A block is kept on a horizontal table. The table is executing simple harmonic motion of time period T in the horizontal plane. The coefficient of static friction between the block and the table is $\mu$. The maximum amplitude of the table for which the block does not slip on the surface of the table is:

 Let the mass of the block be m and let, at a certain instant of time, the direction of acceleration a of the table (executing simple harmonic motion) be along the positive x-direction. As a result, the block will experience a force ma directed along the negative x-axis. Consequently, the force of friction $\mu mg$ will act along the positive x-axis. The weight mg of the block will be balanced by the normal reaction R. The block will not slip on the surface of the table, if the acceleration a of the motion of the table is such that $\mathrm{\mu mg}\ge \mathrm{ma} \mathrm{or} \mathrm{\mu g}\ge \mathrm{a}$

Therefore, for no slipping, the table can have a maximum acceleration $a_{\max }=\mu g$. We know that, for a simple harmonic motion, $a_{\max }={\omega }^{2}A$, where $\omega$ is the angular frequency and A the amplitude of the motion of the table. Therefore, the maximum amplitude is given by ${\omega }^2{A}_{\max }=\mu g$   or $A_{\max }=\frac{\mu g}{{\omega }^2}=\frac{\mu g{T}^2}{4{\mathrm{\pi }}^2}$.