The figure below shows a pair of interlocked gears, where Gear I has a radius of $12$ cm and Gear

Question

The figure below shows a pair of interlocked gears, where Gear I has a radius of $12$ cm and Gear II has a radius of $8$ cm. At the moment $t = 0$ (in seconds), the pair of gears starts rotating, where Gear I rotates clockwise and Gear II rotates anti-clockwise. Let $A$, $B$ be the points on the edges of Gear I and Gear II where the two gears meet 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 vertical distance of point $A$ (respectively, point $B$) from the centre of Gear I (respectively, Gear II). 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 $0 \le t \le 12$. Find the distance between point $A$ and point $B$ at $t = 6$. Let $T$ be the moment when the two gears meet each other at points $A$ and $B$ for the $n^{\text{th}}$ time. Express $T$ in terms of $n$, stating your workings clearly.

Accepted Answer

(i) $4$s (ii) ${{y}_{1}}=-12\sin \left( \frac{\pi }{3}t \right)$, ${{y}_{2}}=-8\sin \left( \frac{\pi }{2}t \right)$ (iv) $16$ cm (v) $T=\left( 12n-12 \right)$ s

Trigonometric Functions and Graphs

Trigonometric Functions and Graphs