NJC Parametric Equations Tutorial Q10

Modelling Movement of Wheel Using Parametric Equations

A particle of negligible size is lodged onto the circumference of a circular wheel of radius \(1\). Initially, the wheel is at rest with its lowest point at the origin. The initial position of the particle between the wheel and the floor (represented by the \(x\)-axis) is at the origin as well, as shown in the diagram below.

The wheel begins to move from rest, rotating towards the positive \(x-\)direction in a clockwise manner. The position of the particle when the wheel has rotated by \(\theta \) radians is denoted by the point \(P\), as shown in the diagram below.

Given that the curve \(C\) is the path of \(P\) on the \(x-y\) plane,

1. show that \(C\) has the parametric equations
   \(x=\theta -\sin \theta \), \(y=1-\cos \theta \).
2. find the range of values of \(y\),
3. sketch \(C\), for \(0\le \theta \le 2\pi \), indicating clearly the exact coordinates of the endpoints and the maximum point.