Chapter 13 — Oscillations

Oscillations is periodic motion in which objects move back and forth about an equilibrium position, with displacement described by sinusoidal functions like x(t) = A cos(ωt + φ) in simple harmonic motion.

Chapter 13 covers oscillatory and periodic motions, with focus on simple harmonic motion (SHM)—the simplest oscillatory form where displacement is sinusoidal. Key concepts include amplitude (A), angular frequency (ω), phase (ωt + φ), period T = 2π/ω, and frequency ν = 1/T. Velocity and acceleration in SHM are also periodic: v(t) = –ωA sin(ωt + φ) and a(t) = –ω²x(t). The chapter demonstrates that SHM connects to uniform circular motion as the projection on a diameter. Energy in SHM oscillates between kinetic K = ½mv² and potential U = ½kx², with total mechanical energy E = ½kA² remaining constant. The simple pendulum executes SHM for small angular displacements, with period T = 2π√(L/g).

1. 01Periodic motion repeats at regular intervals; oscillatory motion is to-and-fro about equilibrium with an equilibrium position where no net force acts
2. 02Simple harmonic motion (SHM) has displacement x(t) = A cos(ωt + φ) where amplitude A is maximum displacement, angular frequency ω = 2π/T, and phase constant φ determines initial conditions
3. 03Velocity v(t) = –ωA sin(ωt + φ) and acceleration a(t) = –ω²x(t) in SHM; both are periodic with period T/2 for velocity magnitude and T for displacement
4. 04Force in SHM is restoring: F = –kx = –mω² x, always directed toward equilibrium; k = mω² relates spring constant to mass and angular frequency
5. 05Energy in SHM conserves total mechanical E = ½kA² = K + U; kinetic energy peaks at equilibrium, potential energy peaks at maximum displacement
6. 06Simple pendulum for small angles executes SHM with period T = 2π√(L/g), independent of amplitude or mass