A small wooden cube is placed on a plank. The plank performs a vertical S.H.M of frequency \(\frac{3}{\pi} \mathrm{Hz}\). The maximum amplitude of the plank so that the wooden block does not leave the plank is [take \(g=10 \mathrm{~m} / \mathrm{s}^2\) ]

Concept: wooden cube can leave the plank at the top extreme location when the plank just begins to move towards the equilibrium position. See the figure below:


For cube not to leave plank the weight of the block should be more than the pseudo force on the cube:
\(\text { ma } \leq \mathrm{mg}\)
Therefore, \(\mathrm{a} \leq \mathrm{g}\).
\(|a|=\omega^2 x\) when, \(x=A\) the amplitude
\(\therefore \omega^2 \mathrm{~A} \leq \mathrm{g} \text { or } \mathrm{A} \leq \frac{\mathrm{g}}{\omega^2}\)
Given, \(\mathrm{f}=\frac{3}{\pi} \mathrm{Hz} \frac{\omega}{2 \pi}=\frac{1}{\mathrm{~T}}\)
\(\Rightarrow \omega=6 \mathrm{~Hz}\)
\(A \leq \frac{10}{6^2} \mathrm{~m}\)
\(\mathrm{A} \leq \frac{5}{18} \mathrm{~m}\)