Past exam of the classical mechanics course of Part B of the physics course of the University of Oxford / 2016 / 1 / b / Solution

Let $\mathsf{R}(\alpha )$ denote the planar rotation matrix, and choose the initial cup center on the positive $x$ axis, as in the requested equations. Its trajectory is $\mathbf{C}(t)=R(\cos \Omega t,\sin \Omega t)$. The handle rotates relative to the radial line at signed rate $\varphi$, so its absolute angular velocity is $\nu =\Omega +\varphi$. A positive recurrence time is $2\pi /|\varphi |$; the sign records rotation direction.

Writing $\mathbf{q}=(x_1,y_1)$, the relation isor explicitlyOne obtains the same result by rotating cup coordinates through $\varphi t$ into the radially aligned axes, then rotating through $\Omega t$ into the inertial axes.

At $\varphi =-\Omega$, $\nu =0$, andThe axes maintain a fixed orientation in inertial space while their origin accelerates around a circle. This is translation without rotation of the cup frame; it is not an inertial frame because the origin accelerates. In particular its inertial correction is a spatially uniform fictitious force, rather than a Coriolis force from rotation of these axes.