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Let 
represent the length of the spring when loaded, but at rest.
In that situation there is equilibrium
which implies that the string tension is equal
but opposite to the weight (mg)
of the object.
Cel
=
Fel(-)
= mg
Let us denote by u
the displacement from the equilibrium situation.
When the system is in movement, u
is a function of time:
u=u(t).
The elastic force is at each instant given by
Fel(-+u)
= - Cel×(-+u)
= Cel
- Celu
= mg - Celu
The resultant force on the object
is the sum
of the upward elastic force
and the downward force of its weight.
We will denote the resultant force just by
F.
F = Fel(-+u) - mg
= - Celu
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