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Applying Newton's Laws of Motion
As a simple example, consider a mass
driven along a frictionless
surface by an ideal spring
, as shown in Fig.B.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 (§B.1.3):
This force is balanced at all times by the inertial force
of
the mass
, i.e.
,
yielding^{B.6}

(B.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:^{B.7}

(B.5) 
where
denotes the frequency of
oscillation in radians per second. More generally, the complete
space of solutions to Eq.
(B.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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