Paper 4, Section II, A

Dynamics and Relativity | Part IA, 2018

The position x=(x,y,z)\mathbf{x}=(x, y, z) and velocity x˙\dot{\mathbf{x}} of a particle of mass mm are measured in a frame which rotates at constant angular velocity ω\omega with respect to an inertial frame. The particle is subject to a force F=9mω2x\mathbf{F}=-9 m|\boldsymbol{\omega}|^{2} \mathbf{x}. What is the equation of motion of the particle?

Find the trajectory of the particle in the coordinates (x,y,z)(x, y, z) if ω=(0,0,ω)\boldsymbol{\omega}=(0,0, \omega) and at t=0,x=(1,0,0)t=0, \mathbf{x}=(1,0,0) and x˙=(0,0,0)\dot{\mathbf{x}}=(0,0,0).

Find the maximum value of the speed x˙|\dot{\mathbf{x}}| of the particle and the times at which it travels at this speed.

[Hint: You may find using the variable ξ=x+iy\xi=x+i y helpful.]

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