Past exam of the mathematical methods I course of the University of Oxford / 2016 resit / 4 / Solution

Choose the cylinder centre's initial position as the coordinate origin, with downhill motion to the right and $0\le \alpha <\pi /2$. For rolling without slipping, the clockwise angular speed is $\omega =v/r$. A point initially vertically above the axis therefore hasA different origin adds constant coordinates without changing the following velocity test. Setting $\theta =vt/r$ gives $\dot{x}/v=\cos \alpha +(a/r)\cos \theta$ and $\dot{y}/v=-\sin \alpha -(a/r)\sin \theta$.

For strictly upward motion, $\dot{x}=0$ and $\dot{y}>0$. The first condition fixes $\cos \theta =-(r/a)\cos \alpha$; choose the negative sine branch to maximize $\dot{y}$. Positivity then requiresAt $a=r$ the relevant point is momentarily stationary, not moving upward. Thus a material point inside or on an ordinary cylinder cannot move exactly vertically upward in this motion. The mathematical extension $a>r$ describes the looping trochoid of a point beyond the rolling radius, illustrated in the figure.