Question

A block is resting on a piston which executes simple harmonic motion with a period 2 s . what is the maximum velocity of the piston , at an amplitude just sufficient for the block to separate from the piston ?

Answer 1

$$\begin{gathered} \mathbf{Maximum}\mathbf{}\mathbf{accleration}\mathbf{}\mathbf{at}\mathbf{}\mathbf{which}\mathbf{}\mathbf{block}\mathbf{}\mathbf{separate}\mathbf{:} \\ {\mathrm{a}}_{\max }=\mathrm{g} \\ {\mathrm{\omega }}^2\mathrm{A}=\mathrm{g} \\ \mathrm{A}=\frac{\mathrm{g}}{{\mathrm{\omega }}^2} \\ =\frac{\mathrm{g}}{{\left({\displaystyle \frac{2\mathrm{\pi }}{\mathrm{T}}}\right)}^2} \\ =\frac{{\mathrm{gT}}^2}{4{\mathrm{\pi }}^2} \\ \mathbf{Maximum}\mathbf{}\mathbf{velocity}\mathbf{}\mathbf{of}\mathbf{}\mathbf{the}\mathbf{}\mathbf{piston}\mathbf{:} \\ {\mathbf{V}}_{\mathbf{max}}\mathbf{=}\mathbf{\omega A} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{2}\mathbf{\pi }}{\mathbf{T}}\mathbf{\times }\frac{{\mathbf{gT}}^{\mathbf{2}}}{\mathbf{4}{\mathbf{\pi }}^{\mathbf{2}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{gT}}{\mathbf{2}\mathbf{\pi }} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{9}\mathbf{.}\mathbf{8}\mathbf{\times }\mathbf{2}}{\mathbf{2}\mathbf{\pi }} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{1}\mathbf{}\mathbf{m}\mathbf{/}\mathbf{s} \end{gathered}$$