Oscillations— Notes, Formulas & Revision

Simple Harmonic Motion (SHM): periodic motion where restoring force is proportional to displacement and opposite in direction. F = −kx.

Equation of motion: m·d²x/dt² = −kx ⇒ d²x/dt² = −ω²·x with ω = √(k/m).

General solution: x(t) = A·cos(ωt + φ). A = amplitude, ω = angular frequency, φ = initial phase.

Period T = 2π/ω = 2π·√(m/k). Frequency f = 1/T.

SHM is the simplest periodic motion — many oscillators (pendulum, spring, sound, LC circuit) reduce to SHM at small amplitudes.

Two SHMs combine to give beats, interference, Lissajous figures.

Phase angle φ: shifts the wave in time; ωt + φ = 0 at maximum displacement.

Mathematical basis: any small oscillation around a stable equilibrium is SHM to leading order.

Displacement in SHM: x(t) = A·cos(ωt + φ) (or equivalently sin form).

Maximum displacement is the AMPLITUDE A.

Crosses zero (equilibrium) twice per cycle. Reaches +A and −A once each per cycle.

Period T = 2π/ω; ωt + φ is the PHASE.

Position depends on initial conditions: x₀ and v₀ at t = 0.

If t = 0 is at extreme position: x(t) = A·cos(ωt).

If t = 0 is at equilibrium moving outward: x(t) = A·sin(ωt).

Both are valid SHM — phase shift relates them by π/2.

Velocity in SHM: v(t) = dx/dt = −Aω·sin(ωt + φ).

Maximum velocity v_max = Aω, occurs at equilibrium (x = 0).

Minimum velocity (= 0) at extremes (x = ±A).

Energy view: KE = ½mv² maximum at center, zero at extremes.

v vs x relation: v² = ω²(A² − x²) ⇒ v = ω·√(A² − x²).

v leads x by π/2 in phase (v = Aω cos shifted by 90° from x = A sin).

On v-x phase plot: ellipse with semi-axes A (x) and Aω (v).

Used in calculating KE distribution and timing of SHM events.

Acceleration in SHM: a(t) = dv/dt = −Aω²·cos(ωt + φ) = −ω²·x.

Maximum |a| = Aω², at the extremes (x = ±A).

ZERO at equilibrium (x = 0).

Always points TOWARD the equilibrium position (restoring nature).

a vs x: linear with negative slope (a = −ω²x). On an a-x graph: straight line through origin with slope −ω².

a is in PHASE with −x: when x is at +A, a is at maximum negative value.

Important relation: a + ω²x = 0 — the defining DE of SHM.

Used in finding force, identifying SHM from given motion equations.

Spring-mass system: a mass m attached to a spring of constant k. Displacement from equilibrium produces a restoring force F = −kx.

Equation of motion: m·d²x/dt² = −kx ⇒ SHM with ω = √(k/m).

Period: T = 2π·√(m/k). Independent of amplitude.

Heavier mass → larger T (slower). Stiffer spring → smaller T (faster).

Total energy: E = ½kA². Oscillates between KE and PE.

Horizontal spring: no gravity to consider; equilibrium at natural length.

Vertical spring: gravity stretches it; new equilibrium x₀ = mg/k. SHM about x₀, with same ω = √(k/m).

Damping (friction) reduces amplitude over time; ω stays approximately the same for light damping.

Vertical spring-mass: gravity adds an extra constant force, shifting the equilibrium position from natural length to L₀ + mg/k.

About the NEW equilibrium, the system undergoes SHM with same ω = √(k/m) — gravity doesn't affect period.

At new equilibrium: spring force kδ exactly balances mg. δ = mg/k = static extension.

Total energy in vertical spring: includes gravitational PE in addition to spring PE.

If we measure displacement y from the new equilibrium: F = −ky and energy = ½ky² (gravity absorbed into the equilibrium shift).

Period unchanged by gravity (in the small-displacement, linear regime).

Static stretch gives a quick way to measure k: k = mg/δ.

Used in: weighing scales, vehicle suspensions, oscillators, accelerometers.

Energy in SHM oscillates between KE and PE but total stays constant.

KE: ½mv² = ½m·ω²·(A² − x²). Maximum (½mω²A²) at x = 0, zero at x = ±A.

PE: ½kx² = ½m·ω²·x². Maximum (½mω²A²) at x = ±A, zero at x = 0.

Total: E = ½kA² = ½mω²A² — depends on amplitude squared.

PE and KE oscillate at TWICE the SHM frequency (sin² and cos² have period T/2).

Time-average of KE over a period = time-average of PE = ½ × Total energy.

Doubling amplitude QUADRUPLES total energy.

Energy conservation: ½kx² + ½mv² = ½kA² always.

For spring-mass SHM: T = 2π·√(m/k). Period proportional to √m.

Doubling mass increases T by √2 (~1.41×).

T² vs m gives a straight line through origin — slope = 4π²/k. Lab method to measure k.

For a SIMPLE PENDULUM: T does NOT depend on mass — period depends only on L and g.

Why difference? Spring force scales with displacement (not mass), so heavier mass takes longer to accelerate. Pendulum gravity scales with mass (= weight), so mass cancels out.

In rotational systems, 'mass' is replaced by moment of inertia I.

Same principle: increasing inertia at constant restoring force/torque ⇒ slower oscillation.

Useful for designing tuning forks, accelerometers, mechanical clocks.

For spring-mass SHM: T = 2π·√(m/k). Period INVERSELY proportional to √k.

Stiffer spring (larger k) ⇒ smaller T (faster oscillation).

Doubling k decreases T by √2 (~0.707×).

T² vs 1/k is linear — slope = 4π²m.

Springs in series: 1/k_eq = 1/k₁ + 1/k₂ ⇒ k_eq SMALLER ⇒ T LARGER (softer combination).

Springs in parallel: k_eq = k₁ + k₂ ⇒ k_eq LARGER ⇒ T SMALLER (stiffer combination).

k is a property of the spring's material and geometry — not amplitude or mass.

Used in tuning forks, watches, accelerometers — selecting k tunes the natural frequency.

Simple pendulum: a point mass m suspended from a massless string of length L, swinging under gravity.

For small angles (θ < ~15°), sin θ ≈ θ ⇒ motion is approximately SHM.

Equation of motion (small angle): θ̈ + (g/L)·θ = 0 ⇒ ω = √(g/L).

Period T = 2π·√(L/g). Independent of mass m and amplitude (for small angles).

T depends on L and g — does NOT depend on mass m of the bob.

Long pendulums swing slowly; short ones swing fast.

At higher altitudes or at the equator: g lower ⇒ T longer.

Foucault pendulum: long pendulum demonstrates Earth's rotation as its swing plane appears to rotate.

Period of a simple pendulum is T = 2π√(L/g) — proportional to √L.

Doubling length increases period by √2 (~1.41×).

T² vs L is LINEAR with slope 4π²/g. Lab method to measure g.

Length L is measured from pivot to center of mass of bob.

For compound pendulum (extended body): T = 2π·√(I/Mgd), where I = moment of inertia about pivot, d = distance from pivot to COM.

Pendulum clock: typical L ≈ 1 m for T = 2 s. Grandfather clocks tune by adjusting L slightly.

Used in seismometers, accelerometers, time-standards before atomic clocks.

Temperature effects: warm air expands the pendulum rod ⇒ longer L ⇒ slower clock.

Period of a pendulum T = 2π√(L/g) — inversely proportional to √g.

Smaller g (higher altitude, equator, Moon) ⇒ LONGER period.

Higher g (poles, Earth's center) ⇒ SHORTER period.

Pendulums historically used to measure g — extremely precise at ~10⁻⁵.

Pendulum at top of mountain runs SLOW: g drops by ~0.3% at 10 km altitude.

Pendulum at equator runs slightly slower than at poles due to Earth's rotation + oblateness.

On the Moon (g ≈ 1.6 m/s²): T_moon ≈ 2.45 × T_earth.

g measurement: g = 4π²L/T². Modern atom-interferometer measurements: g to 10⁻⁹ precision.

Damped oscillation: amplitude decreases over time due to friction or air drag.

Equation of motion: m·ẍ + b·ẋ + kx = 0, where b is the damping constant.

Three regimes: UNDERDAMPED (oscillates with decaying amplitude), CRITICALLY DAMPED (returns to equilibrium fastest, no oscillation), OVERDAMPED (returns slowly, no oscillation).

Underdamped solution: x(t) = A₀·e^(−bt/2m)·cos(ω'·t + φ), where ω' = √(ω₀² − γ²), γ = b/2m, ω₀ = √(k/m).

Amplitude decays exponentially: A(t) = A₀·e^(−bt/2m). Half-amplitude time: t₁/₂ = (2m·ln 2)/b.

Quality factor Q = ω₀·m/b. High Q = light damping, oscillates many times. Low Q = heavy damping.

Examples: pendulum in air, RLC circuit with R, shock absorbers (critically damped by design).

Energy decays: E(t) ∝ e^(−bt/m) (twice as fast as amplitude).

Forced oscillation: a driving force F = F₀·cos(ω_d·t) drives a damped oscillator at frequency ω_d (different from natural ω₀).

Steady-state response: x(t) = A_d·cos(ω_d·t − φ), where A_d depends on ω_d.

Amplitude A_d depends on (i) driving force F₀, (ii) ω_d vs ω₀, (iii) damping b.

Maximum amplitude occurs at RESONANCE ω_d ≈ ω₀ — sharp peak for small damping.

Phase φ between drive and response: 0 at low ω_d, π/2 at resonance, π at high ω_d.

Energy is continuously supplied by drive and dissipated by damping at steady state.

Transient response (initial conditions) dies out exponentially — only steady-state remains.

Examples: child on swing, AC circuit driven by source, building swaying in wind, tuning circuits.

Resonance: when driving frequency ω_d matches natural frequency ω₀, response amplitude is MAXIMUM.

Slightly DOWNSHIFTED by damping: ω_res = √(ω₀² − 2γ²) ≈ ω₀ for light damping.

Resonance amplitude A_res = F₀/(b·ω₀) — inversely proportional to damping.

Examples: tuning a radio (LC resonance), pushing a swing in time with its motion, microwave heating water (molecular resonance ~ 2.45 GHz), MRI scans.

Sharpness: Q-factor measures resonance sharpness; higher Q = sharper, larger amplitude at resonance.

Phase: at resonance, response lags drive by exactly π/2.

Resonance can be CONSTRUCTIVE (musical instruments, lasers) or DESTRUCTIVE (Tacoma Narrows, glass shattered by voice).

Mathematical analog applies to electromagnetic systems (LCR circuits), atoms (absorption spectra), nuclei (Mössbauer effect).