<p>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 origin as well, as shown in the diagram below.</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/converted/1760163498815-ahme85odxfv.png" width="400" class="aligncenter" /></figure>
<p>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.</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/converted/1760163507328-428d7qq1gfa.png" width="400" class="aligncenter" /></figure>
<p>Given that the curve $C$ is the path of $P$ on the $x\mathrm{-}y$plane,</p>
<ol class="i-list"><li><div class="list-item-paragraph">show that $C$ has the parametric equations $x=\theta -\sin \theta $, $y=1-\cos \theta $.</div></li><li><div class="list-item-paragraph">find the range of values of $y$,</div></li><li><div class="list-item-paragraph">sketch $C$, for $0\le \theta \le 2\pi $, indicating clearly the exact coordinates of the endpoints and the maximum point.</div></li></ol>

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 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 $x\mathrm{-}y$plane,

show that $C$ has the parametric equations $x=\theta -\sin \theta $, $y=1-\cos \theta $.

sketch $C$, for $0\le \theta \le 2\pi $, indicating clearly the exact coordinates of the endpoints and the maximum point.

NJC Tutorial Curve Sketching Q10

NJC Tutorial Curve Sketching Q10

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