Question

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 the 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 ≈ π² ≈ 10 m s²)

Answer 1

Given : A = 0.25m, g = π² = 10 m s²

During vertical oscillations, the acceleration is maximum at the turning points at the top and bottom. The block will just lose contact with the piston when its apparent weight is zero at the top,

i.e., when its acceleration is a_max = g, downwards.

|a_max| = ω²A = 4π² η²_max A = g

∴ η²_max  = \(\sqrt{\frac{g}{4π^2A}}\) = \(\sqrt{\frac{10}{4×10×0.25}}\) = 1 Hz

This gives the required frequency of the piston.

E = ½ mω²A² = ½ m(4π² η²)A²

∴ E/m = 2p² h²A² = 2 x 10 x 1² x (1/4)² = 1.25 J/kg