Dynamics | Part IA, 2001

The position x\mathbf{x} and velocity x˙\dot{\mathbf{x}} of a particle of mass mm are measured in a frame which rotates at constant angular velocity ω\boldsymbol{\omega} with respect to an inertial frame. Write down the equation of motion of the particle under a force F=4mω2x\mathbf{F}=-4 m \omega^{2} \mathbf{x}.

Find the motion of the particle in (x,y,z)(x, y, z) coordinates with initial condition

x=(1,0,0) and x˙=(0,0,0) at t=0\mathbf{x}=(1,0,0) \quad \text { and } \quad \dot{\mathbf{x}}=(0,0,0) \quad \text { at } t=0

where ω=(0,0,ω)\boldsymbol{\omega}=(0,0, \omega). Show that the particle has a maximum speed at t=(2n+1)π/4ωt=(2 n+1) \pi / 4 \omega, and find this speed.

[Hint: you may find it useful to consider the combination ζ=x+iy\zeta=x+i y.]

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