The figure below shows a pair of coaxial gears rigidly fixed to the same axle, with a common centre \(O\). Gear I has a radius of \(16\) cm and Gear II has a radius of \(4\) cm. At the moment \(t = 0\) (in seconds), the pair of gears starts rotating anti-clockwise. Let \(A\), \(B\) be the points on the edges of Gear I and Gear II such that \(O\), \(A\) and \(B\) are collinear and form a horizontal line at \(t = 0\).
Given that Gear I completes one full rotation every \(6\) seconds, find the time for Gear II to complete one full rotation.
Let \(y_1\) cm (respectively, \(y_2\) cm) be the signed horizontal displacement, positive to the right, of point \(A\) (respectively, point \(B\)) from centre \(O\). Based on your answer in (i), Express \(y_1\), \(y_2\) as a function of the time \(t\), respectively.
On the same diagram, sketch the graphs of \(y_1\) and \(y_2\) for \(0 \leq t \leq 12\).
Find the distance between point \(A\) and point \(B\) at \(t = 6\).
Let \(T\) be the moment when the points \(O\), \(A\) and \(B\) form a vertical line for the \(n^{\text{th}}\) time. Express \(T\) in terms of \(n\), stating your workings clearly.