Class 11 · Chapter 14

Oscillations

Overview, notes, short notes, formula sheet, daily practice problems, previous year questions, and videos for this chapter — all in one place.

Oscillations Short Notes

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Condensed revision points for Oscillations — for quick recall before exams, not a substitute for the full notes.

Periodic motion: repeats after a fixed time period (e.g. planetary orbits).
Oscillatory motion: to-and-fro about a fixed mean position — all oscillatory motion is periodic, not vice versa.
Harmonic function: constant-amplitude, single-frequency sin/cos only — y = Asinωt or Acosωt.
Non-harmonic: variable amplitude, or built from tan/cot/sec/cosec.
Mean position: restoring force = 0, PE = minimum.
Restoring force F = −kx, always directed toward mean position, opposite to displacement.
Amplitude = max displacement from mean position. One oscillation = mean→extreme→mean→other extreme→mean.

Linear SHM: F ∝ −y (e.g. mass on spring). Angular SHM: τ ∝ −θ (e.g. pendulum bob).
Necessary conditions: oscillatory motion, conserved KE+PE, well-defined extremes.
Linear: F=−kx → d²x/dt²+(k/m)x=0 → ω²=k/m → T=2π√(m/k).
Angular: τ=−Cθ → d²θ/dt²+(C/I)θ=0 → ω²=C/I → T=2π√(I/C).
SHM valid only for small amplitude — that's where restoring force/torque stays linear.

SHM = projection of uniform circular motion (radius A) onto a diameter.
x = Asinωt (from mean) or Acosωt (from extreme).
v = Aωcosωt = ω√(A²−x²); max |v| = Aω at mean position, zero at extremes.
a = −ω²Asinωt = −ω²x; max |a| = ω²A at extremes, zero at mean position.
x²/A² + v²/(ω²A²) = 1 → ellipse (circle if ω=1).
Phase relations: v leads x by π/2; a leads x by π (opposite); a leads v by π/2.
Path length = 2A. Distance per oscillation = 4A. Net displacement & work per oscillation = 0.

U(x) = ½kx² + U₀. U(t) = ½kA²sin²(ωt+φ) — parabolic in x, period = T/2 in time.
K(x) = ½mω²(A²−x²) = ½k(A²−x²). K(t) = ½mω²A²cos²(ωt+φ) — inverted parabola in x.
U max at extremes, K max (=½kA²) at mean position — exactly out of phase with each other.
Total energy E = U+K = ½kA² = ½mω²A² = constant always — depends only on m, ω, A.
Time-averaged: ⟨KE⟩=¼kA², ⟨PE⟩=¼kA²+U₀, ⟨TE⟩=½kA²+U₀.
Frequency of KE & PE oscillation = 2 × frequency of displacement. Frequency of TE = 0 (constant).

ω²=k/m, T=2π√(m/k), n=(1/2π)√(k/m). T independent of g — same on Moon, hill, satellite, any orientation.
k·l = constant for a spring → k ∝ 1/l. Cut into n parts: each part's k → nk, period → (1/√n)×.
Length ×n → k → (1/n)×, T → n×.
Heavier mass → T↑ (T∝√m). Stiffer spring → T↓, frequency↑.
Two masses both oscillating: use reduced mass μ=m₁m₂/(m₁+m₂), T=2π√(μ/k).
Only one mass oscillating (other fixed): T=2π√(m₁/k) — use only the oscillating mass.
Vertical spring stretched y₀ by load mg: ky₀=mg → T=2π√(y₀/g) ≡ 2π√(m/k).

Series (same force, different stretch): 1/k_s = 1/k₁+1/k₂.
Parallel (same stretch, different force): k_p = k₁+k₂.
Stiffer spring (higher k) always gives shorter period than a softer one under the same load.
Horizontal vs vertical mounting: period is identical — gravity shifts equilibrium point only.

Small-angle restoring force: F=−mgsinθ≈−mgθ=−(mg/l)y → k_eff=mg/l → T=2π√(l/g).
T independent of mass/material of bob — depends only on l and g. T ∝ √l.
Standing on a swing raises CM → l effectively decreases → T decreases → faster swing.
Second's pendulum: T=2s exactly; l≈1m at Earth's surface (g≈π² m/s²).
On Moon (g≈g_E/6): T_Moon = √6 × T_Earth ≈ 2.45× — Earth-calibrated clock runs slow on Moon.

General: T=2π√(l/g_eff).
Lift up at a: g_eff=g+a → T↓. Lift down at a: g_eff=g−a → T↑. Free fall: g_eff=0 → T→∞ (no oscillation).
Truck forward at f: g_eff=√(g²+f²) → T↓.
Bob density σ in liquid density ρ: g_eff=g(1−ρ/σ) → T↑.
Charged bob +q, field E downward: g_eff=g+qE/m → T↓. Field upward: g_eff=g−qE/m → T↑.
Pendulum at Earth's centre, in satellite, or in free-fall lift: g_eff=0, never oscillates.

l comparable to R: T=2π√[lR/(g(l+R))].
l<<R: reduces to T=2π√(l/g). l→∞ (infinite pendulum): T→2π√(R/g)≈84.6 min — max possible period.
Two pendulums started together realign when shorter completes exactly 1 more oscillation than longer: N√l_long=(N+1)√l_short.

Free oscillation: constant amplitude/frequency/energy, undamped, natural frequency ω=√(k/m).
Damped: F=−bv drains energy. m(d²x/dt²)+b(dx/dt)+kx=0. x=Ae^(−bt/2m)cos(ω't+φ), ω'=√(ω²−(b/2m)²).
Energy decay: E(t)=½kA²e^(−bt/m) — decays twice as fast (in exponent) as amplitude.
Underdamped (b/2m<ω): oscillates, amplitude decays exponentially.
Critically damped (b/2m=ω): fastest return to equilibrium, no oscillation.
Overdamped (b/2m>ω): slow return, no oscillation — slower than critical.
Forced: F(t)=F₀cos(ω_d t). Steady state at ω_d (not ω): A'=F₀/√[m²(ω²−ω_d²)²+b²ω_d²].
Resonance: ω_d≈ω → maximum amplitude, maximum energy transfer from driver.
Examples: soldiers breaking step on a bridge; tuning fork stem forcing tabletop; pushing a swing in rhythm.

Stretched wire: k=YA/L → n=(1/2π)√(YA/mL).
U-tube liquid (column length h): T=2π√(h/g) — independent of cross-section & density.
V-tube liquid (angles θ₁,θ₂): T=2π√[m/(Aρg(sinθ₁+sinθ₂))]; reduces to U-tube at θ₁=θ₂=90°.
Partially floating body (submerged height h): T=2π√(h/g) — same form as U-tube.
Ball in smooth spherical bowl (radius R): T=2π√(R/g) — behaves like pendulum of length R.
Ball in tunnel through Earth (diameter OR any chord): T=2π√(R/g)≈84.6 min — same period either way, only v_max differs.
Ball simply dropped from height (not released inside a tunnel at rest): NOT SHM — force ∝ 1/r², not ∝ −r.

Don't confuse 'periodic' with 'oscillatory' — uniform circular motion is periodic but never oscillatory.
T = 2π√(l/g) for a simple pendulum is mass-independent — a common wrong-option trap involves mass.
Spring-block period is g-independent; pendulum period is g-dependent — don't mix these up under gravity-change questions (lifts, Moon, satellites).
PE and KE both oscillate at 2× the frequency of displacement — total energy has zero frequency (it's constant), a frequent mix-up.
g_eff sign errors are the #1 trap in lift/field/liquid pendulum problems — always check whether the extra force adds to or opposes gravity.
Critical damping ≠ zero oscillation forever — it's the fastest non-oscillating return; overdamped is slower, not faster.
At resonance, amplitude is large but not infinite unless damping is exactly zero — don't assume infinite amplitude by default.
Tunnel-through-Earth period (≈84.6 min) is identical for a diametric tunnel and a chord tunnel — only maximum speed changes between them.

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