4.II.9A

A horizontal table oscillates with a displacement $\mathbf{A} \sin \omega t$, where $\mathbf{A}=\left(A_{x}, 0, A_{z}\right)$ is the amplitude vector and $\omega$ the angular frequency in an inertial frame of reference with the $z$ axis vertically upwards, normal to the table. A block sitting on the table has mass $m$ and linear friction that results in a force $\mathbf{F}=-\lambda \mathbf{u}$, where $\lambda$ is a constant and $\mathbf{u}$ 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 $(\xi, \eta, \zeta)$ on the table parallel to the axes $(x, y, z)$ in the inertial frame.

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

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

where

$\sin ^{2} \theta=\frac{m^{2} \omega^{2}}{\lambda^{2}+m^{2} \omega^{2}}$

and $\xi_{0}$ 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 $\omega^{2} A_{z}<g$.

*Typos? Please submit corrections to this page on GitHub.*