16.3 Simple Harmonic Motion: A Special Periodic Motion

Simple harmonic motion (SHM) describes the back-and-forth movement of an object around a central equilibrium point, driven by a force that always pulls it back toward that center. Understanding SHM is essential because it forms the foundation for analyzing waves, sound, and many mechanical systems you'll encounter throughout physics.

Simple Harmonic Motion

Characteristics of harmonic motion

Simple harmonic motion occurs when an object oscillates about an equilibrium position and experiences a restoring force that is directly proportional to its displacement but acts in the opposite direction. This restoring force is what makes the motion repeat in a predictable pattern.

The restoring force is expressed as:

$F = -kx$

where $k$ is the force constant (also called the spring constant) and $x$ is the displacement from equilibrium. The negative sign tells you the force always points back toward the equilibrium position. Common examples of harmonic oscillators include a mass on a spring, a pendulum swinging through small angles, and a vibrating guitar string.

Three quantities describe any SHM system:

• Amplitude ($A$): the maximum displacement from equilibrium, measured in meters. If a mass on a spring stretches 0.15 m from its rest position before reversing, the amplitude is 0.15 m.
• Period ($T$): the time for one complete back-and-forth cycle, measured in seconds. A playground swing that takes 3 s to go out and come back has a period of 3 s.
• Frequency ($f$): the number of complete oscillations per second, measured in hertz (Hz). Frequency and period are inverses of each other:

$f = \frac{1}{T}$

A tuning fork vibrating at 440 Hz completes 440 full cycles every second, giving it a period of about 0.00227 s.

Period factors in oscillators

For a mass-spring system, the period depends on two things: the mass ($m$) attached to the spring and the spring's force constant ($k$):

$T = 2\pi\sqrt{\frac{m}{k}}$

Notice what this equation tells you:

• Mass and period have a direct relationship. A larger mass is harder to accelerate, so it oscillates more slowly and the period increases. Doubling the mass doesn't double the period, though, because of the square root. You'd need to quadruple the mass to double the period.
• Force constant and period have an inverse relationship. A stiffer spring (larger $k$) pulls the mass back more forcefully, producing faster oscillations and a shorter period. A spring with $k = 200$ N/m will oscillate faster than one with $k = 50$ N/m, assuming the same mass.

One important detail: the amplitude does not appear in this equation. The period of a mass-spring system is the same whether the object oscillates with a large amplitude or a small one.

Equations for harmonic motion

The position, velocity, and acceleration of an object in SHM can each be written as functions of time.

Position:

$x(t) = A\cos(\omega t + \phi)$

• $A$ is the amplitude
• $\omega$ is the angular frequency, which relates to period and frequency by $\omega = \frac{2\pi}{T} = 2\pi f$
• $\phi$ is the phase constant, set by the initial conditions (where the object starts and how fast it's moving at $t = 0$)

For example, a mass on a spring with amplitude 0.1 m and period 2 s, released from maximum displacement at $t = 0$, has $\omega = \frac{2\pi}{2} = \pi$ rad/s and $\phi = 0$, giving $x(t) = 0.1\cos(\pi t)$.

Velocity:

$v(t) = -A\omega\sin(\omega t + \phi)$

• Velocity is greatest (in magnitude) as the object passes through equilibrium, where displacement is zero.
• Velocity is zero at the extremes of motion, where the object momentarily stops before reversing direction.

Acceleration:

$a(t) = -A\omega^2\cos(\omega t + \phi)$

• Acceleration is greatest at the extremes of motion, where the restoring force is strongest.
• Acceleration is zero at the equilibrium position, where displacement is zero.

Notice the pattern: acceleration is proportional to displacement but opposite in sign. This is just Newton's second law ($F = ma$) combined with $F = -kx$, which gives $a = -\frac{k}{m}x$.

Energy in Simple Harmonic Motion

The total mechanical energy of an SHM system stays constant (assuming no friction), but it continuously converts between two forms:

• Kinetic energy ($\frac{1}{2}mv^2$) is at its maximum when the object passes through equilibrium, where speed is greatest.
• Elastic potential energy ($\frac{1}{2}kx^2$) is at its maximum at the extremes of motion, where displacement is greatest and the object is momentarily at rest.

At every point in the cycle, the sum of these two equals the total energy: $E_{total} = \frac{1}{2}kA^2$. This total depends only on the amplitude and the spring constant.

Two additional concepts to know:

• Damping occurs when friction or air resistance gradually removes energy from the system, causing the amplitude to shrink over time. A pendulum that eventually comes to rest is experiencing damping.
• Resonance happens when a periodic external force drives the system at its natural frequency. The amplitude grows dramatically because energy is added in sync with the oscillation. Pushing a swing at just the right rhythm is a familiar example of resonance.

Connection to Wave Motion

Simple harmonic motion is the building block of wave behavior. A wave traveling through a medium can be thought of as many particles, each undergoing SHM, with each particle slightly out of phase with its neighbor. The same quantities you've learned here (amplitude, frequency, period) carry directly into the study of waves.