A body A of mass m₁ and body B of mass m₂ are interconnected by a massless spring as shown. The body A performs free vertical harmonic oscillations with the amplitude A and frequency f . The maximum value of f such that body B does not leave the surface is

As A oscillates up and down, the normal force between B and surface varies from minimum to maximum. To keep body B in contact with the surface, the minimum $R_{\text{minimum }}\ge 0$ becomes minimum, when A is at topmost position. Free body diagrams are:

$T+m_{1}g=m_{1}A{w}^2$ . . . .(1)

$-T+m_{2}g=R\dots \dots \dots \dots \dots \dots \dots \dots \left(2\right)$

and $R\ge 0$ . . . . . . (3)

$\text{ solve for }{f}_{min}=?$