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Class XI · Chapter 13

Oscillations

A swinging pendulum, a vibrating guitar string, an atom in a crystal — wildly different systems, all governed by the exact same handful of equations.

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1Periodic and Oscillatory Motion

Periodic motion repeats itself at regular time intervals — a planet orbiting, a clock's hands turning, a pendulum swinging. Oscillatory motion is a specific kind of periodic motion where an object moves back and forth around a fixed equilibrium position, rather than continuing endlessly in one direction (like circular motion does). Every oscillation is periodic, but not every periodic motion oscillates back and forth.

2Simple Harmonic Motion

Simple harmonic motion (SHM) is the cleanest, most fundamental type of oscillation — and remarkably, countless real oscillating systems approximate it closely, at least for small displacements.

Displacement in SHM
x(t) = A cos(ωt + φ)
A = amplitude (maximum displacement), ω = angular frequency, φ = phase constant (sets the starting point of the cycle). ω = 2π/T = 2πf, connecting angular frequency to period T and ordinary frequency f.

3SHM and Uniform Circular Motion

SHM has a beautifully simple geometric picture: it's exactly what you get by watching the shadow of a point moving in uniform circular motion (Chapter 3), projected onto any single diameter. As the point sweeps around at constant angular speed, its projection speeds up, slows down, and reverses — tracing out the same cosine pattern that defines SHM.

4Velocity and Acceleration in SHM

Velocity
v = −Aω sin(ωt + φ), vmax = Aω
Velocity is maximum at the mean (equilibrium) position, and momentarily zero at the extremes — exactly where the motion reverses direction.
Acceleration
a = −Aω² cos(ωt + φ) = −ω²x, amax = Aω²
Acceleration is exactly opposite — maximum at the extremes, and zero at the mean position. This a = −ω²x relationship is, in fact, the defining signature of SHM: any motion satisfying it is automatically simple harmonic.

5Force Law for SHM

Combining a = −ω²x with Newton's second law gives the force responsible for any SHM:

Restoring Force
F = −kx, where k = mω² (so ω = √(k/m))
This is exactly Hooke's Law from Chapter 8 — the force always points back toward equilibrium, growing stronger the farther the system is displaced. Any system obeying F = −kx will oscillate in SHM, which is precisely why springs are the go-to example throughout this chapter.

6Energy in SHM

An oscillating system continuously trades kinetic and potential energy back and forth, exactly like the spring problems in Chapter 5 — because that's precisely what SHM is.

Kinetic and Potential Energy
KE = ½mω²(A² − x²), PE = ½mω²x² = ½kx²
Total Energy
E = ½kA² = ½mω²A²
Constant throughout the motion (for undamped SHM) — maximum kinetic energy occurs exactly where potential energy is zero (mean position), and vice versa at the extremes.

7Systems Executing SHM

Spring-Mass System
T = 2π √(m/k)
Heavier mass or a softer (lower k) spring both mean a longer period — matching the everyday intuition that heavier or looser springs oscillate more sluggishly.
Simple Pendulum (Small Angle)
T = 2π √(L/g)
Notice mass doesn't appear at all — a heavy bob and a light bob of the same length swing with the same period, echoing the mass-independence seen with free fall. This approximation only holds for small swing angles; larger swings deviate from true SHM.

8Damped Simple Harmonic Motion

Real oscillators always lose energy to friction or resistance, so their amplitude gradually shrinks over time rather than continuing forever at constant amplitude — this is damped oscillation.

Amplitude Decay
A(t) = A₀ e−bt/2m
b = damping constant. A swinging pendulum in air eventually comes to rest, and a car's shock absorbers are deliberately engineered to damp out oscillations quickly rather than let the car keep bouncing.

9Forced Oscillations and Resonance

Push a system periodically with an external driving force, and it eventually settles into oscillating at the driving frequency, not its own natural frequency. When the driving frequency happens to match the system's natural frequency, the amplitude grows dramatically — this is resonance.

Why It Matters Resonance is why pushing a swing at just the right rhythm sends it soaring with minimal effort — and, far more seriously, why bridges and buildings are engineered to avoid having a natural frequency that matches wind gusts, footsteps, or (in earthquake zones) ground-shaking frequencies. Unchecked resonance has caused real structural failures.

Formula Summary

Displacement
x = A cos(ωt+φ)
Max Velocity
v_max = Aω
Acceleration
a = −ω²x
Restoring Force
F = −kx
Total Energy
E = ½kA²
Spring-Mass Period
T = 2π√(m/k)
Pendulum Period
T = 2π√(L/g)
Damped Amplitude
A(t) = A₀e^(−bt/2m)

Solved Examples

Example 1 · Simple Pendulum

Find the period of a simple pendulum of length 1 m. (g = 9.8 m/s²)

Solution: T = 2π√(L/g) = 2π√(1/9.8) = 2π × 0.319 ≈ 2.0 s.

Example 2 · Maximum Velocity and Acceleration

A particle executes SHM with amplitude 5 cm and angular frequency 2 rad/s. Find its maximum velocity and maximum acceleration.

Solution: vmax = Aω = 0.05 × 2 = 0.1 m/s.

amax = Aω² = 0.05 × 4 = 0.2 m/s².

Quick Check

1. In SHM, the acceleration of the particle is:
Constant throughout the motion
✓ Proportional to displacement, directed toward the mean position (correct)
Maximum at the mean position
Independent of displacement
2. At the mean (equilibrium) position in SHM, the velocity of the particle is:
Zero
✓ Maximum (correct)
Equal to acceleration
Undefined
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