the big ring has rotated 120 degrees counterclockwise; and you are 1/3 of the way done.   But in the problem, the big ring doesn't move, so you have to now rotate the big circle clockwise 1/3 of a turn while holding your fingers pinched.  Aha! the little ring must also rotate an extra 1/3 rotation.   Repeat 3 times and the little ring obviously goes around 4 times.  3+1=4

When the little is inside the big, the two rings both go the same direction (counterclockwise), so when you make the correction you are taking away from the rotations of the little.  3-1 = 2.

I figure that if a thought experiment or gedanken experiment can help Einstein figure out Special relativity, then it can help me figure out lots of things.

And that's my story.

Posted by Larry on 2004-07-05 22:55:47