Class 11 Physics Chapter 13 Oscillations — Formulas & Key Points

Core SHM Displacement, Velocity and Acceleration Formulas

• Displacement lags velocity by π/2 and lags acceleration by π (or equivalently, acceleration is antiparallel to displacement).
• Maximum speed v_max = Aω occurs at mean position (x = 0); maximum acceleration a_max = Aω² occurs at extreme positions (x = ±A).
• Angular frequency ω = 2πf = 2π/T; it is measured in rad/s, not Hz.
• Phase constant φ depends on initial position and velocity; solve x(0) and v(0) simultaneously to find φ.

Spring-Mass and Simple Pendulum Time Period Formulas

• Spring constant k = F/x is measured in N/m; stiffer spring means larger k and shorter T.
• For a vertical spring, equilibrium position shifts by mg/k but time period remains T = 2π√(m/k).
• Simple pendulum formula T = 2π√(l/g) is valid only for small amplitude; large swings require elliptic integral corrections not in CBSE syllabus.
• In an elevator accelerating upward with acceleration a, effective g becomes g + a; T decreases.
• On the moon where g_moon ≈ g_earth/6, pendulum time period increases by factor √6.

Energy in SHM — Kinetic, Potential and Total Energy Formulas

• Total energy is proportional to square of amplitude: doubling A quadruples energy.
• At displacement x = A/2, kinetic energy is three-fourths of total energy and potential energy is one-fourth.
• Energy oscillates between kinetic and potential at frequency 2f, twice the oscillation frequency.
• In damped oscillations, total mechanical energy decreases exponentially with time.
• Power delivered by restoring force averages to zero over one complete cycle in undamped SHM.

Damped Oscillations — Amplitude Decay and Quality Factor

• Damping force is often modeled as F_d = –bv, proportional to velocity, where b is damping coefficient in kg/s.
• Time constant τ = 2m/b: amplitude falls to 1/e of initial value in time τ.
• Energy decays as E(t) = E₀ exp(–bt/m), twice as fast as amplitude decay.
• Critically damped condition: b = 2√(km); system returns to equilibrium in shortest time without oscillating.
• Logarithmic decrement δ = ln(A_n/A_(n+1)) = bT/2m, used to measure damping from successive amplitude ratios.

Forced Oscillations and Resonance Conditions

• Steady-state amplitude A(ω_d) = F₀/[m√{(ω₀² – ω_d²)² + (bω_d/m)²}]; maximum when ω_d ≈ ω₀.
• Phase difference between driving force and displacement varies from 0° (ω_d << ω₀) to 180° (ω_d >> ω₀), passing through 90° at exact resonance.
• Resonance peak height is inversely proportional to damping: narrow sharp peak for low b, broad flat peak for high b.
• Power absorbed from driver is maximum at resonance; half-power bandwidth Δω = b/m.
• In the absence of damping, amplitude at resonance would theoretically become infinite; real systems always have some damping.

Key Definitions and Terminology for Chapter 13

• Periodic motion: motion that repeats itself after equal intervals of time (e.g., rotation of Earth).
• Oscillatory motion: to-and-fro motion about a mean position (e.g., pendulum, spring-mass system).
• Simple Harmonic Motion (SHM): oscillatory motion with restoring force F = –kx and acceleration a = –ω²x.
• Amplitude (A): maximum displacement from equilibrium, measured in metre; determines energy of oscillation.
• Time period (T): time taken for one complete oscillation, measured in second.
• Frequency (f): number of oscillations per second, measured in hertz (Hz); f = 1/T.
• Angular frequency (ω): 2πf, measured in rad/s; relates to time period by ω = 2π/T.
• Phase (φ): argument (ωt + φ) of sine or cosine function; determines initial state and relative position in cycle.
• Epoch or initial phase: value of phase at t = 0, denoted φ₀ or simply φ.

Important Physical Constants and Standard Values

• Acceleration due to gravity: g = 9.8 m/s² (standard); g ≈ 10 m/s² (approximation in quick calculations).
• Seconds pendulum length on Earth: l ≈ 0.993 m or 99.3 cm, giving T = 2 s.
• Value of π: use π ≈ 3.14 or π ≈ 22/7 as specified in the problem; keep at least two decimal places in intermediate steps.
• Radian to degree conversion: 2π rad = 360°; 1 rad ≈ 57.3°.

Common Mistakes in Signs, Units and Notation

• Always include the negative sign in F = –kx and a = –ω²x; it indicates direction opposite to displacement.
• Do not confuse angular frequency ω (rad/s) with frequency f (Hz); ω = 2πf.
• Time period T is in seconds (s), not minutes or milliseconds unless problem states otherwise.
• Spring constant k must be in N/m; if given in N/cm, multiply by 100 to convert.
• Mass must be in kg; convert gram to kg by dividing by 1000.
• Length for pendulum must be in metre; convert cm to m by dividing by 100.
• Phase angle φ is dimensionless but conventionally in radians; if using degrees, convert before applying sine or cosine.
• Amplitude A, being a magnitude, is always positive; displacement x can be negative.

Memory Tricks and Mnemonics for Quick Recall

• T ∝ √m for spring-mass: 'More Mass, More Time'.
• T ∝ 1/√k for spring-mass: 'Stiffer Spring, Shorter Time'.
• T ∝ √l for pendulum: 'Longer Length, Longer Time'.
• Phase: 'V ahead of X by π/2, A opposite to X by π' → remember VXA order.
• Energy: 'E ∝ A²' → double amplitude, quadruple energy.
• Resonance: 'Driver = Natural → Peak Amplitude'.
• Damping: 'b large → A small → Q small → broad peak'.

Three Solved Mini-Examples Applying Core Formulas

One-Glance Last-Minute Revision Box

• **SHM defining equation:** a = –ω²x; restoring force F = –kx.
• **Displacement, velocity, acceleration:** x = A sin(ωt+φ); v = Aω cos(ωt+φ); a = –Aω² sin(ωt+φ).
• **Time period (spring-mass):** T = 2π√(m/k); frequency f = 1/T = (1/2π)√(k/m).
• **Time period (simple pendulum):** T = 2π√(l/g); independent of mass of bob.
• **Energy in SHM:** Total E = ½kA² = ½mω²A²; KE_max = PE_max = E; <KE> = <PE> = E/2.
• **Damped oscillation amplitude:** A(t) = A₀ e^(–bt/2m); energy E(t) = E₀ e^(–bt/m).
• **Resonance condition:** ω_driver = ω_natural; amplitude maximum, phase difference 90°.
• **Quality factor:** Q = ω₀/(Δω) = mω₀/b; high Q means sharp resonance.
• **Key point:** Always write F = –kx (with minus sign) and check units kg, m, s, N/m.
• **Common trap:** Do not confuse ω (rad/s) and f (Hz); remember ω = 2πf.