Full Chapter Summary & Detailed Notes - Oscillations Class 11 NCERT

Overview & Key Concepts

• Chapter Goal: Introduces periodic and oscillatory motions, focusing on simple harmonic motion (SHM) as the simplest oscillatory type. Covers definitions, kinematics (displacement, velocity, acceleration), dynamics (force law), energy, and applications like simple pendulum. Exam Focus: SHM equations, graphs, period calculations, energy conservation, pendulum approximation. 2025 Updates: Reprint includes more examples on real oscillations (e.g., AC circuits, atomic vibrations); emphasizes links to waves (Ch.14). Fun Fact: SHM models everything from clock pendulums to molecular bonds. Core Idea: Restoring force ∝ -displacement leads to sinusoidal motion. Real-World: Earthquake engineering (damped oscillations), musical instruments (string vibrations). Ties: Builds on Ch.3 (projectile/circular motion), leads to waves (Ch.14 propagation).
• Wider Scope: Foundation for wave mechanics, quantum harmonic oscillator; applications in seismology, acoustics, electronics (LC circuits).

13.1 Introduction

Daily motions: Non-repetitive (rectilinear, projectile) vs. periodic (uniform circular, planetary orbits). Oscillatory: To-and-fro about mean position (cradle, swing, pendulum, boat, piston). Depth: Oscillatory subset of periodic; requires equilibrium with restoring force. Study basic to physics: Vibrations in strings (sitar/guitar/violin), membranes (drums), air molecules (sound), atoms in solids (temperature ∝ vibration energy), AC voltage (oscillates about zero). Concepts: Period, frequency, displacement, amplitude, phase. Real-Life: Heartbeat (periodic), tuning fork (oscillatory). Exam Tip: Distinguish periodic (repeats) vs. oscillatory (to-and-fro). Extended: Damped/forced oscillations later; waves as coupled oscillators. Historical: Galileo pendulum isochronism (1583). NCERT: Abounds examples; concepts developed next.

• Examples: Planet periodic but not oscillatory; swing oscillatory.
• Point: Frequency high → vibration (string); low → oscillation (tree branch).

Extended Discussion: Microscale: Quantum zero-point energy. Pitfalls: All periodic oscillatory? No (circle). Applications: GPS atomic clocks (oscillators). Depth: Phase space diagrams. Interlinks: Ch.14 waves from oscillators. Advanced: Nonlinear oscillations (chaos). Real: MRI uses nuclear spin precession (SHM-like). Graphs: Conceptual x-t sinusoidal. Principles: Restoring force key. Scope: Ideal undamped. Errors: Confuse amplitude (max disp) vs. displacement. Historical: Huygens cycloidal pendulum (1673). NCERT: Basic for understanding phenomena.

Principles: Motion repeats → periodic; equilibrium restoring → oscillatory. Errors: Orbital not to-and-fro.

13.2 Periodic and Oscillatory Motions

Motion repeats regular intervals: Periodic (Fig.13.1: insect ramp, child steps, bouncing ball parabolic). T: Smallest repeat time [s]; fast µs (quartz), slow days (Mercury 88d), years (Halley 76y). Frequency ν=1/T [Hz=s⁻¹]; not integer. Depth: Oscillatory: Equilibrium inside path, restoring force for small disp (ball in bowl). Every oscillatory periodic, not vice versa (circle periodic, no equilibrium restoring). No diff oscillations/vibrations (low/high freq). SHM: Simplest, F ∝ -x (towards mean). Damping: Friction stops; forced: External periodic sustains. Medium: Coupled oscillators → waves (water, seismic, EM). Real-Life: Guitar string vibrates SHM segments. Exam Tip: Periodic T fixed; oscillatory has mean position. Extended: Phase-locked loops. Ties: Ch.3 UCM periodic. Graphs: Fig.13.1 h-t periodic.

• Examples: Ball bowl oscillates; circle periodic non-oscillatory.
• SI: T s, ν Hz (Hertz 1857-1894).

Extended: Multidimensional oscillators (Lissajous). Pitfalls: Equilibrium no force, but unstable possible. Applications: Quartz watches (ν=32kHz). Depth: Fourier series any periodic = sum sines/cosines. Interlinks: Ch.14 wave equation from chain oscillators. Advanced: Limit cycles. Real: Heart monitor ECG periodic. Historical: Huygens coupled clocks sync. NCERT: Examples periodic; oscillatory restoring.

Principles: Repeat → periodic; to-fro restoring → oscillatory. Errors: Vibration=high freq oscillation.

13.2.1 Period and Frequency

T: Smallest repeat [s]; ν=1/T [Hz]. Ex: Heart 75/min=1.25 Hz, T=0.8s. Depth: ν not integer (e.g., 1.25). Units: µs quartz, days planets. Real-Life: Radio waves Hz (Hertz). Exam Tip: ν T=1 always. Extended: Angular ω=2πν [rad/s]. Ties: Ch.3 ω=2π/T UCM. Graphs: None specific.

• Example 13.1: Heart ν=1.25 Hz, T=0.8s.

Extended Discussion: Beat frequency diff sources. Pitfalls: T=1/ν inverse. Applications: Ultrasound freq MHz. Depth: Doppler shift Δν. Interlinks: Ch.14 f=v/λ. Advanced: Aliasing sampling. Real: Music beats (ν1-ν2). Historical: Hertz radio. NCERT: Reciprocal gives repeats/unit time.

Principles: T min repeat, ν repeats/s. Errors: Hz=1/s.

13.2.2 Displacement

General: Change physical property/time (position x(t), angle θ(t), voltage, pressure). From equilibrium convenient (spring block Fig.13.2a x=0; pendulum θ=0 vertical Fig.13.2b). Periodic f(t)=A cos ωt; T=2π/ω. Sine same; linear combo A sin ωt + B cos ωt = D sin(ωt + φ), D=√(A²+B²), tanφ=B/A. Fourier: Any periodic = sum sines/cosines. Depth: Displacement ± values. Real-Life: AC V(t)=V0 sin ωt. Exam Tip: Displacement variable, not just position. Extended: Phase space x-v ellipse. Ties: Ch.3 vector displacement. Graphs: f(t) periodic.

• Example 13.2: (i) sinωt + cosωt=√2 sin(ωt+π/4) T=2π/ω; (ii) sum T=2π/ω; (iii) e^{-ωt} non-periodic; (iv) log(ωt) non-periodic.

Extended: Hilbert transform phase. Pitfalls: log diverges non-physical. Applications: Signal processing Fourier. Depth: Orthogonal basis. Interlinks: Ch.14 Fourier waves. Advanced: Laplace non-periodic. Real: ECG Fourier components. Historical: Fourier 1822 heat. NCERT: Many displacement types; periodic functions.

Principles: x(t) any property; sinusoidal simplest periodic. Errors: Phase constant φ at t=0.

13.3 Simple Harmonic Motion

SHM: x(t)=A cos(ωt + φ); between -A, +A (Fig.13.3). Sinusoidal displacement/time. A: Amplitude max |x|; ω: Angular freq; φ: Phase constant (at t=0). Phase=ωt+φ. T=2π/ω, ν=ω/2π. Depth: State (x,v) by phase; A fixed, others vary. Real-Life: Mass-spring ideal SHM. Exam Tip: SHM specific sinusoidal; not all periodic. Extended: General solution x=A cos ωt + B sin ωt. Ties: Ch.3 projection UCM. Graphs: Fig.13.4 positions T/4 intervals; Fig.13.5 x-t continuous; Fig.13.6 symbols; Fig.13.7(a) diff A; (b) diff φ; Fig.13.8 diff T.

• Example 13.3: (1) √2 sin(ωt - π/4) SHM T=2π/ω; (2) sin²ωt=1/2 - 1/2 cos2ωt periodic T=π/ω not SHM (equil 1/2).

Extended Discussion: Damped x=A e^{-γt} cos(ω't + φ). Pitfalls: SHM requires linear force. Applications: Atomic vibrations. Depth: Lissajous 2D SHM. Interlinks: Ch.14 standing waves SHM superpos. Advanced: Anharmonic. Real: Seismograph SHM. Historical: Hooke spring 1678. NCERT: Defines SHM sinusoidal.

Principles: x sinusoidal time. Errors: Period independent initial (Fig.13.4).

13.4 Simple Harmonic Motion and Uniform Circular Motion

Projection UCM on diameter=SHM (Fig.13.9 ball circle shadow to-fro). Math: P uniform circle r=A, ω const anticlockwise, initial φ; x=A cos(ωt + φ), y=A sin(ωt + φ). Reference circle/particle. Depth: Any diameter; y phase π/2 x. Force diff: SHM linear restoring, UCM centripetal. Real-Life: Ferris wheel shadow SHM. Exam Tip: Projection links kinematics. Extended: Ellipse general 2D SHM. Ties: Ch.3 UCM. Graphs: Fig.13.10 vector OP projection P'.

• Example 13.4: (a) x=A cos(πt/2 + π/4) T=4s φ=π/4; (b) Clockwise x=A sin(πt/2) T=4s φ=π/2.

Extended: Phasor diagram. Pitfalls: Force same? No. Applications: Analog computers SHM. Depth: Parametric eq circle. Interlinks: Ch.14 circular polarization. Advanced: Relativistic. Real: Planet projection elliptic SHM approx. Historical: Huygens 1673. NCERT: Visualizes connection.

Principles: UCM projection SHM. Errors: Centripetal ≠ restoring.

13.5 Velocity and Acceleration in Simple Harmonic Motion

v=dx/dt= -A ω sin(ωt + φ); max v_m=Aω at x=0. a=dv/dt= -ω² x; towards mean, max a_m= A ω² at x=±A. Depth: v= ± ω √(A² - x²); a= -ω² x (Hooke-like). Graphs: v-x ellipse area π A² ω /2 kinetic energy. Real-Life: Spring max speed mid, accel ends. Exam Tip: a ∝ -x SHM def. Extended: Phase v= -ω x tan phase. Ties: Ch.3 centrip a=ω² r. Graphs: v-t, a-t sinusoidal phase diff.

• Ex: v(t)= -Aω sin(ωt+φ); a(t)= -Aω² cos(ωt+φ).

Extended Discussion: Power P=F v= -k x (-ω √(A²-x²) sin...) avg zero. Pitfalls: v max x=0. Applications: Accelerometers. Depth: Dimensionless ξ=x/A cos phase. Interlinks: Ch.14 wave v=ω √(T/μ). Advanced: Rel phase space. Real: Car suspension. Historical: Newton cradle. NCERT: Deriv from x(t).

Principles: v= -ω √(A²-x²), a= -ω² x. Errors: Accel direction towards mean.

13.6 Force Law for Simple Harmonic Motion

F= m a= - m ω² x; restoring ∝ -x, k= m ω² spring const. Depth: General any F= -kx SHM ω=√(k/m). Real-Life: Spring F=-kx. Exam Tip: Negative sign direction. Extended: Effective k systems. Ties: Ch.7 rotation analog. Graphs: F-x linear -k slope.

• Ex: ω=√(k/m), T=2π √(m/k).

Extended: Damped F=-kx -b v. Pitfalls: k positive. Applications: Atomic F=-kx harmonic. Depth: Potential U=1/2 k x². Interlinks: Ch.6 work. Advanced: Duffing nonlinear. Real: Balance wheel watch. Historical: Hooke 1678. NCERT: Leads energy.

Principles: F= -kx SHM condition. Errors: Proportionality constant m ω².

13.7 Energy in Simple Harmonic Motion

Total E= 1/2 m ω² A² const; K=1/2 m v²= 1/2 m ω² (A² - x²); U=1/2 k x²= 1/2 m ω² x². Depth: E= K+U conserved; avg K=U=E/2. Real-Life: Spring energy transfer. Exam Tip: E ∝ A². Extended: Power avg zero. Ties: Ch.6 conservation. Graphs: K/U vs x sinusoidal.

• Ex: At x=0 K max, x=A U max.

Extended Discussion: Damped E decays. Pitfalls: Total E=1/2 k A². Applications: Resonance energy build. Depth: virial theorem =. Interlinks: Ch.14 wave energy. Advanced: Quantum E=(n+1/2)ℏω. Real: Oscilloscope trace. Historical: Rayleigh 1900. NCERT: Mean position zero PE.