Mass on a Spring

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

Here, is the so-called

Newton's second law of motion leads to the following time evolution equation for the system,

where . This differential equation is known as the

where , , and are constants. We can demonstrate that Equation (3) is indeed a solution of Equation (2) by direct substitution. Plugging the right-hand side of Equation (3) into Equation (2), and recalling from standard calculus that and (see Appendix B), so that and , where use has been made of the

(4) |

we obtain

(5) |

It follows that Equation (3) is the correct solution provided

Figure 2 shows a graph of
versus
derived from Equation (3). The type of motion displayed here is
called *simple harmonic oscillation*.
It can be seen that
the displacement
oscillates between
and
. Here,
is termed the *amplitude*
of the oscillation. Moreover, the motion is repetitive in time (i.e., it repeats exactly after
a certain time period has elapsed). The repetition *period* is

(7) |

This result can be obtained from Equation (3) by noting that is a periodic function of with period : that is, . It follows that the motion repeats each time increases by . In other words, each time increases by . The

(8) |

It is apparent that is the motion's

(9) |

Varying the phase angle shifts the pattern of oscillation backward and forward in time. (See Figure 3.)

Table 1 lists the displacement, velocity, and acceleration of the mass at various key points on the simple harmonic oscillation cycle. The information contained in this table is derived from Equation (3). All of the non-zero values shown in the table represent either the maximum or the minimum value taken by the quantity in question during the oscillation cycle.

As we have seen, when a mass on a spring is disturbed it executes simple harmonic oscillation about its equilibrium position. In physical terms, if the mass's initial displacement is positive ( ) then the restoring force is negative, and pulls the mass toward the equilibrium point ( ). However, when the mass reaches this point it is moving, and its inertia thus carries it onward, so that it acquires a negative displacement ( ). The restoring force then becomes positive, and pulls the mass toward the equilibrium point. However, inertia again carries it past this point, and the mass acquires a positive displacement. The motion subsequently repeats itself ad infinitum. The angular frequency of the oscillation is determined by the spring constant, , and the system inertia, , via Equation (6). On the other hand, the amplitude and phase angle of the oscillation are determined by the initial conditions. To be more exact, suppose that the instantaneous displacement and velocity of the mass at are and , respectively. It follows from Equation (3) that

(10) | ||

(11) |

Here, use has been made of the trigonometric identities and . (See Appendix B.) Hence, we deduce that

(12) |

and

because and . (See Appendix B.)

The kinetic energy of the system, which is the same as the kinetic energy of the mass, is written

(14) |

The potential energy of the system, which is the same as the potential energy of the spring, takes the form (Fitzpatrick 2012)

(15) |

Hence, the total energy is

because and . According to the previous expression, the total energy is a constant of the motion, and is proportional to the amplitude squared of the oscillation. Hence, we deduce that the simple harmonic oscillation of a mass on a spring is characterized by a continual back and forth flow of energy between kinetic and potential components. The kinetic energy attains its maximum value, and the potential energy its minimum value, when the displacement is zero (i.e., when ). Likewise, the potential energy attains its maximum value, and the kinetic energy its minimum value, when the displacement is maximal (i.e., when ). The minimum value of is zero, because the system is instantaneously at rest when the displacement is maximal.