From Eq. 27 it is clear that springs support harmonic oscillations. k is simply the spring constant there and so
stiffer springs and lighter objects produce higher harmonic
oscillation frequencies. We verify the formulae qualitatively and quantitatively by measuring the period
of oscillations for carts on the air-track and comparing to
with known values for the mass, m, and the spring constant, k.
The resonance frequency stays the same if we consider a mass hanging from a spring because that system has the same magnitude
of restoring force if we have not driven the spring beyond its elastic
limit. Specifically if 
is the relaxed location of the spring
end when no mass is suspended and 
is the location when the
mass is suspended then we can write
and hence if x is the location of the spring end during the putative oscillations then
From this expression we immediately conclude that
the system can support harmonic oscillations with
Note that the amplitude, A, and phase, 
, are not
determined by the restoring force but by the nature of the
external disturbance which sets of the oscillation.
is in effect uninteresting since it just corresponds to a
shift in choice of origin that is when did the oscillation start.
The amplitude, A, can take on any value
consistent with the condition that the restoring force
remains linear in the displacement from equilibrium. In any real
physical system there are limits for the linear regime
beyond which the system may still oscillate but the oscillations
are no-longer perfectly harmonic.