Q.A wooden block of mass m is kept on a piston that can perform vertical vibrations of adjustable frequency and amplitude. During vibrations, we don't want the block to leave contact with the piston. How much maximum frequency is possible if the amplitude of vibrations is restricted to 25 cm? In this case, how much is the energy per unit mass of the block? (g $\approx\pi^2\approx10$ m/s$^2$)

[!TLDR] Condition: $a_{max}=\omega^2A\le g\Rightarrow\omega_{max}=\sqrt{g/A}$. With $g\approx\pi^2\approx10$ m/s$^2$, A=0.25 m: $\omega_{max}=2\pi$ rad/s, so $n_{max}=1$ Hz. Energy/mass $=\ …

For the block to stay in contact with the vibrating piston at all times, the piston's downward acceleration (during the part of the cycle where it accelerates downward fastest, i.e. at the upper extreme) must never exceed g -- otherwise the piston would accelerate away faster than gravity can pull the block down, and the block would leave the surface. Since the maximum acceleration in S.H.M. is $a_{max}=\omega^2A$ (section 5.5), the limiting condition is $\omega^2A\le g$, i.e. maximum allowed $\omega$ is $\omega_{max}=\sqrt{g/A}$. Using $A=25$ cm $=0.25$ m and $g\approx\pi^2\approx10$ m/s$^2$ (as given), $\omega_{max}=\sqrt{\frac{\pi^2}{0.25}}=\sqrt{4\pi^2}=2\pi\text{ rad/s}$ so the maximum frequency is $n_{max}=\frac{\omega_{max}}{2\pi}=\frac{2\pi}{2\pi}=1$ Hz (1 oscilla …

Set the S.H.M. maximum acceleration $\omega^2A$ equal to g (the limiting no-lift-off condition), solve for $\omega_{max}$ and hence …

Assuming contact is lost at the MEAN position (where speed, not acceleration, is maximum) instead of correctly identifying the extreme position …