Energy in the Mass-Spring Oscillator

Summarizing the previous sections, 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.B.2, the potential energy in the spring is converted to kinetic energy in the moving mass.

We can draw some inferences from the oscillatory motion of the mass-spring system written in Eq.(B.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 $\cos(\omega_0t)$ 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 (§B.2.5), 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:
  $$E = E_k(t) + E_m(t) =$$   $$\mbox{constant [$E_k(0)$]}$$

Regarding the last point, the potential energy, $E_k(t)=k\,x^2(t)/2$ was defined as the work required to displace the spring by $x$ meters, where work was defined in Eq.(B.6). The kinetic energy of a mass $m$ moving at speed $v$ was found to be $E_m(t)=m\,v^2(t)/2=m\,{\dot x}^2/2$ . The constance of the potential plus kinetic energy at all times in the mass-spring oscillator is easily obtained from its equation of motion using the trigonometric identity $\cos^2(\theta)+\sin^2(\theta)=1$ (see Problem 3).