← Physics — Std 12
⚡

Oscillations

Ch. 5Std 12

Easy Overview

The world is full of things that go back and forth — a swinging pendulum, a vibrating guitar string, a bouncing spring, the swaying of a tall building in the wind, even the alternating current that powers your home. These repetitive motions are oscillations, and the simplest, most fundamental type is simple harmonic motion (SHM). Understanding oscillations is not just about pendulums and springs — it is a gateway to understanding waves, sound, light, alternating currents, and even quantum mechanics. Simple harmonic motion occurs when the restoring force on an object is directly proportional to its displacement from equilibrium and acts in the opposite direction. Mathematically: F = -kx. This is Hooke's law, but it describes much more than springs. A pendulum swinging through small angles, a molecule vibrating in a crystal, a tuning fork — all follow SHM for small displacements. The 'simple' in SHM refers to the fact that the motion can be described by a single sine or cosine function. The 'harmonic' part comes from the connection to musical harmony — pure tones are sinusoidal oscillations. The key quantities describing SHM are: amplitude (A) — the maximum displacement; angular frequency (ω) — how fast the oscillation occurs (ω = 2πf); period (T) — the time for one complete oscillation (T = 1/f = 2π/ω); and phase (φ) — where in the cycle the motion starts. The displacement as a function of time is x(t) = A sin(ωt + φ). The velocity is v(t) = Aω cos(ωt + φ), and the acceleration is a(t) = -ω²x. At the extremes (x = ±A), velocity is zero and acceleration is maximum. At equilibrium (x = 0), velocity is maximum and acceleration is zero. Energy in SHM constantly changes form. When the mass passes through equilibrium, all energy is kinetic. At the extremes, all energy is potential. The total energy E = ½kA² is constant. Real oscillators lose energy to friction — these are damped oscillations. If we push the oscillator periodically at just the right frequency, we get resonance — the amplitude grows dramatically. Resonance can be beautiful (a singer shattering a glass) or destructive (a bridge collapsing in the wind). From the swing in a playground to the tuning of a radio, oscillations are the heartbeat of the physical world.

Periodic Motion and Oscillations

Periodic motion is any motion that repeats itself at regular intervals. The time for one complete repetition is the period T, and the number of repetitions per unit time is the frequency f = 1/T. Oscillatory motion is a special type of periodic motion where a body moves back and forth about a mean (equilibrium) position. Examples include a pendulum swinging, a mass on a spring, a vibrating tuning fork, and the pistons in an engine. Not all periodic motion is oscillatory — uniform circular motion is periodic but not oscillatory.

Simple Harmonic Motion — Definition and Equation

Simple harmonic motion (SHM) occurs when the restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction: F = -kx. Using Newton's second law, ma = -kx, or d²x/dt² + (k/m)x = 0. This is the differential equation of SHM. Its general solution is x(t) = A sin(ωt + φ) or x(t) = A cos(ωt + φ), where ω = √(k/m) is the angular frequency. The solution is sinusoidal — a smooth, continuous wave.

Amplitude, Angular Frequency, Phase, Period, and Frequency

Amplitude (A) is the maximum displacement from equilibrium. Angular frequency ω = 2π/T = 2πf, measured in rad/s. Frequency f is the number of complete oscillations per second (hertz). Period T = 1/f = 2π/ω. Phase (φ) determines where in the cycle the motion starts at t = 0. If φ = 0, motion starts at equilibrium moving positively. Phase difference between two oscillations determines whether they reinforce or cancel.

Velocity and Acceleration in SHM

Given x(t) = A sin(ωt + φ), v = dx/dt = Aω cos(ωt + φ), and a = dv/dt = -Aω² sin(ωt + φ) = -ω²x. Key features: (1) Velocity leads displacement by π/2. When x = 0, v is maximum (±Aω). When x = ±A, v = 0. (2) Acceleration is opposite in phase to displacement (π out of phase). (3) v_max = Aω, a_max = Aω². (4) v = ±ω√(A² − x²).

Energy in Simple Harmonic Motion

Potential energy U = ½kx² = ½kA² sin²(ωt + φ). Kinetic energy K = ½mv² = ½kA² cos²(ωt + φ). Total energy E = K + U = ½kA² — constant. At any position, v = ±ω√(A² − x²). Average KE over one cycle equals average PE, each equal to ¼kA². The total energy is proportional to the square of the amplitude.

The Spring-Block System

A mass m attached to a spring of force constant k executes SHM with ω = √(k/m), f = (1/2π)√(k/m), and T = 2π√(m/k). The period increases with mass and decreases with spring stiffness. The period is independent of amplitude. For a vertical spring, gravity shifts equilibrium but does not change the period. Springs in series: 1/k_eff = 1/k₁ + 1/k₂. Springs in parallel: k_eff = k₁ + k₂.

The Simple Pendulum

A simple pendulum with a point mass bob and massless string. For small angular displacements (θ < 15°), the restoring torque τ = -mgL sinθ ≈ -mgLθ. Using τ = Iα = mL²α, we get d²θ/dt² + (g/L)θ = 0, giving ω = √(g/L) and T = 2π√(L/g). The period is independent of mass and amplitude (for small angles). Pendulums are used in timekeeping.

The Compound (Physical) Pendulum

A compound pendulum is any rigid body free to oscillate about a horizontal axis not through its centre of mass. T = 2π√(I/(Mgd)), where d is the distance from pivot to centre of mass. Using the parallel axis theorem, I = I_cm + Md². The equivalent length of a simple pendulum with the same period is L_eq = I/(Md). Kater's reversible pendulum uses this for precise measurement of g.

Oscillations of a Liquid in a U-Tube

A liquid column in a U-tube, when disturbed, executes SHM. If displaced by h in one arm, the restoring force F = -(ρA)(2h)g. The mass of the liquid column is m = ρAL, where L is the total length. Using F = ma: d²h/dt² + (2g/L)h = 0, giving ω = √(2g/L) and T = 2π√(L/(2g)). This is analogous to a spring-mass system.

Damped Oscillations

Real oscillators experience damping forces F_d = -bv. The equation: m(d²x/dt²) + b(dx/dt) + kx = 0. Underdamping (b² < 4mk): x(t) = A₀e^(-bt/2m) cos(ω_d t + φ), where ω_d = √(ω₀² − (b/2m)²). Critical damping (b² = 4mk): fastest return to equilibrium without oscillation. Overdamping (b² > 4mk): slow return to equilibrium. Car shock absorbers use critical damping.

Quality Factor and Logarithmic Decrement

Quality factor Q = ω₀m/b = 2π × (energy stored)/(energy lost per cycle). A high-Q oscillator (tuning fork, Q ≈ 1000) rings for many cycles. Logarithmic decrement δ = ln(A_n/A_{n+1}) = bT/2m ≈ π/Q for high Q. Energy decays as E(t) = E₀e^(-ω₀t/Q). Relaxation time τ = m/b is the time for amplitude to drop to 1/e of its initial value.

Forced Oscillations and Resonance

An external periodic force F(t) = F₀ cos(ωt) drives the oscillator. Steady-state amplitude A = F₀/√((k − mω²)² + (bω)²). At ω = ω₀ = √(k/m), resonance occurs — amplitude is maximum. Sharper peak for higher Q. At resonance, phase difference δ = 90°. Resonance is used in radio tuning, musical instruments, microwave ovens, and MRI.

Applications of SHM — Timekeeping and Metrology

Pendulum clocks use SHM isochronism. Quartz watches use a vibrating quartz crystal at 32,768 Hz. Seismometers use a mass-spring system to measure ground motion. Atomic clocks use cesium-133 oscillations at 9,192,631,770 Hz to define the second. The metre is defined via the speed of light and the second, so timekeeping underpins length measurement.

Angular SHM and Torsional Pendulum

Angular SHM occurs when τ = -κθ (restoring torque proportional to angular displacement). For a torsional pendulum, I(d²θ/dt²) = -κθ, giving ω = √(κ/I) and T = 2π√(I/κ). This is directly analogous to the spring-mass system. Torsional pendulums are used in mechanical watches and for measuring moment of inertia.

Combining SHMs — Lissajous Figures

When a particle is subjected to two perpendicular SHMs, it traces Lissajous figures. If ω₁ = ω₂ and phase difference = 0, the figure is a straight line. For phase π/2, it is a circle (if amplitudes equal) or an ellipse. For other frequency ratios (1:2, 2:3), intricate patterns emerge. Used to measure unknown frequencies on an oscilloscope.

Key Points

  • •SHM: restoring force proportional to displacement, F = -kx.
  • •General equation: x(t) = A sin(ωt + φ) or x(t) = A cos(ωt + φ).
  • •Angular frequency ω = √(k/m). For a pendulum: ω = √(g/L).
  • •Period T = 2π/ω = 2π√(m/k). Frequency f = 1/T.
  • •Velocity in SHM: v = Aω cos(ωt + φ). v_max = Aω. v = ±ω√(A² − x²).
  • •Acceleration: a = -ω²x. a_max = Aω².
  • •Total energy: E = ½kA² = constant. K_avg = U_avg = E/2.
  • •Simple pendulum: T = 2π√(L/g). Independent of mass and amplitude (small angles).
  • •Compound pendulum: T = 2π√(I/(Mgd)).
  • •Springs in series: 1/k_eff = 1/k₁ + 1/k₂. Parallel: k_eff = k₁ + k₂.
  • •Damped oscillations: amplitude decays as e^(-bt/2m).
  • •Critical damping: b² = 4mk.
  • •Resonance: ω_driving = ω₀ gives maximum amplitude.
  • •Quality factor Q = ω₀m/b = 2π × energy stored/energy lost per cycle.
  • •Torsional pendulum: T = 2π√(I/κ). τ = -κθ.

Practice Questions

  • Define SHM. Derive the differential equation for SHM and show that x = A sin(ωt + φ) satisfies it.
  • A particle executes SHM of amplitude A. At what displacement is its KE equal to its PE? Derive.
  • Derive the expression for time period of a simple pendulum. How does it change if length is increased by 44%?
  • Explain damped oscillations. Derive amplitude as a function of time in the underdamped case. Condition for critical damping?
  • What is resonance in forced oscillations? Explain with amplitude vs frequency graph. Two examples each of constructive and destructive resonance.
  • A block of mass 0.5 kg is attached to a spring of force constant 50 N/m. The block is pulled 4 cm from equilibrium and released. Find period, maximum speed, total energy, and speed at displacement 2 cm.
  • Derive expression for period of a compound pendulum. A uniform rod of length 1 m is suspended from one end. Find its period of oscillation (g = 9.8 m/s²).
  • A particle is subjected to two perpendicular SHMs: x = A sin(ωt) and y = A sin(ωt + π/2). Find the resultant path.