Oscillations

An oscillation is a periodic back-and-forth motion about an equilibrium position. Oscillations are fundamental to many phenomena — from pendulums and springs to sound and light.

Periodic Motion

Periodic motion: Motion that repeats itself after a fixed time interval called the period (T).
Frequency (f): Number of oscillations per second, f = 1/T (unit: Hz = s^-1).
Angular frequency: omega = 2 pi f = 2 pi / T

Simple Harmonic Motion (SHM)

A special type of oscillation where the restoring force is proportional to displacement and directed toward equilibrium:
F = -k x (Hooke's law form)

Equation of SHM: a = -omega² x, where omega = √(k/m)

Displacement in SHM: x(t) = A cos(omega t + phi)

where A = amplitude, phi = initial phase.

Velocity: v(t) = -A omega sin(omega t + phi); maximum speed v_max = A omega (at mean position)
Acceleration: a(t) = -A omega² cos(omega t + phi); maximum at extremes = A omega²

Energy in SHM

Kinetic energy: KE = (1/2) m v² = (1/2) m omega² (A² - x²)
Potential energy: PE = (1/2) k x² = (1/2) m omega² x²
Total energy: E = (1/2) k A² = (1/2) m omega² A² = constant

Energy oscillates between KE and PE, total remains constant in the absence of damping.

Simple Pendulum

For small angles of swing (< 15 degrees), a simple pendulum undergoes SHM:
T = 2 pi √(L/g)

where L = length of pendulum, g = acceleration due to gravity.
Note: Period does NOT depend on mass or amplitude (for small angles).

Spring-Mass System

For a mass m on a spring of spring constant k:
T = 2 pi √(m/k)

Frequency increases with stiffer spring and decreases with greater mass.

Damped and Forced Oscillations

Damping: In real systems, energy is lost to friction/resistance. The amplitude decreases exponentially over time.

Resonance: When the frequency of the driving force equals the natural frequency of the system, the amplitude is maximum. This is resonance. Examples: Tacoma bridge collapse, tuning of radio receivers.

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Example 1

A mass of 0.5 kg on a spring (k = 200 N/m) oscillates. Find period and frequency.
omega = √(k/m) = √(200/0.5) = √(400) = 20 rad/s
T = 2 pi / omega = 2 pi / 20 = 0.314 s; f = 1/T = 3.18 Hz

Example 2

A particle in SHM has amplitude 5 cm and frequency 2 Hz. Find maximum speed.
omega = 2 pi f = 4 pi rad/s
v_max = A omega = 0.05 x 4 pi = 0.628 m/s

Example 3

A simple pendulum of length 1 m. Find period at g = 9.8 m/s².
T = 2 pi √(L/g) = 2 pi √(1/9.8) = 2 pi x 0.319 = 2.006 s ~ 2 s

Example 4

In SHM, x = 0.1 cos(10t) m. Find amplitude, angular frequency, and time period.
Comparing: A = 0.1 m, omega = 10 rad/s, T = 2 pi / 10 = 0.628 s

Example 5

Find KE and PE in SHM when displacement = A/2.
KE = (1/2) m omega² (A² - x²) = (1/2) m omega² (A² - A²/4) = (3/4) E_total
PE = (1/2) m omega² x² = (1/4) E_total

Example 6

Total energy of SHM with A = 0.1 m, k = 100 N/m.
E = (1/2) k A² = (1/2) x 100 x (0.1)² = 0.5 J

Example 7

A pendulum of length 4 m is transferred from Earth (g = 9.8) to Moon (g = 1.6). How does the period change?
T_Earth = 2 pi √(4/9.8) = 4.01 s
T_Moon = 2 pi √(4/1.6) = 9.93 s
Period increases on the Moon because g is smaller.

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Common mistakes

The period of a simple pendulum depends on L and g only, not mass or amplitude (for small angles).
At equilibrium (x = 0): KE is maximum, PE is minimum.
At extremes (x = A): KE is zero, PE is maximum.
SHM requires small oscillations for the pendulum approximation to be valid.

Summary

SHM is characterised by a restoring force proportional to displacement. Energy alternates between kinetic and potential, total remaining constant. Period of a spring-mass system depends on k and m; pendulum period depends on L and g. Resonance occurs when driving frequency matches natural frequency.