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?

very interesting. at first it seemed too easy, but it’s deceptive. my first thought was that it makes three revolutions but that is not the case.

the tangent point of the small circle arrives back on the perimeter of the large circle after 120 degrees of the large circle. but that is MORE than one revolution. it makes one revolution after travelling 90 degrees of the large circle. so after 360 degrees it has made FOUR REVOLUTIONS.

travelling on the inside, the small circle turns in the other direction. it’s tangent point again arrives after 120 degrees of the large circle, but it only makes a full revolution after 180 degrees. so after 360 degrees it has made TWO REVOLUTIONS.

interestingly, the circle formed from the travel of the CENTERPOINT of the small circle is radius 4 when rolling around the outside of the large circle and radius 2 when rolling around the inside of the circle.

(this situation occurs with planetary motion. the revolution of the earth on its axis relative to the stars is a few minutes shorter than its revolution relative to the sun. it’s called the SIDEREAL day)

Posted by rixar on 2004-06-27 22:37:02