Paper 4, Section II, C

Dynamics and Relativity | Part IA, 2014

A reference frame SS^{\prime} rotates with constant angular velocity ω\boldsymbol{\omega} relative to an inertial frame SS that has the same origin as SS^{\prime}. A particle of mass mm at position vector x\mathbf{x} is subject to a force F\mathbf{F}. Derive the equation of motion for the particle in SS^{\prime}.

A marble moves on a smooth plane which is inclined at an angle θ\theta to the horizontal. The whole plane rotates at constant angular speed ω\omega about a vertical axis through a point OO fixed in the plane. Coordinates (ξ,η)(\xi, \eta) are defined with respect to axes fixed in the plane: OξO \xi horizontal and OηO \eta up the line of greatest slope in the plane. Ensuring that you account for the normal reaction force, show that the motion of the marble obeys

ξ¨=ω2ξ+2ωη˙cosθ,η¨=ω2ηcos2θ2ωξ˙cosθgsinθ\begin{aligned} \ddot{\xi} &=\omega^{2} \xi+2 \omega \dot{\eta} \cos \theta, \\ \ddot{\eta} &=\omega^{2} \eta \cos ^{2} \theta-2 \omega \dot{\xi} \cos \theta-g \sin \theta \end{aligned}

By considering the marble's kinetic energy as measured on the plane in the rotating frame, or otherwise, find a constant of the motion.

[You may assume that the marble never leaves the plane.]

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