Paper 4, Section II, A

Dynamics and Relativity | Part IA, 2017

(a) Consider an inertial frame SS, and a frame SS^{\prime} which rotates with constant angular velocity ω\boldsymbol{\omega} relative to SS. The two frames share a common origin. Identify each term in the equation

(d2rdt2)S=(d2rdt2)S2ω×(drdt)Sω×(ω×r)\left(\frac{d^{2} \mathbf{r}}{d t^{2}}\right)_{S^{\prime}}=\left(\frac{d^{2} \mathbf{r}}{d t^{2}}\right)_{S}-2 \boldsymbol{\omega} \times\left(\frac{d \mathbf{r}}{d t}\right)_{S^{\prime}}-\boldsymbol{\omega} \times(\boldsymbol{\omega} \times \mathbf{r})

(b) A small bead PP of unit mass can slide without friction on a circular hoop of radius aa. The hoop is horizontal and rotating with constant angular speed ω\omega about a fixed vertical axis through a point OO on its circumference.

(i) Using Cartesian axes in the rotating frame SS^{\prime}, with origin at OO and xx^{\prime}-axis along the diameter of the hoop through OO, write down the position vector of PP in terms of aa and the angle θ\theta shown in the diagram (12πθ12π)\left(-\frac{1}{2} \pi \leqslant \theta \leqslant \frac{1}{2} \pi\right).

(ii) Working again in the rotating frame, find, in terms of a,θ,θ˙a, \theta, \dot{\theta} and ω\omega, an expression for the horizontal component of the force exerted by the hoop on the bead.

(iii) For what value of θ\theta is the bead in stable equilibrium? Find the frequency of small oscillations of the bead about that point.

Typos? Please submit corrections to this page on GitHub.