14.1 Simple Harmonic Motion

Simple harmonic motion (SHM) describes the back-and-forth movement of objects that oscillate around a central equilibrium point, like a swinging pendulum or a bouncing spring. Understanding SHM is the foundation for studying waves, vibrations, and energy transfer across all of physics.

Fundamentals of Simple Harmonic Motion

Characteristics of harmonic motion

Simple Harmonic Motion is periodic oscillation around an equilibrium position, where the restoring force is directly proportional to how far the object has been displaced. The further you pull it from equilibrium, the harder it gets pulled back.

This restoring force follows Hooke's Law:

$F = -kx$

where $k$ is the spring constant (a measure of stiffness) and $x$ is the displacement from equilibrium. The negative sign tells you the force always points back toward equilibrium, opposing the displacement.

Under ideal conditions (no friction or air resistance), SHM produces sinusoidal motion with constant frequency, constant period, and constant amplitude.

Motion of springs and pendulums

A mass on a spring is the classic SHM system. Whether the spring hangs vertically or stretches horizontally, the motion is governed by $F = -kx$. A stiffer spring (higher $k$) produces a stronger restoring force and faster oscillations.

A simple pendulum approximates SHM, but only for small angles (roughly less than about 15°). The "simple" part means you treat the bob as a point mass on a massless, inextensible string. At larger angles, the restoring force is no longer proportional to displacement, and the SHM approximation breaks down.

Both systems continuously convert between kinetic energy (maximum at equilibrium) and potential energy (maximum at the extremes of motion).

Measures of harmonic motion

Three quantities define any oscillation:

• Period ($T$) is the time for one complete back-and-forth cycle. For a mass-spring system: $T = 2\pi\sqrt{\frac{m}{k}}$. For a simple pendulum: $T = 2\pi\sqrt{\frac{L}{g}}$. Notice that the pendulum's period depends on string length $L$ and gravitational acceleration $g$, but not on mass. The spring's period depends on mass $m$ and spring constant $k$, but not on gravity (for horizontal oscillation).
• Frequency ($f$) is the number of oscillations per second, measured in hertz (Hz). It's the inverse of period: $f = \frac{1}{T}$.
• Amplitude ($A$) is the maximum displacement from equilibrium. In ideal SHM (no damping), amplitude stays constant over time. Amplitude does not affect period or frequency.

Relationships in harmonic motion

The position, velocity, and acceleration of an object in SHM are all sinusoidal functions of time, but they're shifted relative to each other.

• Displacement: $x = A\cos(\omega t)$, where $\omega = 2\pi f$ is the angular frequency (in rad/s)
• Velocity: $v = -A\omega\sin(\omega t)$
• Acceleration: $a = -A\omega^2\cos(\omega t)$

Each equation is the time derivative of the one above it. Velocity is the derivative of displacement; acceleration is the derivative of velocity.

The phase relationships are worth memorizing:

• Velocity leads displacement by 90° (a quarter cycle). When displacement is zero (object passes through equilibrium), velocity is at its maximum. When displacement is at its maximum (the extremes), velocity is zero.
• Acceleration is 180° out of phase with displacement. That means acceleration is greatest at the extremes of motion, pointing back toward equilibrium, and zero when the object passes through equilibrium.