Paper 4, Section II, C

Dynamics and Relativity | Part IA, 2021

Consider an inertial frame of reference SS and a frame of reference SS^{\prime} which is rotating with constant angular velocity ω\boldsymbol{\omega} relative to SS. Assume that the two frames have a common origin OO.

Let A\mathbf{A} be any vector. Explain why the derivative of A\mathbf{A} in frame SS is related to its derivative in SS^{\prime} by the following equation

(dAdt)S=(dAdt)S+ω×A.\left(\frac{d \mathbf{A}}{d t}\right)_{S}=\left(\frac{d \mathbf{A}}{d t}\right)_{S^{\prime}}+\omega \times \mathbf{A} .

[Hint: It may be useful to use Cartesian basis vectors in both frames.]

Let r(t)\mathbf{r}(t) be the position vector of a particle, measured from OO. Derive the expression relating the particle's acceleration as observed in S,(d2rdt2)SS,\left(\frac{d^{2} \mathbf{r}}{d t^{2}}\right)_{S}, to the acceleration observed in S,(d2rdt2)SS^{\prime},\left(\frac{d^{2} \mathbf{r}}{d t^{2}}\right)_{S^{\prime}}, written in terms of r,ω\mathbf{r}, \boldsymbol{\omega} and (drdt)S\left(\frac{d \mathbf{r}}{d t}\right)_{S^{\prime}}

A small bead of mass mm is threaded on a smooth, rigid, circular wire of radius RR. At any given instant, the wire hangs in a vertical plane with respect to a downward gravitational acceleration g\mathbf{g}. The wire is rotating with constant angular velocity ω\boldsymbol{\omega} about its vertical diameter. Let θ(t)\theta(t) be the angle between the downward vertical and the radial line going from the centre of the hoop to the bead.

(i) Show that θ(t)\theta(t) satisfies the following equation of motion

θ¨=(ω2cosθgR)sinθ.\ddot{\theta}=\left(\omega^{2} \cos \theta-\frac{g}{R}\right) \sin \theta .

(ii) Find any equilibrium angles and determine their stability.

(iii) Find the force of the wire on the bead as a function of θ\theta and θ˙\dot{\theta}.

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