Simple Harmonic Motion - Definition, Formula

First sight the eight physical systems :

• a simple pendulum, a mass $m$ swinging at the end of a light rigid rod of length $l$
  
• a flat disc supported by a rigid wire through its centre and oscillating through small angles in the plane of its circumference
  
• a mass fixed to a wall via a spring of stiffness $s$ sliding to and fro in the $x$ direction on a frictionless plane
• a mass $m$ at the centre of a light string of length $2l$ fixed at both ends under a constant tension $T$. The mass vibrates in the plane of the paper
  
• a frictionless U-tube of constant cross-sectional area containing a length $l$ of liquid, density $\rho$, oscillating about its equilibrium position of equal levels in each limb
  
• an open flask of volume $V$ and a neck of length $l$ and constant cross-sectional area $A$ in which the air of density $\rho$ vibrates as sound passes across the neck
  
• a hydrometer, a body of mass $m$ floating in a liquid of density $\rho$ with a neck of constant cross-sectional area cutting the liquid surface. When depressed slightly from its equilibrium position it performs small vertical oscillations
  
• an electrical circuit, an inductance $L$ connected across a capacitance $C$ carrying a charge $q$
  

All of these systems are simple harmonic oscillators which, when slightly disturbed from their equilibrium or rest postion, will oscillate with simple harmonic motion. This is the most fundamental vibration of a single particle or one-dimensional system. A small displacement $x$ from its equilibrium position sets up a restoring force which is proportional to $x$ acting in a direction towards the equilibrium position.

Thus, this restoring force $F$ may be written
$$F=-sx$$
where $s$, the constant of proportionality, is called the stiffness and the negative sign shows that the force is acting against the direction of increasing displacement and back towards the equilibrium position. A constant value of the stiffness restricts the displacement $x$ to small values (this is Hooke’s Law of Elasticity). The stiffness s is obviously the restoring force per unit distance (or displacement) and has the dimensions $$\frac{force}{distance}=\frac{MLT^{-2}}{L}$$
The equation of motion of such a disturbed system is given by the dynamic balance between the forces acting on the system, which by Newton’s Law is $$mass\,times\,acceleration=restoring\,force$$
or $$m\ddot{x}=-sx$$
where the acceleration $$\ddot{x}=\frac{d^{2}x}{dt^{2}}$$
This gives $$m\ddot{x}+sx=0$$

Or $$\ddot{x}+\frac{s}{m}x=0$$

where the dimensions of $$\frac{s}{m}\,\,are\,\,\frac{MLT^{-2}}{ML}=T^{-2}=\nu^{2}$$

Here $T$ is a time, or period of oscillation, the reciprocal of $v$ which is the frequency with which the system oscillates.

However, when we solve the equation of motion we shall find that the behaviour of $x$ with time has a sinusoidal or cosinusoidal dependence, and it will prove more appropriate to consider, not $\nu$, but the angular frequency $\omega$ = $2\pi\nu$ so that the period $$T=\frac{1}{\nu}=2\pi \sqrt{\frac{m}{s}}$$

where $s/m$ is now written as $\omega^{2}$. Thus the equation of simple harmonic motion  $$\ddot{x}+\frac{s}{m}x=0$$

becomes $$\ddot{x}+\omega ^{2}x=0$$

Reference [11]