Simple Harmonic Motion and Oscillations

Period ($T$)

The time taken for an oscillating object to complete one full cycle of oscillation.

Formula:$T = \frac{\Delta t}{N} = \frac{2\pi}{\omega} = \frac{1}{f}$ .

SI unit: second ($\text{s}$).

Displacement ($x$)

The signed distance of an oscillating object from its equilibrium position at a given moment in time. Formula: $x = A \cos(\omega t + \varphi)$. SI unit: meter ($\text{m}$). It equals $0$ at the equilibrium position.

Amplitude ($A$)

The maximum magnitude of displacement of an oscillating object from its equilibrium position ($A = |x|_{\max}$). It is always positive ($A > 0$). SI unit: meter ($\text{m}$).

Phase ($\omega t + \varphi$)

The quantity that determines the state of oscillation (position and direction of motion) of an object at a given time $t$. Formula: $\theta = \omega t + \varphi$. SI unit: radian ($\text{rad}$).

Initial Phase ($\varphi$)

The phase of oscillation at the time origin ($t = 0$), which determines the initial state of oscillation. It depends on the choice of the time reference. SI unit: radian ($\text{rad}$).

Angular Frequency ($\omega$)

The rate of change of the phase of an oscillating object with respect to time. Formula: $\omega = 2\pi f = \frac{2\pi}{T}$. SI unit: radian per second ($\text{rad/s}$).

Simple Harmonic Motion (SHM)

A periodic motion in which the displacement varies sinusoidally with time according to the equation $x = A \cos(\omega t + \varphi)$, and the acceleration is proportional and opposite to the displacement: $a = -\omega^2 x$.

Equilibrium Position

The position where the net force acting on the oscillating object is zero ($F_{\text{net}} = 0$). At this position, displacement is zero ($x = 0$), speed reaches its maximum ($|v| = v_{\max} = \omega A$), and acceleration is zero ($a = 0$).

Extreme Position

The position farthest from the equilibrium position ($x = \pm A$). At this position, instantaneous velocity is zero ($v = 0$) and acceleration magnitude reaches its maximum ($|a| = a_{\max} = \omega^2 A$).

Velocity (in SHM)