Unit 6: Oscillations

Oscillation

Back-and-forth motion about an equilibrium position.

Stable equilibrium position

The position where the net restoring influence is zero and nearby displacements produce forces/torques back toward this position.

Simple Harmonic Motion (SHM)

Oscillation in which the restoring force/torque points toward equilibrium and is proportional to displacement from equilibrium (leading to sinusoidal motion).

Restoring force

A force that acts to return a system to equilibrium (typically opposite the displacement).

Linear restoring force

A restoring force proportional to displacement, of the form F = −kx (or torque proportional to angle), which produces SHM.

Hooke’s law

For an ideal spring, the restoring force is F_x = −kx, proportional to displacement x from equilibrium.

Spring constant (k)

A positive constant measuring spring stiffness in Hooke’s law; larger k means a stiffer spring.

Displacement (x)

The signed position measured from equilibrium (the correct “zero” for SHM analysis).

Periodic motion (non-SHM)

Motion that repeats in time but does not satisfy a restoring influence proportional to displacement, so it is not simple harmonic.

Nonlinear force

A force not proportional to displacement (e.g., depends on x^2); can produce periodic motion but not SHM.

Amplitude (A)

Maximum displacement from equilibrium in SHM.

Period (T)

Time required for one full cycle of oscillation.

Frequency (f)

Number of cycles per second; f = 1/T.

Hertz (Hz)

Unit of frequency equal to s⁻¹ (one cycle per second).

Angular frequency (ω)

SHM constant related to period and frequency: ω = 2πf = 2π/T; for a spring-mass, ω = √(k/m).

Phase

The quantity (ωt + φ) that specifies where the oscillator is in its cycle.

Phase constant (φ or φ0)

The phase at t = 0 that sets the oscillator’s starting point; determined from initial conditions.

Reference circle model

Model where SHM is viewed as the projection of uniform circular motion; helps explain why x(t) is sinusoidal and the phase shifts between x, v, and a.

Newton’s second law (translation)

ΣF_x = m d²x/dt²; combined with Hooke’s law to derive the SHM differential equation.

SHM differential equation

The governing equation d²x/dt² + ω²x = 0 (for a spring, ω² = k/m).

General SHM solution

A sinusoidal position function such as x(t) = A cos(ωt + φ) (or A sin(ωt + φ)).

Initial conditions

Values like x(0) and v(0) used to determine amplitude A and phase constant φ in the SHM solution.

Velocity function in SHM

Time derivative of position; e.g., if x(t)=Acos(ωt+φ), then v(t)=−Aω sin(ωt+φ).

Acceleration function in SHM

Second derivative of position; e.g., if x(t)=Acos(ωt+φ), then a(t)=−Aω² cos(ωt+φ).

Acceleration–displacement relation (a = −ω²x)

Key SHM relationship showing acceleration is proportional to and opposite displacement.

Turning point

Endpoint of SHM where |x| = A, velocity is zero, and acceleration magnitude is maximum.

Maximum speed (v_max)

Largest speed in SHM, occurring at equilibrium: v_max = ωA.

Maximum acceleration (a_max)

Largest acceleration magnitude in SHM, occurring at turning points: a_max = ω²A.

Speed–displacement identity (v² = ω²(A² − x²))

Relationship giving speed magnitude at position x without solving for time (two times per cycle for a given |x|).

Elastic potential energy (spring) (U_s)

Energy stored in a spring: U_s = (1/2)kx² (with x measured from equilibrium).

Kinetic energy (K)

Energy of motion: K = (1/2)mv².

Total mechanical energy in SHM (E)

Constant energy in ideal SHM: E = K + U_s = (1/2)mv² + (1/2)kx².

Energy at amplitude (spring oscillator)

At x = A, v = 0, so E = (1/2)kA² (all spring potential energy).

Energy bar chart (SHM)

Conceptual chart showing constant total energy while kinetic and potential energy trade off; for a spring, U_s is minimum at equilibrium (x=0).

Vertical spring equilibrium stretch (d = mg/k)

Static stretch of a vertical spring when weight balances spring force: kd = mg, so d = mg/k.

Displacement measured from equilibrium

Choosing the coordinate origin at equilibrium (x=0 there) so the net force becomes purely restoring, avoiding extra constant terms.

Equivalent spring constant (k_eq)

Single spring constant that replaces multiple springs acting along the same coordinate to give the same force–displacement behavior.

Springs in parallel

Combination where displacements are equal and forces add: keq = k1 + k_2 (system becomes stiffer).

Springs in series

Combination where forces are equal and displacements add: 1/keq = 1/k1 + 1/k_2 (system becomes less stiff).

Simple pendulum

Point mass on a massless string/rod of length L swinging under gravity about a pivot; displacement described by angle θ.

Angular displacement (θ)

Pendulum’s angle from the vertical equilibrium line; θ = 0 at the bottom position.

Tangential component of gravity

Restoring influence along the swing: F_t = −mg sin(θ), pointing toward equilibrium.

Torque (τ)

Rotational analog of force; for a simple pendulum τ = −mgL sin(θ) about the pivot.

Moment of inertia (I)

Rotational inertia; for a point mass m at distance L, I = mL².

Angular acceleration (α)

Second time derivative of angle: α = d²θ/dt²; appears in Στ = Iα.

Small-angle approximation (sinθ ≈ θ)

Approximation valid for small θ in radians that linearizes pendulum motion, making it (approximately) SHM.

Simple pendulum period (small-angle)

For small oscillations, T = 2π√(L/g), independent of mass and approximately independent of amplitude.

Height-angle relation (h = L(1 − cosθ))

Exact geometry for a pendulum bob’s vertical rise h at angle θ, used with energy to find speeds.

Physical (compound) pendulum

Any rigid body swinging about a pivot under gravity; depends on mass distribution via I_p and the COM distance d.

Parallel-axis theorem

Relates moment of inertia about pivot to center-of-mass inertia: Ip = Icm + md².