Past exam of the mathematics course of the University of Cambridge / 2013 / ia / Paper 4 / 11B (Dynamics and Relativity)

Write the position in the rotating basis as $\mathbf{r}={\sum }_i{x}_i^{\prime }{\mathbf{e}}_i^{\prime }$. Differentiating with respect to time in the inertial frame,Here ${\mathbf{v}}^{\prime }={\sum }_i{\dot{x}}_i^{\prime }{\mathbf{e}}_i^{\prime }$ is the relative velocity measured in the rotating reference frame. ConsequentlyThis is the vector differentiation rule for a rotating reference frame with a common origin: the extra term accounts for the changing basis, not a translational motion of that origin.

The PDF gives initial position $(x^{\prime },y^{\prime },z^{\prime })=(1,0,0)$ and initial rotating-frame velocity zero. Use those values; the TeX erroneously places dots over the position coordinates.

For rotation about the third axis, the supplied acceleration relation and Newton's second law give the equations of motion in a rotating frameThus $z^{\prime }=0$. Set $\zeta =x^{\prime }+i{y}^{\prime }$ to combine the planar equations:The two characteristic roots are $i\omega$ and $-3i\omega$. From $\zeta (0)=1$ and $\dot{\zeta }(0)=0$,The triple-angle formulas therefore give the particularly simple astroid motion of a harmonic oscillator in a rotating frameThe orbit satisfies $|x^{\prime }{|}^{2/3}+|y^{\prime }{|}^{2/3}=1$; the zero speed at each cusp is consistent with the smooth time-parametrized motion.

Its rotating-frame speed isHence, for $\omega >0$, the first maximum occurs at $t=\pi /(4\omega )$, andThe maximum repeats periodically. For a signed negative angular frequency, use $\pi /(4|\omega |)$ and $3|\omega |/2$; at $\omega =0$ the particle remains at rest.