Oscillations and Simple Harmonic Motion

A comprehensive collection of vocabulary flashcards covering kinematics, dynamics, energy equations, and oscillation types in simple harmonic motion.

Displacement (in oscillation)

The signed distance of an oscillating object from its equilibrium position, given by $x = A \cos(\omega t + \varphi)$. Measured in meters ($\text{m}$), it can be positive, negative, or zero (when passing through the equilibrium position).

Velocity in simple harmonic motion

The rate of change of displacement with respect to time, given by $v = x' = -\omega A \sin(\omega t + \varphi) = \omega A \cos\left(\omega t + \varphi + \frac{\pi}{2}\right)$. Measured in $\text{m/s}$, it leads displacement in phase by $\frac{\pi}{2}$, equals zero at both extreme positions, and reaches maximum speed $|v|_{\max} = \omega A$ at the equilibrium position.

Amplitude

The maximum magnitude of displacement from the equilibrium position ($A = |x|_{\max}$). Measured in meters ($\text{m}$), it is always positive ($A > 0$), and the total trajectory length of the oscillation is $L = 2A$.

Phase and Initial Phase

The phase ($\omega t + \varphi$) describes the state of oscillation at time $t$, while the initial phase ($\varphi$) determines the state at time $t = 0$. Both are measured in radians ($\text{rad}$), with $\varphi$ depending on the choice of time origin ($t = 0$) and the positive direction of the coordinate axis.

Angular frequency

The rate of change of the phase of an oscillating object with respect to time, given by $\omega = 2\pi f = \frac{2\pi}{T}$. Measured in $\text{rad/s}$, it is a positive constant for a given simple harmonic motion.

Simple harmonic motion (SHM)

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

Equilibrium position

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

Extreme position

The two positions farthest from the equilibrium position during oscillation. At these points, $|x| = A$, velocity $v = 0$, and acceleration reaches its maximum magnitude ($|a| = a_{\max} = \omega^2 A$) directed toward the equilibrium position.

Period and Frequency

Period ($T = \frac{\Delta t}{N} = \frac{1}{f} = \frac{2\pi}{\omega}$) is the time required to complete one full cycle, measured in seconds ($\text{s}$). Frequency ($f = \frac{N}{\Delta t} = \frac{1}{T} = \frac{\omega}{2\pi}$) is the number of complete cycles per unit time, measured in hertz ($\text{Hz}$).

Time-independent relation between displacement and velocity

The formula $\frac{x^2}{A^2} + \frac{v^2}{\omega^2 A^2} = 1$, which gives the magnitude of velocity as $|v| = \omega \sqrt{A^2 - x^2}$. At the equilibrium position ($x = 0$), $|v| = \omega A$, and at the extreme positions ($x = \pm A$), $v = 0$.

Speed in simple harmonic motion

The magnitude of an oscillating object's instantaneous velocity ($v_{\text{speed}} = |v| \ge 0$). Measured in $\text{m/s}$ or $\text{cm/s}$, it reaches a maximum at the equilibrium position ($v_{\max} = \omega A$) and a minimum of zero at the extreme positions ($v_{\min} = 0$).

Average Speed (in SHM)

The total distance travelled divided by the total elapsed time ($v_{tb} = \frac{S}{\Delta t}$). Over one complete period ($T$), the distance covered is $S = 4A$, yielding an average speed of $v_{tb} = \frac{4A}{T} = \frac{2\omega A}{\pi} = \frac{2v_{\max}}{\pi}$, measured in $\text{m/s}$.

Acceleration in simple harmonic motion

The rate of change of an object's velocity with respect to time, given by $a = v' = -\omega^2 x = \omega^2 A \cos(\omega t + \varphi + \pi)$. Measured in $\text{m/s}^2$, it always points toward the equilibrium position, is in antiphase with displacement, is zero at $x = 0$, reaches an algebraic minimum $a_{\min} = -\omega^2 A$ at $x = A$, and an algebraic maximum $a_{\max} = \omega^2 A$ at $x = -A$.

Kinetic energy (in SHM)

The energy an oscillating object possesses due to its motion, defined by $W_đ = \frac{1}{2}mv^2 = \frac{1}{2}m\omega^2 (A^2 - x^2)$. Measured in joules ($\text{J}$), it varies periodically with angular frequency $\omega' = 2\omega$, reaching its maximum at the equilibrium position and zero at the extreme positions.

Potential energy (in SHM)

The stored energy of an oscillating system associated with its displacement, defined by $W_t = \frac{1}{2}m\omega^2 x^2 = \frac{1}{2}kx^2$. Measured in joules ($\text{J}$), it varies periodically with period $T' = \frac{T}{2}$, reaching its maximum at the extreme positions ($x = \pm A$) and vanishing at the equilibrium position ($x = 0$).

Mechanical energy (in SHM)

The sum of kinetic energy and potential energy of the oscillating system: $W = W_đ + W_t = \frac{1}{2}kA^2 = \frac{1}{2}m\omega^2 A^2$. Measured in joules ($\text{J}$), it remains constant when friction is neglected and is directly proportional to the square of the amplitude.

Spring pendulum

A system consisting of a mass $m$ attached to a spring of stiffness $k$ with the other end fixed. It oscillates with angular frequency $\omega = \sqrt{\frac{k}{m}}$ and period $T = 2\pi \sqrt{\frac{m}{k}}$. For a vertically suspended spring, $\Delta l_0 = \frac{mg}{k} \implies \omega = \sqrt{\frac{g}{\Delta l_0}}$.

Simple pendulum

A system consisting of a small bob of mass $m$ suspended from an inextensible string of length $l$ with negligible mass. For small angular displacements, its angular frequency is $\omega = \sqrt{\frac{g}{l}}$, its period is $T = 2\pi \sqrt{\frac{l}{g}}$, and its potential energy is $W_t \approx \frac{1}{2}mgl\alpha^2$.

Damped oscillation

An oscillatory motion whose amplitude and mechanical energy decrease over time due to environmental resistance and friction converting mechanical energy into thermal energy. Practical applications include shock absorbers in motorcycles and cars, as well as automatic door-closing devices.

Forced oscillation

An oscillation that occurs under the continuous influence of an external periodic driving force $F = F_0 \cos(\Omega t + \varphi_F)$. Its oscillation frequency equals the driving force frequency ($f = f_{\text{driving}}$), and its amplitude depends on the driving force amplitude $F_0$, medium damping, and the frequency detuning $|\Omega - \omega_0|$.

Resonance

The phenomenon in which the amplitude of a forced oscillation increases rapidly and reaches a maximum value when the driving frequency equals the system's natural frequency ($f = f_0$). It is utilized in sound boxes of guitars and violins to amplify acoustic waves, and must be avoided in bridges and buildings to prevent structural damage.

Free oscillation

An oscillation of a system that occurs solely under the influence of internal restoring forces after being disturbed from equilibrium. Its oscillation frequency depends only on the inherent structural properties of the system ($\omega_0$, $f_0$) rather than external factors.

Resistive force

The force opposing the relative motion of an oscillating body through its surrounding medium. The change in mechanical energy equals the work performed by the resistive force ($\Delta W = W_2 - W_1 = A_{\text{resistive}} < 0$), measured in newtons ($\text{N}$); greater resistive forces cause faster damping.