QUESTION: Consider the circle with radius $1$ and cente at the point $(0,1)$. From this initial position the circle is rolled along the positive x-axis without slipping. Find the locus of the point $P$ on the circumference of the circle which is on the origin at the initial position of the circle.
MY ATTEMPT: Pardon me, for this question has been asked before. But I was unable to understand the explanation.. I applied the concept that no matter where the circle is, the distance of $P$ from the (then) centre of the circle is always equal to the radius, $1$. For this I first found out the locus of the centre, which is clearly $y=1$. But, this does not help. I don't understand how to approach this problem. I think there must be some smarter way to solve this. Can anyone please help me out?
Thank you.
Best Answer
Let the center be at $(\theta,1)$. To reach this position, the circle must have rolled by the angle $\theta$, clockwise. Hence the absolute position of $P$ is the position of the center plus the position of $P$ relative to the center,
$$(x,y)=(\theta,1)+\left(-\sin\theta,-\cos\theta\right)=\left(\theta-\sin\theta,1-\cos\theta\right).$$
You can eliminate $\theta$ using
$$\cos\theta=1-y$$ and
$$x=\pm\arccos(1-y)\mp\sqrt{1-(1-y)^2}+2k\pi.$$