Paper 4, Section II, B

Dynamics and Relativity | Part IA, 2012

For any frame SS and vector A\mathbf{A}, let [dAdt]S\left[\frac{d \mathbf{A}}{d t}\right]_{S} denote the derivative of A\mathbf{A} relative to SS. A frame of reference SS^{\prime} rotates with constant angular velocity ω\omega with respect to an inertial frame SS and the two frames have a common origin OO. [You may assume that for any vector A,[dAdt]S=[dAdt]S+ω×A.]\left.\mathbf{A},\left[\frac{d \mathbf{A}}{d t}\right]_{S}=\left[\frac{d \mathbf{A}}{d t}\right]_{S^{\prime}}+\boldsymbol{\omega} \times \mathbf{A} .\right]

(a) If r(t)\mathbf{r}(t) is the position vector of a point PP from OO, show that

[d2rdt2]S=[d2rdt2]S+2ω×v+ω×(ω×r)\left[\frac{d^{2} \mathbf{r}}{d t^{2}}\right]_{S}=\left[\frac{d^{2} \mathbf{r}}{d t^{2}}\right]_{S^{\prime}}+2 \boldsymbol{\omega} \times \mathbf{v}^{\prime}+\boldsymbol{\omega} \times(\boldsymbol{\omega} \times \mathbf{r})

where v=[drdt]S\mathbf{v}^{\prime}=\left[\frac{d \mathbf{r}}{d t}\right]_{S^{\prime}} is the velocity in SS^{\prime}.

Suppose now that r(t)\mathbf{r}(t) is the position vector of a particle of mass mm moving under a conservative force F=ϕ\mathbf{F}=-\nabla \phi and a force G\mathbf{G} that is always orthogonal to the velocity v\mathbf{v}^{\prime} in SS^{\prime}. Show that the quantity

E=12mvv+ϕm2(ω×r)(ω×r)E=\frac{1}{2} m \mathbf{v}^{\prime} \cdot \mathbf{v}^{\prime}+\phi-\frac{m}{2}(\boldsymbol{\omega} \times \mathbf{r}) \cdot(\boldsymbol{\omega} \times \mathbf{r})

is a constant of the motion. [You may assume that [(ω×r)(ω×r)]=2ω×(ω×r)\nabla[(\boldsymbol{\omega} \times \mathbf{r}) \cdot(\boldsymbol{\omega} \times \mathbf{r})]=-2 \boldsymbol{\omega} \times(\boldsymbol{\omega} \times \mathbf{r}).]

(b) A bead slides on a frictionless circular hoop of radius aa which is forced to rotate with constant angular speed ω\omega about a vertical diameter. Let θ(t)\theta(t) denote the angle between the line from the centre of the hoop to the bead and the downward vertical. Using the results of (a), or otherwise, show that

θ¨+(gaω2cosθ)sinθ=0.\ddot{\theta}+\left(\frac{g}{a}-\omega^{2} \cos \theta\right) \sin \theta=0 .

Deduce that if ω2>g/a\omega^{2}>g / a there are two equilibrium positions θ=θ0\theta=\theta_{0} off the axis of rotation, and show that these are stable equilibria.

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