Dynamics | Part IA, 2004

A horizontal table oscillates with a displacement Asinωt\mathbf{A} \sin \omega t, where A=(Ax,0,Az)\mathbf{A}=\left(A_{x}, 0, A_{z}\right) is the amplitude vector and ω\omega the angular frequency in an inertial frame of reference with the zz axis vertically upwards, normal to the table. A block sitting on the table has mass mm and linear friction that results in a force F=λu\mathbf{F}=-\lambda \mathbf{u}, where λ\lambda is a constant and u\mathbf{u} is the velocity difference between the block and the table. Derive the equations of motion for this block in the frame of reference of the table using axes (ξ,η,ζ)(\xi, \eta, \zeta) on the table parallel to the axes (x,y,z)(x, y, z) in the inertial frame.

For the case where Az=0A_{z}=0, show that at late time the block will approach the steady orbit

ξ=ξ0Axsinθcos(ωtθ),\xi=\xi_{0}-A_{x} \sin \theta \cos (\omega t-\theta),


sin2θ=m2ω2λ2+m2ω2\sin ^{2} \theta=\frac{m^{2} \omega^{2}}{\lambda^{2}+m^{2} \omega^{2}}

and ξ0\xi_{0} is a constant.

Given that there are no attractive forces between block and table, show that the block will only remain in contact with the table if ω2Az<g\omega^{2} A_{z}<g.

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