The equilibrium state of the system corresponds to the situation where
the mass is at rest, and the spring is unextended (*i.e.*, ).
In this state, zero net force acts on the mass, so there is no reason for it to start to move.
If the system is perturbed from this equilibrium state (*i.e.*, if the mass is moved, so that the
spring becomes extended) then the mass experiences a *restoring force* given by Hooke's law:

(503) |

Newton's second law gives following equation of motion for the system:

where , , and are constants. We can demonstrate that Eq. (505) is indeed a solution of Eq. (504) by direct substitution. Substituting Eq. (505) into Eq. (504), and recalling from calculus that and , we obtain

(506) |

Figure 95 shows a graph of versus obtained from Eq. (505). The type of motion shown here is
called *simple harmonic motion*.
It can be seen that
the displacement *oscillates* between and . Here, is termed the *amplitude*
of the oscillation. Moreover, the motion is *periodic* in time (*i.e.*, it repeats exactly after
a certain time period has elapsed). In fact, the *period* is

(508) |

(509) |

(510) |

Table 4 lists the displacement, velocity, and acceleration of the mass at various phases of the simple harmonic cycle. The information contained in this table can easily be derived from the simple harmonic equation, Eq. (505). Note that all of the non-zero values shown in this table represent either the maximum or the minimum value taken by the quantity in question during the oscillation cycle.

We have seen that when a mass on a spring is disturbed from equilibrium it executes *simple harmonic
motion* about its equilibrium state. In physical terms, if the initial displacement is positive () then the
restoring force *overcompensates*, and sends the system past the equilibrium state () to
negative displacement states (). The restoring force again overcompensates, and sends the
system back through to positive displacement states. The motion then repeats itself *ad infinitum*.
The frequency of the oscillation is determined by the spring stiffness, , and the system
inertia, , via Eq. (507).
In contrast, the amplitude and phase angle of the oscillation are determined by the *initial conditions*.
Suppose that the instantaneous displacement and velocity of the mass at are and ,
respectively. It follows from Eq. (505) that

(511) | |||

(512) |

Here, use has been made of the well-known identities and . Hence, we obtain

(513) |

(514) |

The kinetic energy of the system is written

(515) |

(516) |

(517) |