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Solutions were already suggested, in the form
u(t) = A sin(ωt)
Indeed, the second order time derivative of such expression gives
du/dt = ωA cos(ωt)
and
d2u/dt2
= -ω2A sin(ωt)
= -ω2 u(t)
This solution relates the angular frequency
ω
with the elasticity constant
Cel
and the mass m
of the object:
ω2
= Cel/m
or
ω
= √(Cel/m)
So, when we know the mass m
of the object and the elasticity constant
Cel
of the spring,
then we can predict the period
T=2π/ω
or the frequency f=ω/2π
of the oscillations of the mass-spring system.
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