Question

Obtain an expression for the frequency of vertical oscillations ofa loaded spring.

Answer 1

mass  m  is attached to the upper end of a mass less spring with a constant of k.
The lower end of spring is attached to a firm surface of a rigid body.
 
 under the weight, the spring gets compressed a little, the displacement being α.  The mass stays in equilibrium and stationary.  Let us call the displacement from this equilibrium position = x, positive upwards.

     m g = k α      =>  α = m g / k

   Now if the mass is displaced a little from this equilibrium, then it will oscillated in SHM.  Let X be the displacement of the mass from its unextended length.  Let x be  the displacement from the equilibrium position.
                X = x - α

  Forces acting on the mass:   - m g downwards. and  - k x  (negative as when x is positive, the force is downwards).
           F = m a = m d²X / dt² = - m g - k X
           m d²x/ dt² =  - m g - k x + k α  = - k x  = - ω²  x

        so angular frequency = ω = √(k/m)
         frequency = f = ω/2π  = 1/2π √(k/m)