Applying Newton's Laws of Motion

Figure E.2: Mass-spring system.

As a simple example, consider a mass $m$ driven along a frictionless surface by an ideal spring $k$, as shown in Fig. E.2. Assume that the mass position $x=0$ corresponds to the spring at rest, i.e., not stretched or compressed. The force necessary to compress the spring by a distance $x$ is given by Hooke's law (§E.1.4):

$$f(t) = k\,x(t)$$

This force is balanced at all times by the inertial force $m{\ddot x}$ of the mass $m$, yielding^E.7

$$m{\ddot x}(t) + k\,x(t) = 0\, \quad \forall t\ge 0, \quad x(0)=A, \quad {\dot x}(0)=0. \protect$$ (E.4)

where we have defined $A$ as the initial displacement of the mass along $x$. This is a differential equation whose solution gives the equation of motion of the mass-spring junction for all time:^E.8

$$x(t) = A\cos(\omega_0 t), \quad \forall t\ge 0, \protect$$ (E.5)

where $\omega_0\isdef \sqrt{k/m}$ 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 $x(0)$ and initial velocities ${\dot x}(0)$, is the set of all sinusoidal oscillations at frequency $\omega_0$:

$$x(t) = A\cos(\omega_0 t + \phi), \quad \forall A,\phi\in{\bf R}$$

The amplitude of oscillation $A$ and phase offset $\phi$ are determined by the initial conditions, i.e., the initial position $x(0)$ and initial velocity ${\dot x}(0)$ of the mass (its initial state) when we ``let it go'' or ``push it off'' at time $t=0$.