Simple Harmonic Motion (SHM): Definition, Formula, Working Principle, Equations, and More

• Displacement - It is defined as the distance of the particle from the mean position at that instant. Displacement in SHM at time t is given by $\mathrm{x}=\mathrm{A}\mathrm{s}\mathrm{i}\mathrm{n}(\omega \mathrm{t}+\varphi )$

• Amplitude : It is the maximum value of the displacement of the particle from its equilibrium position. Amplitude $=\frac{1}{2}$ [distance between extreme points or positions]. It depends on the energy of the system.
• Angular Frequency is $\left(\omega \right):\omega =\frac{2\pi }{\mathrm{T}}=2\pi \mathrm{f}$ and its unit is $\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}\mathrm{e}\mathrm{c}$ .
• Frequency (f): The number of oscillations completed in unit time interval is called the frequency of oscillations, $f=\frac{1}{T}=\frac{\omega }{2\pi }$ , its unit is ${\mathrm{s}\mathrm{e}\mathrm{c}}^{-1}$ or Hz .
• Time period (T) : This is the smallest time interval after which the oscillatory motion gets repeated is called time period, $T=\frac{2\pi }{\omega }=2\pi \sqrt[]{\frac{m}{k}}$
• Phase: The physical quantity which represents the state of motion of particle (eg. its position and direction of motion at any instant). The argument ( $\omega t+\varphi$ ) of sinusoidal function is called instantaneous phase of the motion.
• Phase constant ( $\varphi$ ): Constant $\varphi$ in equation of SHM is called the phase constant or the initial phase. It depends on the initial position and direction of velocity.
• Velocity(v): Velocity at an instant is the rate of change of the particle's position w.r.t time at that instant.
  Let the displacement from the mean position be given by

$\begin{array}{rr}& \mathrm{}x=A\mathrm{s}\mathrm{i}\mathrm{n}(\omega t+\varphi )\\ & \text{Velocity,}\mathrm{}v=\frac{dx}{dt}=\frac{d}{dt}[A\mathrm{s}\mathrm{i}\mathrm{n}(\omega t+\varphi \left)\right]\\ & v=A\omega \mathrm{c}\mathrm{o}\mathrm{s}(\omega t+\varphi )\mathrm{}\text{or,}\mathrm{}v=\omega \sqrt[]{A^2-x^2}\end{array}$

At the mean position ( $\mathrm{x}=0$ ), the velocity is maximum.

$V_{\mathrm{m}\mathrm{a}\mathrm{x}}=\omega A$

At the extreme position ( $x=A$ ), velocity is minimum.

$V_{\mathrm{m}\mathrm{i}\mathrm{n}}=zero$

GRAPH WOULD BE AN ELLIPSE

Acceleration at an instant is the rate of change of the particle's velocity with respect to time at that instant.

Acceleration, $\mathrm{}a=\frac{dv}{dt}=\frac{d}{dt}[A\omega \mathrm{c}\mathrm{o}\mathrm{s}(\omega t+\varphi \left)\right]$

$\begin{array}{rr}& a=-{\omega }^{2}A\mathrm{s}\mathrm{i}\mathrm{n}(\omega t+\varphi )\\ & a=-{\omega }^{2}x\end{array}$

Note

A negative sign shows that acceleration is always directed towards the mean position. At the mean position ( $x=0$ ), acceleration is minimum. $a_{\mathrm{m}\mathrm{i}\mathrm{n}}=\text{zero}$

At the extreme position ( $\mathrm{x}=\mathrm{A}$ ), acceleration is maximum.

$a_{\mathrm{m}\mathrm{a}\mathrm{x}}={\omega }^2\mathrm{A}$

GRAPH OF ACCELERATION (A) vs DISPLACEMENT (x)

$a=-{\omega }^{2}x$