Oscillations

2.1 Core concepts

• Motion that repeats itself at regular intervals of time is called periodic motion; to-and-fro motion about a mean (equilibrium) position is called oscillatory motion. Every oscillatory motion is periodic, but every periodic motion need not be oscillatory — uniform circular motion is periodic but not oscillatory (NCERT §13.1–13.2, p. 259–260).
• At the equilibrium position no net external force acts on the body; a small displacement brings a restoring force into play, producing oscillations or vibrations. The terms oscillation and vibration are essentially synonymous; oscillation is used at low frequency (tree branch), vibration at high frequency (musical string) (NCERT §13.2, p. 260).
• The period T is the smallest time interval after which the motion repeats; its SI unit is the second. The frequency ν = 1/T is measured in hertz (1 Hz = 1 oscillation per second = 1 s⁻¹); ν need not be an integer (NCERT §13.2.1, Eqs. 13.1–13.2, p. 260–261).
• In this chapter displacement is generalised: it refers to the change with time of any physical property (position of a block on a spring, angle of a pendulum from the vertical, voltage across a capacitor, pressure in a sound wave). Displacement can be positive or negative (NCERT §13.2.2, p. 261).
• A periodic displacement is conveniently written as f(t) = A cos ωt (or A sin ωt). The period is T = 2π/ω; any linear combination A sin ωt + B cos ωt is also periodic with the same T. By Fourier's theorem any periodic function can be written as a superposition of sines and cosines (NCERT §13.2.2, Eqs. 13.3a–13.3d, p. 261–262).
• Simple Harmonic Motion (SHM) is the oscillation in which the displacement is a sinusoidal function of time: x(t) = A cos(ωt + φ), where A is the amplitude, ω the angular frequency, (ωt+φ) the phase, and φ the phase constant. SHM is not just any periodic motion — it is the sinusoidal one (NCERT §13.3, Eq. 13.4, p. 262).
• The amplitude A is the magnitude of maximum displacement and is taken positive without loss of generality; the cosine varies between +1 and −1, so x varies between +A and −A. The phase (ωt + φ) determines the instantaneous state of motion; at t = 0 the phase equals the phase constant φ (NCERT §13.3, p. 263).
• ω is related to T by ω = 2π/T (Eq. 13.7), and to frequency by ω = 2πν. ω is called the angular frequency and has SI units of rad s⁻¹ (NCERT §13.3, Eqs. 13.5–13.7, p. 264).
• SHM ↔ uniform circular motion: the projection of a particle moving uniformly on a circle of radius A with angular speed ω on a diameter of that circle executes SHM with amplitude A and angular frequency ω. The circle is called the reference circle and the moving point the reference particle (NCERT §13.4, p. 264–265).
• Velocity in SHM: v(t) = −Aω sin(ωt + φ), obtained either by projecting the tangential velocity v = ωA of the reference particle or by differentiating x(t). Hence vmax = Aω, occurring at x = 0 (mean position), and v = 0 at x = ±A. Eliminating time gives v = ±ω√(A² − x²) (NCERT §13.5, Eqs. 13.8–13.10, p. 266).
• Acceleration in SHM: a(t) = −ω²A cos(ωt + φ) = −ω² x(t), obtained from the centripetal acceleration ω²A of the reference particle, or by differentiating v(t). Thus acceleration is proportional to displacement and always directed towards the mean position; |a|max = ω²A at the extremes, a = 0 at x = 0 (NCERT §13.5, Eqs. 13.11–13.12, p. 266–267).
• Phase relations: with respect to displacement, velocity has a phase lead of π/2 and acceleration has a phase difference of π (i.e., is in anti-phase). All three quantities have the same period T (NCERT §13.5, Fig. 13.13, p. 267).
• Force law for SHM: by Newton's second law F = ma = −mω² x = −kx, where k = mω² (so ω = √(k/m)). The restoring force is linearly proportional to displacement and directed towards the mean position; the system is a linear harmonic oscillator. Forces with extra x², x³ terms give non-linear oscillators (NCERT §13.6, Eqs. 13.13–13.14, p. 267).
• Energy in SHM: kinetic energy K = ½ mv² = ½ mω²A² sin²(ωt+φ) = ½ k A² sin²(ωt+φ); potential energy U = ½ k x² = ½ k A² cos²(ωt+φ). Both are periodic with period T/2 (NCERT §13.7, Eqs. 13.15–13.17, p. 268).
• The total mechanical energy of the harmonic oscillator is E = K + U = ½ k A² = ½ mω²A² — constant in time, independent of where the particle is. At x = 0 the energy is entirely kinetic; at x = ±A it is entirely potential; in between K and U exchange (NCERT §13.7, Eq. 13.18 and Fig. 13.16, p. 269).
• Spring–mass oscillator: F = −kx gives ω = √(k/m) and T = 2π√(m/k) (NCERT §13.6 and Summary point 8, Eqs. 13.13–13.14, p. 267 and p. 272).
• Simple pendulum: for a bob of mass m on an inextensible massless string of length L, the tangential restoring torque is τ = −L (mg sinθ). Newton's law of rotation gives α = −(mgL/I) sinθ. For small θ, sinθ ≈ θ (Table 13.1 shows sinθ ≈ θ even up to ~20°), so α = −(mgL/I) θ — SHM in angular displacement (NCERT §13.8, Eqs. 13.19–13.24, p. 270–271).
• With I = mL², the angular frequency is ω = √(g/L) and the period is T = 2π√(L/g), independent of mass and amplitude (NCERT §13.8, Eqs. 13.25–13.26, p. 271).
• Two-spring effective stiffness (NCERT Example 13.6, p. 267–268): when a mass is connected to two identical springs of constant k on opposite sides, both contribute restoring force, so the effective k is 2k and T = 2π√(m/2k). The same logic gives k_series = k₁k₂/(k₁ + k₂) and k_parallel = k₁ + k₂ in standard combinations encountered in NCERT exercises.
• Energy interconversion is continuous in SHM: at any displacement x, K = ½ mω²(A² − x²) and U = ½ mω² x²; sum K + U = ½ mω² A² is conserved. Plots of K and U against time are sin² and cos² curves of period T/2, mirror images of each other around the half-energy line (NCERT §13.7, Fig. 13.16, p. 268–269).
• SHM as projection of uniform circular motion explains why the kinematic equations of SHM have the same form as the components of uniform circular motion: x = A cos θ, vₓ = −Aω sin θ, aₓ = −Aω² cos θ, with θ = ωt + φ — this geometric mapping is what NCERT exploits to derive every SHM relation without calculus (NCERT §13.4, p. 264–266).

2.2 Definitions to memorise

2.3 Diagrams / processes to remember

• Fig. 13.1 (p. 260) — three examples of periodic motion: insect climbing a ramp; child climbing a step; a ball bouncing (parabolic arcs).
• Fig. 13.2 (p. 261) — (a) block on a spring, the prototype linear SHM system; (b) simple pendulum with angular displacement θ.
• Fig. 13.3 / 13.4 / 13.5 (p. 262–263) — particle vibrating between +A and −A; snapshots at t = 0, T/4, T/2, 3T/4, T; continuous x-vs-t cosine curve.
• Fig. 13.6 (p. 263) — table mapping the symbols A, ω, ωt+φ, φ to their meanings in x(t) = A cos(ωt+φ).
• Fig. 13.7 (p. 263) — two SHMs with same ω, φ but different A (a); same A, ω but different φ (b).
• Fig. 13.8 (p. 264) — two SHMs with same A, φ but different periods T (and hence different ω).
• Fig. 13.9 / 13.10 (p. 265) — ball on a string in horizontal circular motion; its projection on a wall is SHM. Reference-circle construction.
• Fig. 13.11 / 13.12 (p. 266) — projecting the tangential velocity v = ωA of reference particle gives v(t) = −ωA sin(ωt+φ); projecting the centripetal acceleration ω²A gives a(t) = −ω²A cos(ωt+φ) = −ω² x(t).
• Fig. 13.13 (p. 267) — stacked plots of x(t), v(t), a(t): same period T but velocity leads displacement by π/2, acceleration is anti-phase to displacement.
• Fig. 13.14 / 13.15 (p. 268) — Example 13.6: a mass between two identical springs k each gives effective force −2kx and T = 2π√(m/2k).
• Fig. 13.16 (p. 269) — K, U, and total E plotted vs time and vs displacement: K and U both repeat with period T/2; total E = ½kA² is a horizontal line.
• Fig. 13.17 (p. 270) — simple pendulum free-body diagram: T along string; mg cosθ (radial) and mg sinθ (tangential, restoring).
• Table 13.1 (p. 271) — values of θ (degrees, radians) vs sinθ, showing sinθ ≈ θ to good accuracy up to ~20°.

2.4 Common confusions / NTA trap points

• Periodic vs SHM. Every SHM is periodic, but the reverse is false. Earth's rotation is periodic, not SHM. NTA distractors will offer a periodic non-sinusoidal motion (e.g., a bouncing ball, planetary orbit) as "SHM" (NCERT §13.2 and Points to Ponder #2, p. 273).
• Phase of v and a. Students often write "velocity is in phase with displacement". It isn't — v leads x by π/2 and a is in anti-phase (phase difference π), not π/2 (NCERT §13.5, Fig. 13.13, p. 267).
• vmax and amax locations. vmax = Aω occurs at the mean position (x = 0), not at the extremes; amax = ω²A occurs at the extremes (x = ±A), not at the mean. NTA loves swapping these (NCERT §13.5, p. 266–267).
• KE/PE at mean and extreme. At x = 0 energy is entirely kinetic; at x = ±A entirely potential. KE and PE individually have period T/2, but total E is constant — not periodic in any non-trivial sense (NCERT §13.7, Fig. 13.16, p. 268–269).
• Pendulum period independent of mass and amplitude. T = 2π√(L/g) — students sometimes write it as depending on mass m or on initial amplitude θ₀. Both are wrong for small oscillations (NCERT §13.8 and Points to Ponder #7, Eq. 13.26, p. 271 and p. 273).
• Two-spring trap (Example 13.6). With identical springs k on either side of a mass, both pull it back: F = −2kx, so ω = √(2k/m), T = 2π√(m/2k) — not T = 2π√(m/k). Students forget to add the spring constants (NCERT §13.6, Example 13.6, p. 267–268).
• Small-angle approximation for pendulum. SHM holds only for small θ where sinθ ≈ θ. For large amplitudes the pendulum is still periodic, but not simple harmonic (NCERT §13.8 and Points to Ponder #8, p. 271, 273).
• Confusing T (period) with t (time). T is the smallest interval for one full repetition; t is the running time. The phase ωt + φ uses the running t, not T.
• Energy quadratic in amplitude. Doubling A multiplies E by 4 (not 2), because E ∝ A². Many students miss the square.
• Pendulum on the Moon. Since T ∝ 1/√g and g_Moon ≈ g_Earth/6, the same pendulum has period √6 ≈ 2.45 times longer on the Moon — NTA loves this swap.
• k for a vertically hanging spring. The natural length shifts to a new equilibrium under gravity; the SHM about this new equilibrium still has T = 2π√(m/k) — gravity does not enter the period.

2.5 Key formulas table