Simple harmonic motion is all about back-and-forth movement. Think of a swinging pendulum or a bouncing spring - they oscillate around a central point, following a predictable pattern. This type of motion is everywhere, from vibrating guitar strings to earthquake waves.
Understanding simple harmonic motion helps us grasp how energy moves in waves and vibrations. We'll explore key concepts like frequency, period, and amplitude, and see how they relate to forces and energy in oscillating systems.
Fundamentals of Simple Harmonic Motion
Characteristics of harmonic motion
- Simple Harmonic Motion (SHM) describes periodic oscillation about equilibrium position with restoring force proportional to displacement
- Sinusoidal motion exhibits constant frequency and period while amplitude remains constant under ideal conditions (no friction)
- Restoring force always directed towards equilibrium position follows Hooke's Law $F = -kx$ (k is spring constant, x is displacement)
Motion of springs and pendulums
- Mass on spring oscillates horizontally or vertically governed by $F = -kx$ where spring constant determines stiffness
- Simple pendulum approximates SHM for small angles using point mass suspended by weightless, inextensible string
- Both systems demonstrate oscillatory motion converting between potential and kinetic energy
Measures of harmonic motion
- Period (T) measures time for one complete oscillation (mass-spring: $T = 2\pi\sqrt{\frac{m}{k}}$, pendulum: $T = 2\pi\sqrt{\frac{L}{g}}$)
- Frequency (f) counts oscillations per unit time, inverse of period $f = \frac{1}{T}$
- Amplitude (A) represents maximum displacement from equilibrium position, constant in ideal SHM
Relationships in harmonic motion
- Displacement (x) varies sinusoidally with time $x = A\cos(\omega t)$, $\omega = 2\pi f$ (angular frequency)
- Velocity (v) derived from displacement $v = -A\omega\sin(\omega t)$, reaches maximum at equilibrium
- Acceleration (a) derived from velocity $a = -A\omega^2\cos(\omega t)$, peaks at extreme positions
- Phase relationships show velocity 90° out of phase with displacement, acceleration 180° out of phase