AP Calculus BC · Differential equations

A circular wheel of radius 2 meters rolls to the right along the x-axis without slipping

Problem

A circular wheel of radius $2$ meters rolls to the right along the $x$-axis without slipping. The center of the wheel travels at the constant speed $2$ meters per second. At time $t=0$ seconds a marked point $P$ on the rim is at the origin, and the parameter $t$ is the rotation angle in radians (which equals elapsed time for this motion). The position of $P$ is \[ x(t)=2(t-\sin t),\qquad y(t)=2(1-\cos t), \] with $x$ and $y$ in meters. (a) Explain, from the no-slip condition, why the term $2t$ appears in $x(t)$ and why the remaining trigonometric terms have coefficient $2$. (b) Find the velocity and speed of $P$. Determine every time in $[0,2\pi]$ at which $P$ is instantaneously at rest, and find the speed of $P$ at the unique time in that interval when $P$ is farthest from the $x$-axis. Include units. (c) Find the exact distance traveled by $P$ during one full revolution of the wheel, $0\le t\le 2\pi$.

Hint

This is a cycloid whose radius equals the given wheel radius; speed and arc length scale by that radius relative to the unit cycloid.

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