A body `A` of mass `m_(1)` and body `B` of mass `m_(2)` 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

Options

A. `(1)/(2pi) sqrt(((m_(1)+m_(2))/(m_(2)))(g)/(A))`
B. `(1)/(2pi)sqrt(((m_(1)+m_(2))/(m_(1)))(g)/(A))`
C. `(1)/(2pi) sqrt((g)/(A))`
D. `(1)/(2pi) sqrt((m_(2))/(m_(1)).(g)/(A))`

Answer

B

Explanation

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 `R_("minimum") ge 0` and `R` becomes minimum, when `A` is at topmost position. Free body diagrams are:

`T + m_(1)g = m_(1)Aw^(2) .............(1)`
`-T +m_(2)g = R..........(2)`
and `R ge 0 ...............(3)`
solve for `f_("min") = ?`