Complete Solutions and Summary of Oscillations – NCERT Class 11, Physics, Chapter 13 – Summary, Questions, Answers, Extra Questions

Detailed concepts of periodic and oscillatory motion, simple harmonic motion, velocity and acceleration in SHM, force law, energy in SHM, simple pendulum, damping and resonance with solved NCERT problems.

Updated: 8 months ago

Categories: NCERT, Class XI, Physics, Summary, Oscillations, Simple Harmonic Motion, Pendulum, Damped Oscillations, Resonance, Chapter 13

Tags: Oscillations, Periodic Motion, Simple Harmonic Motion, SHM Velocity, SHM Acceleration, Force Law, Energy in SHM, Simple Pendulum, Damping, Resonance, NCERT, Class 11, Physics, Chapter 13, Answers, Extra Questions

Oscillations Class 11 NCERT Chapter 13 - Ultimate Study Guide, Notes, Questions, Quiz 2025

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.

Principles: E conserved, K+U=const. Errors: U=1/2 k x² not A.

13.8 The Simple Pendulum

Ideal: Small θ, massless inextensible string, point mass bob, no air friction. T=2π √(l/g); independent mass, θ small. Depth: Approx sinθ≈θ; large nonlinear. Real-Life: Clock seconds pendulum l=1m. Exam Tip: Small angle approx. Extended: Physical l to CG, T=2π √(I/m g d). Ties: Ch.3 g. Graphs: T-θ increasing large.

Ex: l=1m T≈2s.

Extended: Foucault latitude. Pitfalls: T indep amplitude only small. Applications: Seismometers. Depth: Anharmonic correction. Interlinks: Ch.14 torsion pendulum. Advanced: Chaos large amp. Real: Grandfather clock. Historical: Galileo 1581. NCERT: Simple approx.

Principles: T=2π √(l/g) small θ. Errors: sinθ=θ rad.

Summary

Periodic repeats T, ν=1/T; oscillatory to-fro restoring. SHM x=A cos(ωt+φ) ω=2π/T; v=-Aω sin, a=-ω²x. F=-kx k=mω². E=1/2 mω²A². Pendulum T=2π√(l/g) small θ.

Why This Guide Stands Out

Complete: All subtopics (10+), examples solved (3+), Q&A exam-style, 30 numericals. Physics-focused with tables/eqs/graphs. Free for 2025.

Key Themes & Tips

SHM: Sinusoidal, F∝-x.
Energy: Conserved K+U.
Tip: Memorize eqs; practice graphs; units rad/s.

Exam Case Studies

Ex13.1 heart; Ex13.4 projection.

Project & Group Ideas

Simple pendulum: Vary l, measure T, plot √l vs T.
Spring-mass: Find k from T, energy calc.

Key Definitions & Terms - Complete Glossary

All terms from chapter; detailed with examples, relevance, formulas. Expanded: 30+ terms, derivations, applications. Focus: Motion types, SHM parameters.

Periodic Motion

Motion repeats regular intervals T. Relevance: Basis oscillations. Deriv: Graph repeats. Ex: Planet orbit. Vs Oscillatory: No to-fro req. Depth: Fourier decompose. Applications: Clocks.

Oscillatory Motion

To-and-fro about mean, restoring force. Relevance: Vibrations/sound. Ex: Swing. Depth: Damped/forced. Applications: Instruments.

Period (T)

Smallest repeat time [s]. Relevance: Cycle measure. Ex: Heart 0.8s. Depth: µs-days. Applications: Watches.

Frequency (ν)

Repeats/unit time=1/T [Hz]. Relevance: Rate. Ex: 1.25 Hz heart. Depth: Not integer. Applications: Radio.

Displacement

Change property/time from mean. Relevance: State. Ex: x(t) spring. Depth: General (θ, V). Applications: Sensors.

Amplitude (A)

Max |displacement|. Relevance: Extent. Ex: 5cm swing. Depth: Positive. Applications: Energy ∝A².

Angular Frequency (ω)

ω=2π/T [rad/s]. Relevance: Phase rate. Ex: 2π rad/s T=1s. Depth: UCM link. Applications: Waves.

Phase (ωt + φ)

Argument, state. Relevance: Position/vel. Ex: 0 at max x. Depth: Mod 2π. Applications: Sync.

Phase Constant (φ)

Phase at t=0. Relevance: Initial cond. Ex: π/2 sin start. Depth: Determines curve. Applications: Timing.

Simple Harmonic Motion (SHM)

x=A cos(ωt+φ) sinusoidal. Relevance: Simplest osc. Ex: Spring. Depth: F∝-x. Applications: Models many.

Restoring Force

Towards mean ∝ -disp. Relevance: Osc cause. Ex: Gravity pendulum. Depth: -kx. Applications: Stability.

Equilibrium Position

Mean, no net force. Relevance: Osc center. Ex: x=0 spring. Depth: Stable. Applications: Bowl bottom.

Velocity in SHM (v)

v= -Aω sin(ωt+φ); v_m=Aω. Relevance: Speed. Ex: Max mid. Depth: √(A²-x²) ω. Applications: Kinetics.

Acceleration in SHM (a)

a= -ω² x; a_m=Aω². Relevance: Towards mean. Ex: Max ends. Depth: Proportional -x. Applications: Force.

Force Law (F)

F= -kx; k=mω². Relevance: Dynamics. Ex: Spring const. Depth: Linear. Applications: Design.

Kinetic Energy (K)

K=1/2 m v²=1/2 m ω² (A²-x²). Relevance: Motion energy. Ex: Max x=0. Depth: Var. Applications: Total E.

Potential Energy (U)

U=1/2 k x²=1/2 m ω² x². Relevance: Position energy. Ex: Max x=A. Depth: Min mean. Applications: Conservation.

Total Energy (E)

E=1/2 m ω² A² const. Relevance: Invariant. Ex: Spring total. Depth: K+U. Applications: Amplitude.

Simple Pendulum

Mass on string, small θ. Relevance: Timekeeper. Ex: Clock. Depth: T=2π√(l/g). Applications: Gravity meas.

Reference Circle

Circle whose projection SHM. Relevance: Visualize. Ex: UCM shadow. Depth: Radius A. Applications: Phasors.

Damping

Friction reduces amp. Relevance: Real stop. Ex: Air resistance. Depth: e^{-γt}. Applications: Controlled.

Forced Oscillation

External periodic drive. Relevance: Sustain. Ex: Push swing. Depth: Resonance. Applications: Radio tuning.

Coupled Oscillators

Multiple interacting. Relevance: Waves. Ex: Loaded string. Depth: Normal modes. Applications: Ch.14.

Fourier Theorem

Any periodic = sum harmonics. Relevance: Decompose. Ex: Square wave. Depth: Coefficients. Applications: Signals.

Phase Difference

φ1 - φ2. Relevance: Sync. Ex: π/2 v-x. Depth: Lead/lag. Applications: Interference.

Isochronous

T indep amplitude. Relevance: Clocks. Ex: Small θ pendulum. Depth: Approx. Applications: Accurate time.

Mean Position

Equilibrium. Relevance: Center. Ex: Vertical pendulum. Depth: Zero force. Applications: Rest.

Tip: Memorize: T=2π/ω, v_m=Aω, E=1/2 m ω² A². Depth: All [rad] angles. Applications: Quantum SHO. Errors: ν=ω/2π. Historical: Fourier. Interlinks: Ch.14. Advanced: Parametric resonance. Real-Life: Smartphone haptics. Graphs: x-t cos. Symbols: A amp, ω ang freq, φ phase. Coherent SI s, Hz. Extended: Relativistic SHM. Non-SHM: Exponential decay.

Additional: Restoring opp disp. Isotropic no. Atomic: Harmonic approx. Calculus: d²x/dt² = -ω² x.

60+ Questions & Answers - NCERT Based (Class 11) Theory Only

Part A (1 mark short: 1-2 sentences), B (4 marks medium ~6 lines/detailed explanation), C (8 marks long: Detailed with examples/derivations/graphs). Based on NCERT Exercises 13.1-13.20 theory.

Part A: 1 Mark Questions (Short Answers - From NCERT Exercises)

13.1 Period of motion that repeats after 10s?

1 Mark Answer: 10s.

13.2 Frequency if T=2s?

1 Mark Answer: 0.5 Hz.

13.3 Unit of angular frequency?

1 Mark Answer: rad/s.

13.4 SHM displacement form?

1 Mark Answer: A cos(ωt + φ).

13.5 Amplitude in SHM?

1 Mark Answer: Max displacement.

13.6 Phase at t=0?

1 Mark Answer: φ.

13.7 ω relation to T?

1 Mark Answer: 2π/T.

13.8 v max in SHM?

1 Mark Answer: Aω.

13.9 a in SHM?

1 Mark Answer: -ω² x.

13.10 Force law SHM?

1 Mark Answer: F = -kx.

13.11 Total E in SHM?

1 Mark Answer: ½ m ω² A².

13.12 Pendulum period?

1 Mark Answer: 2π √(l/g).

13.13 Small angle approx?

1 Mark Answer: sinθ ≈ θ.

13.14 UCM projection?

1 Mark Answer: SHM.

13.15 Damping effect?

1 Mark Answer: Reduces amplitude.

13.16 Fourier theorem?

1 Mark Answer: Periodic = sum sines/cosines.

13.17 Mean position?

1 Mark Answer: Equilibrium.

13.18 v-x relation SHM?

1 Mark Answer: v = ±ω √(A² - x²).

13.19 Avg K in SHM?

1 Mark Answer: E/2.

13.20 T pendulum indep?

1 Mark Answer: Mass, small θ.

13.21 Circular periodic?

1 Mark Answer: Yes, not oscillatory.

13.22 Phase diff v and x?

1 Mark Answer: π/2.

13.23 k in F=-kx?

1 Mark Answer: m ω².

13.24 U max at?

1 Mark Answer: x=±A.

13.25 sin²ωt SHM?

1 Mark Answer: No, periodic.

13.26 Reference circle radius?

1 Mark Answer: A.

13.27 e^{-t} periodic?

1 Mark Answer: No.

13.28 log t periodic?

1 Mark Answer: No.

13.29 Hertz named after?

1 Mark Answer: Heinrich Hertz.

13.30 Isochronous means?

1 Mark Answer: T indep amplitude.

Part B: 4 Marks Questions (Medium Length ~6 Lines - From NCERT)

13.1 Explain periodic vs oscillatory.

4 Marks Answer: Periodic repeats regular T (ex: circle); oscillatory to-fro mean restoring (ex: pendulum). Oscillatory periodic but not reverse; circle no equilibrium restoring. Damping stops real; forced sustains. Waves coupled osc.

13.2 Derive ν=1/T.

4 Marks Answer: T min repeat; ν repeats/s =1/T. Ex: Heart 75/min=75/60=1.25 Hz. Units Hz=s^{-1}. ω=2πν for angular.

13.3 Displacement general sense.

4 Marks Answer: Change property/time from mean (x spring, θ pend, V AC). Periodic f(t)=A cos ωt T=2π/ω. Combo D sin(ωt+φ).

13.4 SHM define, parameters.

4 Marks Answer: x=A cos(ωt+φ); A amp, ω ang freq T=2π/ω, φ phase const. Sinusoidal time.

13.5 UCM projection SHM.

4 Marks Answer: Shadow diameter to-fro; x=A cos(ωt+φ). y sin phase π/2. Visualize reference circle.

13.6 Derive v, a SHM.

4 Marks Answer: v=dx/dt=-Aω sin(ωt+φ) v_m=Aω; a=d²x/dt²=-ω² x a_m=Aω². Towards mean.

13.7 Force law derive.

4 Marks Answer: F=ma=-mω² x; k=mω². Restoring ∝-x.

13.8 Energy SHM expressions.

4 Marks Answer: K=½mω²(A²-x²), U=½mω²x², E=½mω²A². Avg K=U=E/2.

13.9 Simple pendulum approx.

4 Marks Answer: Small θ sinθ≈θ; T=2π√(l/g). Indep mass, θ.

13.10 Damped vs forced.

4 Marks Answer: Damped amp decay friction; forced external periodic sustains, resonance.

13.11 Fourier importance.

4 Marks Answer: Any periodic sum sines/cosines diff T. Decompose complex.

13.12 v-x graph SHM.

4 Marks Answer: Ellipse v=±ω√(A²-x²); area παω kinetic.

13.13 Pendulum isochronous why?

4 Marks Answer: Small θ approx indep amp; large T increases.

13.14 Phase role SHM.

4 Marks Answer: ωt+φ determines state; diff φ shifts curve.

13.15 Energy conservation SHM.

4 Marks Answer: K+U=const no dissipation; transfers mid-ends.

13.16 Reference particle.

4 Marks Answer: UCM particle whose proj SHM; circle ref.

13.17 sinωt + cosωt SHM?

4 Marks Answer: Yes √2 sin(ωt+π/4) T=2π/ω.

13.18 Coupled oscillators waves.

4 Marks Answer: Medium atoms osc interact propagate waves.

13.19 a max when?

4 Marks Answer: x=±A, a_m=Aω².

13.20 T pendulum factors.

4 Marks Answer: l, g; indep m, θ small.

Part C: 8 Marks Questions (Detailed Long Answers - From NCERT)

13.1 Detailed periodic oscillatory diff, examples.

8 Marks Answer: Periodic: Repeats exact after T (Fig.13.1 insect ramp h-t, child steps square, ball bounce parabola u-gt²/2). T min interval [s]; ν=1/T [Hz] ex heart 1.25Hz T=0.8s. Oscillatory: Subset, equilibrium path, small disp restoring force (bowl ball). Every osc periodic (T defined), not reverse (UCM periodic T=2πr/v no to-fro). Ex: Planet periodic non-osc; pendulum osc. Diff: Osc has mean pos zero net F, disp → restoring; periodic any repeat. Real: String vibrate osc sound; AC osc zero mean. Graphs: Periodic arbitrary shape repeat; osc sinusoidal approx. Ties: Ch.3 UCM periodic. Advanced: Quasi-periodic. Errors: Circle osc? No. Historical: Galileo period pend. NCERT: Basic physics, vibrations temp.

13.2 Detailed period freq, derive relation, ex.

8 Marks Answer: T smallest repeat time [s]; ν=1/T repeats/s [Hz=s^{-1}] named Hertz radio discoverer. Deriv: In T one cycle, ν cycles/s=1/T. Ex: Quartz µs T high ν; Mercury 88d low ν; Halley 76y. Heart 75/min=75/60=1.25 Hz T=0.8s. ω=2π/T rad/s phase advance. Graphs: f(t+T)=f(t). Physical: Scale seconds too fast/slow other units. Ties: Ch.3 f=1/T UCM. Advanced: Beat ν1-ν2. Errors: ν integer? No. Real: FM radio MHz. Historical: Rømer light speed Jupiter moons period. NCERT: Reciprocal unit time.

13.3 Detailed displacement, periodic functions, Fourier.

8 Marks Answer: Displacement: Change pos/property time from mean (x spring Fig.13.2a, θ pend Fig.13.2b vertical, V AC, P sound). General not just pos. Periodic f(t)=A cos ωt T=2π/ω f(t+T)=f(t); sin same. Combo A sin + B cos = D sin(ωt+φ) D=√(A²+B²) tanφ=B/A. Fourier: Any periodic superposition sines/cosines suitable coeff periods (Jean Fourier 1768-1830). Ex13.2: sin+cos=√2 sin(ωt+π/4) T=2π/ω; sum diff freq lcm T=2π/ω; e^{-ωt} mono dec no repeat; log inc no. Graphs: Cos periodic 2π. Physical: Displacement ±. Ties: Ch.14 displacement wave. Advanced: Spectrum analysis. Errors: Log physical? No diverge. Real: Music Fourier harmonics. Historical: Fourier heat conduction. NCERT: Simplest cos; importance superposition.

13.4 Detailed SHM define, parameters, graphs.

8 Marks Answer: SHM: Oscillatory disp sinusoidal time x(t)=A cos(ωt+φ) between -A +A Fig.13.3. Not any periodic, specific form. A amp max |x| (>0); ω ang freq T=2π/ω ν=ω/2π; φ phase const (t=0 phase). Phase ωt+φ time-dep state. Graphs: Fig.13.4 positions t=0,T/4,... max v x=0 zero v ends; Fig.13.5 x-t cont; Fig.13.6 symbols; Fig.13.7(a) diff A same ωφ; (b) diff φ same Aω shift; Fig.13.8 diff T (b half T twice ν). Ex13.3: sin-cos=√2 sin(ωt-π/4) SHM; sin²=½-½cos2ωt periodic T=π/ω equil ½ not SHM. Physical: Initial φ from x(0). Ties: Ch.3 proj. Advanced: Complex e^{i(ωt+φ)}. Errors: Period dep initial no. Real: Balance SHM. Historical: Hooke. NCERT: Characterizes given SHM.

13.5 Detailed UCM SHM link, derive, ex.

8 Marks Answer: Projection UCM diameter SHM (Fig.13.9 ball horizontal circle shadow to-fro). Ex: Tie ball string rotate const ω, side view/ shadow horizontal line mid rotation. Math Fig.13.10: P circle r=A ω anticlock, initial OP φ x-pos; t angle ωt total φ+ωt, proj x=A cos(ωt+φ) SHM. y=A sin(ωt+φ) phase π/2. Ref circle r=A particle P. Any diameter. Force diff: SHM linear -kx, UCM centrip mω²r inward. Ex13.4: (a) Initial 45°=π/4, anticlock T=4s angle πt/2 total πt/2 + π/4 x=A cos(πt/2 + π/4) φ=π/4; (b) Clockwise initial 45° proj sin form phase π/2. Graphs: Vector OP proj P'. Physical: Visualize kinematics. Ties: Ch.3 UCM. Advanced: Elliptical orbits approx SHM. Errors: Force same no. Real: Moon tide proj. Historical: Huygens. NCERT: Connection uniform circular.

13.6 Detailed v a SHM derive, graphs, significance.

8 Marks Answer: From x=A cos(ωt+φ); v=dx/dt= -Aω sin(ωt+φ) v_m=Aω x=0; a=dv/dt= -Aω² cos(ωt+φ)= -ω² x a_m=Aω² x=±A. v=±ω √(A²-x²) ellipse v-x area πA²ω/2. Graphs: v-t sin phase π x; a-t cos phase 0 x amp ω² A. Significance: a ∝ -x def SHM; v max mean zero ends; energy K=½mv² max mean. Ex: Spring mid fast accel ends. Physical: Phase space conserved. Ties: Ch.3 a= -ω² r UCM. Advanced: Lyapunov stability. Errors: v phase sin not cos. Real: Accelerometer a. Historical: Lagrange eqs. NCERT: Basic kinematics.

13.7 Detailed force law, derive k, general.

8 Marks Answer: Newton's F=ma= m d²x/dt²= m (-ω² x)= - (mω²) x; k=mω² restoring ∝ -x towards mean. General: Any F= -kx SHM ω=√(k/m) T=2π√(m/k). Ex: Spring F=-kx ideal. Graphs: F-x straight -k slope thru origin. Physical: Linear force sinusoidal sol. Ties: Ch.7 torsional k. Advanced: Effective k parallel/series. Errors: Sign positive no. Real: Molecular bonds harmonic approx. Historical: Hooke ut tensio sic vis. NCERT: Basis dynamics.

13.8 Detailed energy SHM, conservation, avg.

8 Marks Answer: Total E= K + U = ½ m v² + ½ k x² = ½ m ω² (A² - x²) + ½ m ω² x² = ½ m ω² A² const. K max x=0 v=Aω; U max x=A v=0. Avg over T: ==E/2 since sin² avg ½. Graphs: K/U vs t sinusoidal; E const. Deriv: dE/dt= F v + dU/dt= -kx (-ω √...) wait power avg 0. Physical: No diss transfer. Ties: Ch.6 mech energy. Advanced: Virial =. Errors: E ∝ A not A². Real: Oscillator circuits. Historical: Helmholtz. NCERT: Basic conserved.

13.9 Detailed simple pendulum, derive T, approx, limitations.

8 Marks Answer: Ideal: Point mass bob, massless string, small θ, no friction, pivot fixed. Eq θ'' + (g/l) sinθ=0 approx sinθ≈θ θ'' + (g/l) θ=0 SHM ω=√(g/l) T=2π √(l/g). Indep m, θ small (error <1% θ<10°). Limitations: Large θ T increases anharmonic; finite size physical l to CG; air damping. Graphs: T vs θ flat small rise large. Ex: l=1m T=2s g=10. Physical: Isochronous small. Ties: Ch.3 g meas. Advanced: Foucault precess. Errors: sinθ=θ deg no rad. Real: Clocks compensate. Historical: Galileo Pisa. NCERT: Simplest time meas.

Tip: Diagrams graphs long; derive eqs.

Key Concepts - In-Depth Exploration

Core ideas with derivations, examples, pitfalls, interlinks. Substantial content.

Periodic vs Oscillatory

Periodic repeat T; osc to-fro restoring. Deriv: Eq motion periodic sol. Pitfall: Circle osc? No. Ex: Ball ramp periodic, bowl osc. Interlink: Ch.14 periodic waves.

Period Frequency

T repeat, ν=1/T. Deriv: Cycles/s. Pitfall: Integer? No. Ex: Heart. Interlink: Ch.3 f UCM.

Displacement Periodic

Property change; f=A cos ωt. Deriv: f(t+2π/ω)=f. Pitfall: General not pos. Ex: θ pend. Interlink: Fourier signals.

SHM Definition

Sinusoidal x(t). Deriv: Sol d²x/dt²=-ω²x. Pitfall: All periodic SHM? No. Ex: Spring. Interlink: Harmonic osc.

Parameters A ω φ

A extent, ω rate, φ initial. Deriv: x(0)=A cosφ. Pitfall: T dep φ no. Ex: Diff φ shift. Interlink: Phasors.

UCM SHM Projection

x= A cos θ θ=ωt+φ. Deriv: Proj vector. Pitfall: Force link no. Ex: Shadow. Interlink: Ch.3.

Velocity Acceleration

v=-Aω sin, a=-ω²x. Deriv: Diff x. Pitfall: v max ends no. Ex: v-x ellipse. Interlink: Energy.

Force Law

F=-kx k=mω². Deriv: ma. Pitfall: +x no. Ex: Spring. Interlink: Ch.7 torsion.

Energy Conservation

E=½mω²A² K+U. Deriv: Integrate F dx. Pitfall: Avg K≠U no equal. Ex: Transfer. Interlink: Ch.6.

Simple Pendulum

T=2π√(l/g). Deriv: θ''+(g/l)θ=0. Pitfall: Large θ no. Ex: Clock. Interlink: g meas.

Damped Forced

Damp decay; forced drive. Deriv: + -b v; resonance. Pitfall: Undamped ideal. Ex: Push swing. Interlink: Ch.14.

Coupled Waves

Interact propagate. Deriv: Chain eq. Pitfall: Single no wave. Ex: String. Interlink: Ch.14.

Advanced: Nonlinear Duffing. Pitfalls: rad not deg. Interlinks: Ch.12 EM osc. Real: 3D print vib. Depth: Phase portraits. Examples: Ex13.2 functions. Graphs: x v a t. Calculus: SHM diff eq. Errors: φ deg. Tips: Derive T pend; energy plots.

Extended: Quantum SHO. Math: Characteristic eq. Applications: NMR. Common: Forget - sign F.

Principles: Linear restoring SHM. Advanced: Stability analysis. Vector: No, scalar 1D.

Force Law for SHM - Detailed Guide

Deriv details.

F=ma Apply

-mω² x.

k=mω²

Spring const.

General F=-kx

ω=√k/m.

Graph F-x

Linear -k.

Restoring Nature

Towards mean.

Ex Spring

Ideal linear.

Tip: Negative sign. Depth: Potential. Examples: Pend approx. Graphs: Slope. Advanced: Nonlinear.

Principles: Linear prop. Errors: k sign.

Extended: Magnetic force. Historical: Hooke.

Energy in SHM - Detailed Guide

Conservation details.

Total E

½ m ω² A².

K Expression

½ m ω² (A²-x²).

U Expression

½ m ω² x².

Avg Values

E/2 each.

Conservation Deriv

dE/dt=0.

Graphs Energy

Parabola U, const E.

Tip: ∝ A². Depth: Virial. Examples: Transfer. Graphs: vs x. Advanced: Quantum.

Principles: K+U const. Errors: Avg.

Extended: Radiation damping. Historical: Lagrange.

The Simple Pendulum - Detailed Guide

Approx details.

Ideal Conditions

Small θ, massless string.

T Derivation

2π √(l/g) from θ'' + (g/l)θ=0.

Approx sinθ=θ

Rad small.

Indep Factors

m, θ small.

Limitations

Air, large θ.

Graphs T-θ

Flat then rise.

Tip: l to bob. Depth: Physical pend. Examples: Clock. Graphs: vs l linear √. Advanced: Compound.

Principles: SHM approx. Errors: Deg.

Extended: Tides double pend. Historical: Huygens.

Interactive Quiz - Master Oscillations

20 MCQs; 85%+ goal. SHM, pendulum, energy.

Quick Revision Notes & Mnemonics

13.1 Introduction

Periodic repeat; osc to-fro restoring. Ex: Pendulum, AC.

13.2 Periodic Oscillatory

T min repeat [s]; ν=1/T [Hz]; osc equil restoring. Ex: Circle periodic non-osc.

13.2.1 Period Freq

ν=1/T; ω=2π/T. Ex: Heart 1.25 Hz T=0.8s.

13.2.2 Displacement

From mean x(t); f=A cos ωt T=2π/ω; Fourier sum.

13.3 SHM

x=A cos(ωt+φ); A amp, ω ang, φ const. T=2π/ω.

13.4 UCM SHM

Proj x=A cos(ωt+φ); ref circle r=A.

13.5 v a SHM

v=-Aω sin v_m=Aω; a=-ω² x a_m=Aω².

13.6 Force Law

F=-kx k=mω²; restoring -disp.

13.7 Energy SHM

K=½mω²(A²-x²); U=½mω²x²; E=½mω²A² const; avg E/2.

13.8 Simple Pendulum

T=2π√(l/g) small θ sinθ≈θ; indep m.

Summary

SHM sinusoidal F∝-x; pend approx; energy cons.

Points Ponder

Period indep init; osc subset periodic; a always mean; E∝A²; small θ rad; damping real; coupled waves; Fourier any periodic.

Mnemonics: SHM "Sinusoidal Harmonic Motion" (SHM). T=2π√(l/g) "Two Pi Square Root Length Gravity" (TPSRLG). Energy "Half Mass Omega Square A Square" (HMOSAS). v_m "A Omega" (AO). Phase "Omega T Plus Phi" (OTPF).

Tip: Graphs key; derive T pend; g=9.8 approx.

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10 Qs · ~10 min
#158
Class 11 Physics — Oscillations (Practice Quiz)
10 Qs · ~10 min
#159
Class 11 Physics — Kinetic Theory (Practice Quiz)
10 Qs · ~10 min
#160
Class 11 Chemistry — Thermodynamics (Practice Quiz)
10 Qs · ~10 min
#161
Class 11 Physics — Thermal Properties of Matter (Practice Quiz)
10 Qs · ~10 min
#162
Class 11 Physics — Mechanical Properties of Fluids (Practice Quiz)
10 Qs · ~10 min
#163
Class 11 Physics — Mechanical Properties of Solids (Practice Quiz)
10 Qs · ~10 min
#164
Class 11 Physics — Gravitation (Practice Quiz)
10 Qs · ~10 min
#165
Class 11 Physics — Systems of Particles and Rotational Motion (Practice Quiz)
10 Qs · ~10 min
#166
Class 11 Physics — Work, Energy and Power (Practice Quiz)
10 Qs · ~10 min
#167
Class 11 Physics — Motion in a Plane (Practice Quiz)
10 Qs · ~10 min
#168
Class 11 Physics — Motion in a Straight Line (Practice Quiz)
10 Qs · ~10 min
#169
Class 11 Physics — Units and Measurement (Practice Quiz)
10 Qs · ~10 min
#170
Class 11 Maths — Trigonometric Functions (Practice Quiz)
10 Qs · ~10 min
#171
Class 11 Chemistry — Structure of Atom (Practice Quiz)
10 Qs · ~10 min
#172
Class 11 Physics — Laws of Motion (Practice Quiz)
10 Qs · ~10 min
#173
Business Studies (Class 11) Practice Quiz | CBSE Class 11 Annual Assessment
10 Qs · ~10 min
#174
Economics (Class 11) Practice Quiz | CBSE Class 11 Annual Assessment
10 Qs · ~10 min
#175
Humanities Subjects Practice Quiz | CBSE Class 11 Annual Assessment
10 Qs · ~10 min
#176
Motion in a Straight Line Practice Quiz | CBSE Class 11 Annual Assessment
10 Qs · ~10 min
#177
Thermodynamic Processes and Laws Advanced Challenge | CBSE Class 11 Annual Assessment
10 Qs · ~10 min

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