Oscillations
'There is geometry in the humming of the strings... there is music in the spacing of the spheres.' — Pythagoras
1. Chapter Overview
OSCILLATORY motion is REPETITIVE motion back and forth about a MEAN position. This chapter focuses on SIMPLE HARMONIC MOTION (SHM) — the simplest form of oscillation — and its applications in SPRING-MASS systems and the SIMPLE PENDULUM. You will also study DAMPED oscillations, FORCED oscillations, and the IMPORTANT phenomenon of RESONANCE.
2. Periodic and Oscillatory Motion
- Periodic Motion: Motion that REPEATS itself at regular intervals
- Oscillatory Motion: To-and-fro motion about a MEAN position
- Period (T): Time for ONE complete oscillation
- Frequency (f): Number of oscillations per second (f = 1/T)
- SI unit: hertz (Hz), 1 Hz = 1 s⁻¹
Amplitude and Phase
- Amplitude (A): Maximum displacement from mean position
- Phase (φ): Describes the STATE of the oscillator (position + direction)
- Phase Constant (φ₀): Initial phase at t = 0
3. Simple Harmonic Motion (SHM)
Definition
A particle is in SHM if its acceleration is DIRECTLY proportional to its displacement from the mean position and ALWAYS directed TOWARD the mean position.
- a = -ω²x (where ω = angular frequency)
Equation of SHM
- x(t) = Acos(ωt + φ₀) or x(t) = Asin(ωt + φ₀)
- ω = 2π/T = 2πf
Velocity and Acceleration in SHM
- v = dx/dt = -Aω sin(ωt + φ₀)
- a = d²x/dt² = -Aω² cos(ωt + φ₀) = -ω²x
Important Relations
- |v_max| = Aω (at mean position, x = 0)
- |a_max| = Aω² (at extreme positions, x = ±A)
- v = ω√(A² — x²) (velocity in terms of displacement)
Worked Problem
Q: A particle in SHM has amplitude 5 cm and time period 2 s. Find maximum velocity and acceleration. A: ω = 2π/T = π rad/s. v_max = Aω = 0.05π = 0.157 m/s. a_max = Aω² = 0.05×π² = 0.493 m/s².
4. Energy in SHM
Kinetic Energy
- K = ½mω²(A² — x²)
- K_max = ½mω²A² (at x = 0)
Potential Energy (Spring-type force)
- U = ½mω²x²
- U_max = ½mω²A² (at x = ±A)
Total Energy
- E = K + U = ½mω²A² = CONSTANT
- Energy interconversion: KE ↔ PE, but total remains CONSTANT
Worked Problem
Q: A 0.2 kg mass oscillates with A = 10 cm and T = 0.5 s. Find total energy. A: ω = 2π/0.5 = 4π rad/s. E = ½mω²A² = ½×0.2×(4π)²×(0.1)² = 0.1×16π²×0.01 = 0.016π² = 0.158 J.
5. Spring-Mass System
Horizontal Spring
- Restoring force: F = -kx (Hooke's law)
- Angular frequency: ω = √(k/m)
- Time period: T = 2π√(m/k)
Vertical Spring
- Same ω = √(k/m)
- The Mean position shifts due to gravity (extension x₀ = mg/k)
Series and Parallel Combinations
| Combination | Effective Spring Constant |
|---|---|
| Series (same force, different extensions) | 1/k_eff = 1/k₁ + 1/k₂ |
| Parallel (same extension, different forces) | k_eff = k₁ + k₂ |
6. Simple Pendulum
- Restoring torque: τ = -mgLθ (for small θ)
- Angular frequency: ω = √(g/L)
- Time period: T = 2π√(L/g)
Key Points
- Period is INDEPENDENT of MASS (Galileo's discovery)
- Period is INDEPENDENT of AMPLITUDE (for small angles, θ < 10°)
- T ∝ √L: longer pendulum → longer period
- T ∝ 1/√g: weaker gravity → longer period
Worked Problem
Q: A simple pendulum has length 1 m. Find its period at Earth's surface (g = 9.8 m/s²). A: T = 2π√(L/g) = 2π√(1/9.8) = 2π×0.319 = 2.006 s.
7. Damped and Forced Oscillations
Damped Oscillations
- Real oscillations have FRICTION or DRAG → amplitude DECREASES with time
- x(t) = Ae^(-bt/2m)cos(ω't + φ₀) (exponentially decaying amplitude)
- b = damping constant
- Underdamped (b² < 4mk): Oscillations with decreasing amplitude
- Critically damped (b² = 4mk): Fastest return to equilibrium (NO oscillation)
- Overdamped (b² > 4mk): Slow return without oscillation
Forced Oscillations and Resonance
- Periodic DRIVING force applied to an oscillator
- Amplitude is MAXIMUM when driving frequency = NATURAL frequency of the system
- This is RESONANCE
Examples of Resonance
- Mechanical: Soldier breaking step while crossing a bridge
- Sound: Opera singer shattering a glass
- Electrical: Tuning a radio to a specific station (LC circuit)
- Tacoma Narrows Bridge: Wind-induced resonance caused collapse (1940)
8. Common Mistakes
- SHM acceleration is NOT constant: a = -ω²x, which depends on x. It is maximum at extremes, zero at mean
- Pendulum T is independent of mass: Many students think heavier pendulums swing slower
- T = 2π√(m/k) for spring, T = 2π√(L/g) for pendulum: Don't confuse the formulas
- v_max = Aω, not ω²: Acceleration maximum has ω², velocity maximum has ω
- Resonance amplitude is NOT infinite: In real systems, damping limits the maximum amplitude
9. CBSE Exam Focus
- SHM definition — acceleration proportional to -displacement (1-mark)
- Derivation of velocity and acceleration in SHM (3-mark)
- Energy in SHM — KE, PE, total energy (5-mark)
- Spring-mass system — time period derivation (3-mark)
- Simple pendulum — time period (3/5-mark)
- Graphs of x, v, a versus time in SHM
- Resonance — definition and examples (1/3-mark)
10. Key Formulas
- x = Acos(ωt + φ₀)
- v = -Aω sin(ωt + φ₀), v = ω√(A² — x²)
- a = -Aω² cos(ωt + φ₀) = -ω²x
- K = ½mω²(A² — x²), U = ½mω²x²
- E = ½mω²A² (constant)
- T_spring = 2π√(m/k)
- T_pendulum = 2π√(L/g)
11. Self-Test (5+ Q&A)
Q1: A particle in SHM has amplitude 10 cm and frequency 2 Hz. Write the equation of motion assuming it starts from the mean position moving positively. A: ω = 2πf = 4π rad/s. At t = 0, x = 0 moving positive → use x = Asin(ωt). x = 0.1sin(4πt) m.
Q2: At what displacement is the kinetic energy equal to potential energy in SHM? A: K = U → ½mω²(A² — x²) = ½mω²x² → A² — x² = x² → 2x² = A² → x = ±A/√2.
Q3: A spring of constant k = 100 N/m has a 4 kg mass attached. Find period. A: T = 2π√(m/k) = 2π√(4/100) = 2π×0.2 = 1.257 s.
Q4: A pendulum clock runs slow. Should the pendulum length be increased or decreased? A: T = 2π√(L/g). If the clock runs slow, T is too large. Decrease L to decrease T.
Q5: What is the difference between damped and forced oscillations? A: Damped oscillations have amplitude decreasing due to energy loss (friction). Forced oscillations are driven by an external periodic force, and their amplitude depends on the driving frequency.
12. Conclusion
Oscillations are EVERYWHERE — from the vibrating string of a guitar to the swaying of a building in an earthquake. SHM is the FOUNDATIONAL model because its mathematics (sine/cosine functions) describes small oscillations around equilibrium in ALL physical systems. The spring-mass system and pendulum are the CLASSIC examples. Understanding resonance is CRITICAL for both avoiding disasters (bridge collapse) and enabling technology (radio tuning, MRI).
