From the heartbeat of a pendulum to the resonance of molecules — a complete pillar guide to periodic motion, SHM, and wave phenomena.
An oscillation (or vibration) is a repetitive back-and-forth motion of an object about a central equilibrium position. Unlike translational or rotational motion, the object periodically reverses its direction, spending equal time on either side of the mean position.
Oscillatory motion is everywhere in nature — from the pendulum of a grandfather clock to the vibration of atoms in a crystal lattice. Understanding oscillations is fundamental to optics, acoustics, quantum mechanics, and electrical engineering.
Any motion that repeats itself at regular time intervals. The interval of repetition is the time period. Not all periodic motion is oscillatory (e.g., uniform circular motion).
The body moves back and forth about a fixed equilibrium point. The net force is always directed toward the mean position. Examples: pendulum, spring-mass system, LC circuit.
The most fundamental oscillation where the restoring force is directly proportional to displacement and directed opposite to it. Described by a sinusoidal function.
SHM is defined as the motion in which the acceleration is always directed toward the mean position and is proportional to the displacement from it. It is the simplest form of oscillatory motion and serves as the building block for all complex periodic phenomena.
IMPORTANT RELATIONS
The quantity (ωt + φ) is called the phase. It determines the state of the oscillating particle — its position and direction of motion at any instant.
The value of phase at t = 0. It depends on the initial conditions — where the particle starts and in which direction it moves at t = 0.
For two SHM oscillators: Δφ = 0 (in phase), Δφ = π (antiphase). Phase difference determines superposition behavior and beats phenomena.
A block of mass m attached to a spring of force constant k executes SHM. The restoring force is Hooke's Law: F = −kx. For a vertical spring, gravity shifts the equilibrium position but does not affect the time period.
A point mass suspended by a massless, inextensible string of length L from a fixed support. For small oscillations (θ < 15°), the motion approximates SHM. The restoring force is the component of gravity along the arc.
In SHM, energy continuously transforms between kinetic and potential forms. The total mechanical energy remains constant (for ideal, undamped SHM) and is proportional to the square of the amplitude.
Maximum at mean position (x = 0). KE = ½mω²A².
Zero at extreme positions (x = ±A).
Maximum at extreme positions (x = ±A). PE = ½kA².
Zero at mean position (x = 0).
Constant throughout motion. Independent of position. E ∝ A² — doubling amplitude quadruples energy.
| Position | Displacement (x) | KE | PE | Total (E) |
|---|---|---|---|---|
| Mean Position | x = 0 | ½mω²A² (max) | 0 | ½mω²A² |
| Extreme Position | x = ±A | 0 | ½mω²A² (max) | ½mω²A² |
| Mid-way | x = A/√2 | E/2 | E/2 | ½mω²A² |
In real oscillators, the amplitude decreases over time due to resistive forces (air drag, friction). The damping force is typically proportional to velocity: Fd = −bv.
When an external periodic force is applied to a damped oscillator, the system undergoes forced oscillations. The system oscillates at the driving frequency, not its natural frequency.
⚡ Resonance
When driving frequency (ωd) equals natural frequency (ω₀ = √k/m), amplitude is maximum. This is resonance. Examples: Tacoma Narrows bridge collapse, tuning a radio, MRI machines, breaking a glass with sound.
| Type | External Force | Amplitude | Frequency | Energy |
|---|---|---|---|---|
| Free (Ideal) | None | Constant (A) | ω₀ (natural) | Conserved |
| Damped | Resistive (−bv) | Decreases (Ae^−γt) | ω′ < ω₀ | Decreases |
| Forced | External (F₀cosω_dt) | Steady state | ω_d (driving) | Input = Dissipated |
| Resonance | External at ω₀ | Maximum | ω_d = ω₀ | Max transfer |
All critical formulas for Chapter 13 — Oscillations. Bookmark this for JEE/NEET revision.
| # | Quantity | Formula | Notes |
|---|---|---|---|
| 1 | SHM condition | a = −ω²x | Restoring, directed to mean |
| 2 | Displacement | x = A cos(ωt + φ) | Or A sin(ωt + φ) depending on initial cond. |
| 3 | Velocity | v = ±ω√(A² − x²) | v_max = Aω at x = 0 |
| 4 | Acceleration | a = −ω²x | a_max = Aω² at x = ±A |
| 5 | Angular frequency | ω = 2π/T = 2πf | Unit: rad/s |
| 6 | Time period | T = 2π/ω = 1/f | Unit: seconds |
| 7 | Spring-mass period | T = 2π√(m/k) | Independent of amplitude, g |
| 8 | Pendulum period | T = 2π√(L/g) | Small oscillations only |
| 9 | Kinetic energy | KE = ½mω²(A² − x²) | Max at x = 0 |
| 10 | Potential energy | PE = ½mω²x² | Max at x = ±A |
| 11 | Total energy | E = ½mω²A² = ½kA² | E ∝ A² |
| 12 | Springs in series | 1/k_eff = 1/k₁ + 1/k₂ | k_eff < k₁, k₂ |
| 13 | Springs in parallel | k_eff = k₁ + k₂ | k_eff > k₁, k₂ |
| 14 | Damping force | F_d = −bv | b = damping coefficient (kg/s) |
| 15 | Damped amplitude | A(t) = A₀e^(−bt/2m) | Exponential decay |
| 16 | Damped frequency | ω′ = √(ω₀² − b²/4m²) | Always < ω₀ |
| 17 | Resonance condition | ω_d = ω₀ = √(k/m) | Amplitude maximum |
| 18 | Second's pendulum | T = 2 s, L ≈ 0.993 m | At Earth's surface |
Sinusoidal curve. Amplitude = A, period = T. x = A cos(ωt) starts at max displacement.
Cosine leads x by 90°. v = −Aω sin(ωt). Velocity is max when displacement is zero.
Exactly antiphase to x–t. a = −Aω²cos(ωt). Acceleration max at extreme positions.
Both are parabolas in x. KE + PE = constant. They intersect at x = ±A/√2.
Both oscillate with double the frequency (2ω) of displacement. Always positive.
The essential ideas from Chapter 13 that you must internalize — not just memorise.
Curated resources for Chapter 13 — Oscillations. Work through them in order for the best results.
Suggested Study Order
Read the Detailed Notes → Attempt MCQs → Check concepts with True-False → Solve Textbook Exercises → Challenge yourself with PYQs. This sequence builds from understanding to application to exam readiness.