Dynamics | Part IA, 2002

An inertial reference frame SS and another reference frame SS^{\prime} have a common origin O. SS^{\prime} rotates with constant angular velocity ω\omega with respect to SS. Assuming the result that

(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}}+\boldsymbol{\omega} \times \mathbf{a}

for an arbitrary vector a(t)\mathbf{a}(t), show that

(d2xdt2)S=(d2xdt2)S+2ω×(dxdt)S+ω×(ω×x)\left(\frac{d^{2} \mathbf{x}}{d t^{2}}\right)_{\mathcal{S}}=\left(\frac{d^{2} \mathbf{x}}{d t^{2}}\right)_{\mathcal{S}^{\prime}}+2 \boldsymbol{\omega} \times\left(\frac{d \mathbf{x}}{d t}\right)_{\mathcal{S}^{\prime}}+\boldsymbol{\omega} \times(\boldsymbol{\omega} \times \mathbf{x})

where x\mathbf{x} is the position vector of a point PP measured from the origin.

A system of electrically charged particles, all with equal masses mm and charges ee, moves under the influence of mutual central forces Fij\mathbf{F}_{i j} of the form

Fij=(xixj)f(xixj)\mathbf{F}_{i j}=\left(\mathbf{x}_{i}-\mathbf{x}_{j}\right) f\left(\left|\mathbf{x}_{i}-\mathbf{x}_{j}\right|\right)

In addition each particle experiences a Lorentz force due to a constant weak magnetic field B\mathbf{B} given by

edxidt×Be \frac{d \mathbf{x}_{i}}{d t} \times \mathbf{B}

Transform the equations of motion to the rotating frame S\mathcal{S}^{\prime}. Show that if the angular velocity is chosen to satisfy

ω=e2mB\boldsymbol{\omega}=-\frac{e}{2 m} \mathbf{B}

and if terms of second order in B\mathbf{B} are neglected, then the equations of motion in the rotating frame are identical to those in the non-rotating frame in the absence of the magnetic field B.

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