Harmonic Oscillator: Introduction

Harmonic Oscillator

Introduction

In nature, we may come across motions which repeat itself over time, like the motion of the earth around the sun. It is called periodic motion. A special type of periodic motion is the motion which is bounded over a certain range, like the motion of a pendulum. Such motion is called a Simple Harmonic Motion (SHM). This kind of motion occurs all over nature; pendulum, wired musical instruments, charges in certain circuits all show SHM. The objects undergoing SHM are termed as harmonic oscillators.

Potential Energy Curve

This curve shows the variation of PE of a particle with its position. For a conservative system, the potential energy is the function of position co-ordinate alone. PE is independent of velocity and time. At positions of equilibrium,

$$\begin{align*}\frac{\mathrm{d} U}{\mathrm{d} x}=0 \end{align*}$$ $$\begin{align*} F=- \frac{\mathrm{d} U}{\mathrm{d} x}=0 \end{align*}$$ Except at positions of equilibrium, the motion of the particles is confined between certain regions called potential well.

Properties of Harmonic Oscillators

The harmonic oscillator's acceleration at any point is directly proportional to its displacement but directed opposite. So, the force acting on the oscillator at any point can also be written as, $$\begin{align*} F=-Cx \end{align*}$$ But, from Newton's Second Law of motion, $F=ma$ and $a=\frac{\mathrm{d^2x} }{\mathrm{d} t^2}$, $$\begin{align*} m\frac{\mathrm{d^2x} }{\mathrm{d} t^2}=-Cx \end{align*}$$ $$\begin{align*}\frac{\mathrm{d^2x} }{\mathrm{d} t^2}+\frac{C}{m}x=0 \end{align*}$$ This is the differential equation of SHM.

$$\begin{align*} \arcsin \frac{x}{a}=\omega t+\phi\end{align*}$$ $$\begin{align*} x=a\sin (\omega t+\phi ) \end{align*}$$ It gives the displacement of particle at any instant.

A Harmonic Oscillator is an example of the conservative system; the total mechanical energy remains conserved. Assuming zero potential energy at equilibrium, the PE of a harmonic oscillator with displacement x is $$\begin{align*}V=-\int f\mathrm{dx}=\int kx\mathrm{dx}=\frac{kx^2}{2} \end{align*}$$ If we plot V versus x, it will be a parabola.

The kinetic energy of the harmonic oscillator is, $$\begin{align*}KE=T=\frac{1}{2}mv^2=\frac{1}{2}m\omega ^2(a^2-x^2)\end{align*}$$ Thus, the total energy of the oscillator is given by, $$\begin{align*}E=T+V=\frac{1}{2}m\omega ^2(a^2-x^2)+\frac{1}{2}kx^2 \end{align*}$$ $$\begin{align*}E=\frac{1}{2}m\omega ^2a^2 \end{align*}$$ Thus, we see that the total mechanical energy of a harmonic oscillator remains conserved.

The average PE of the oscillator over the period T is given by, $$\begin{align*} \overline{V}=\frac{1}{T}\int_{0}^{T}\frac{1}{2}ka^2\sin ^2(\omega t+\phi )\mathrm{dt}=\frac{1}{4}ka^2\end{align*}$$ The average KE of the oscillator is, $$\begin{align*} \overline{T}=\frac{1}{T}\int_{0}^{T}\frac{1}{2}ma^2\omega ^2\cos ^2(\omega t+\phi )\mathrm{dt}=\frac{1}{4}ka^2\end{align*}$$ Hence, it is seen that $E=\overline{T}+\overline{V}=\frac{1}{2}m\omega ^2a^2$

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