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As a simple example, consider a mass driven along a frictionless
surface by an ideal spring , as shown in Fig. E.2.
Assume that the mass position corresponds to the spring at rest,
i.e., not stretched or compressed. The force necessary to compress the
spring by a distance is given by Hooke's law (§E.1.4):
This force is balanced at all times by the inertial force
of
the mass , yielding^{E.7}

(E.4) 
where we have defined as the initial displacement of the mass
along . This is a differential equation whose solution
gives the equation of motion of the massspring junction for all
time:^{E.8}

(E.5) 
where
denotes the frequency of
oscillation in radians per second. More generally, the complete
space of solutions to Eq. (E.4), corresponding to all possible
initial displacements and initial velocities
, is the
set of all sinusoidal oscillations at frequency :
The amplitude of oscillation and phase offset are
determined by the initial conditions, i.e., the initial position
and initial velocity
of the mass (its initial
state) when we ``let it go'' or ``push it off'' at time .
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