Energy in the Mass-Spring Oscillator

We say that a compressed spring holds a potential energy equal to the work required to compress the spring from rest to its current displacement. If a compressed spring is allowed to expand by pushing a mass, as in the system of Fig.E.2, the potential energy in the spring is converted to kinetic energy $E_m$, which is ``stored'' in the moving mass.

We can draw some inferences from the oscillatory motion of the mass-spring system written in Eq.$\,$(E.5):

$$x(t) = A\cos(\omega_0 t), \quad t\ge 0$$

• From a global point of view, we see that energy is conserved, since the oscillation never decays.
• At the peaks of the displacement $x(t)$ (when $\cos(\omega_0 t)$ is either $1$ or $-1$), all energy is in the form of potential energy, i.e., the spring is either maximally compressed or stretched, and the mass is momentarily stopped as it is changing direction.
• At the zero-crossings of $x(t)$, the spring is momentarily relaxed, thereby holding no potential energy; at these instants, all energy is in the form of kinetic energy, stored in the motion of the mass.
• Since total energy is conserved (§E.2.4), the kinetic energy of the mass at the displacement zero-crossings is exactly the amount needed to stretch the spring to displacement $-A$ (or compress it to $+A$) before the mass stops and changes direction. At all times, the total energy $E$ is equal to the sum of the potential energy $E_k(t)$ stored in the spring, and the kinetic energy $E_m(t)$ stored in the mass:
  $$E = E_k(t) + E_m(t) =$$   $$\mbox{constant [$E_k(0)$]}$$

The potential energy, $E_k(t)=kx^2/2$ is defined as the work required to displace the spring by $x$ meters, where work was defined in Eq.$\,$(E.6). The next section derives that the kinetic energy $E_m(t)$ associated with a mass $m$ moving at speed $v$ is given specifically by

$$E_m(t) = \frac{1}{2}m v^2(t) = \frac{1}{2}{\dot x}^2(t).$$

This result is easily obtained from the equation of motion for the mass-spring oscillator in conjunction with the above remarks (see Problem 3).