### 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):

*i.e.*, yielding

^{B.6}

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 mass-spring junction for all time:

^{B.7}

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 :

*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 .

**Next Section:**

Potential Energy in a Spring

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Hooke's Law