P is a point on the circumference of a circle radius r which rolls,
without slipping, inside a circle of radius 2r. What is the locus
Two semicircle sit on the diameter of a semicircle centre O of
twice their radius. Lines through O divide the perimeter into two
parts. What can you say about the lengths of these two parts?
Find the perimeter and area of a holly leaf that will not lie flat
(it has negative curvature with 'circles' having circumference
greater than 2πr).
When coin A rolls once all the way around coin
B (which is stationary) then coin A makes two revolutions. No-one
came up with a satisfactory reason why this should be. Is it really
so counter intuitive? Perhaps you may like to consider points at
the centre and circumference of coin A as it rolls through a
specific arc around coin B.
However, Thomas from Madras College, St
Andrews had convinced himself that 2 is the correct solution
because "as it travels around the stationary coin the
distance it goes accounts for 1 revolution and as it moves around
it is rotating so it ends up putting in an extra revolution".
He built upon this insight, as well as that
concerning the centres of the coins involved, to generalise what
happens when a coin rolls around a line of 'n' similar coins.
When the rolling coin rolls around two stationary coins it
revolves through an angle of 16($\pi$/3) or 8/3 revolutions.
The number of revolutions R, when n is the number of stationary
identical coins, is given by the formula:
R = 2(n+2)/3.