Energy in the harmonic oscillator

The total energy of the oscillator (mass plus spring, without friction) is conserved. During the motion it is continually exchanged between two forms: kinetic, maximum at equilibrium where the mass whizzes past, and elastic, maximum at the extremes where the spring is most deformed and the body is momentarily at rest.

Let us write the two energies using $x(t) = A\cos(\omega t + \varphi)$ and $v(t) = -A\omega\sin(\omega t + \varphi)$, recalling that $m\omega^2 = K$:

$\begin{aligned} E_k(t) &= \tfrac{1}{2} m v^2 = \tfrac{1}{2} m A^2\omega^2 \sin^2(\omega t + \varphi) = \tfrac{1}{2}K A^2\sin^2(\omega t + \varphi) \\ E_p(t) &= \tfrac{1}{2}K x^2 = \tfrac{1}{2}K A^2\cos^2(\omega t + \varphi) \end{aligned}$

Adding them together and using the identity $\sin^2 + \cos^2 = 1$, time disappears:

$\ev{E_\text{tot} = E_k + E_p = \tfrac{1}{2}\,K\,A^2}$

The total energy is constant and depends only on the amplitude.

Principle — Total energy of the harmonic oscillator

The total energy of a harmonic oscillator is proportional to the square of the amplitude. If we double the amplitude, the energy quadruples; if we want to halve the energy, we must divide the amplitude by $\sqrt{2}$.

A universal signature

The square of the amplitude is the signature of all wave phenomena: from the energy of a mechanical wave to the intensity of sound, right up to the energy carried by photons. Wherever there is an oscillation, the energy scales as $A^2$.

Example — Maximum speed from energy

For a mass-spring system with $K = 50$ N/m and $A = 8{,}1$ cm, the total energy is $E = \tfrac{1}{2}K A^2 = 0{,}5\cdot 50\cdot (0{,}081)^2 \approx 0{,}164$ J. The maximum speed is $v_\text{max} = A\omega = 0{,}081\cdot 15{,}8 \approx 1{,}28$ m/s. Check: $\tfrac{1}{2}m v_\text{max}^2 = 0{,}5\cdot 0{,}2\cdot (1{,}28)^2 \approx 0{,}164$ J. The balance closes: at equilibrium all the energy is kinetic.

Collegamenti

Argomenti: Oscillations and harmonic motion Concetti: Simple harmonic motion · Elastic potential energy · Kinetic energy · Conservation of mechanical energy Competenze: Conservation of energy Oggetti: Spring

Esercizi collegati: Worked exercise — Maximum speed from energy · Problem — From amplitude to k · Worked exercise — Vertical spring with energy balance