Roll on
Weekly Problem 5 - 2006
How many times does the inside disc have to roll around the inside of the ring to return to its initial position?
How many times does the inside disc have to roll around the inside of the ring to return to its initial position?
Age
14 to 16
Challenge level
Problem
Image
A circular disc of diameter $d$ rolls without slipping around the inside of a ring of internal diameter $3d$, as shown in the diagram. By the time that the centre of the inner disc returns to its original position for the first time, how many times will the inner disc have turned about its centre? What if the disc rolls around the outside of the ring?
Student Solutions
$3$ times
In returning to its original position, the disc rolls a distance $3\pi d$. As the disc does not slip, it rolls out a distance of $\pi d$ when the disc makes one complete turn about its centre, so $3$ complete turns are made.
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.