Periodic Motion

Periodic motion bridges mechanics, waves, sound, and physiology: your heart beats rhythmically, sound waves carry information to your cochlea, and ultrasound imaging relies on wave reflection. The MCAT tests it heavily because it ties these together.

Priority labels: Must know = cold; Know the logic = mechanism not names; Passage-level = recognize, don't memorize; Optional = skippable.

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Simple Harmonic Motion: The Foundation

Must know

Amplitude, frequency, and phase are all properties of simple harmonic motion (SHM), so build that model first.

Picture a mass on a spring on a frictionless surface. Pull it right and release: it accelerates back toward the equilibrium position, overshoots left, turns around, and repeats. Whenever a restoring force always points back toward equilibrium, you get SHM.

Hooke's Law and the Restoring Force

Must know

The restoring force follows Hooke's Law:

$F = -kx$

where $k$ is the spring constant (stiffness, N/m) and $x$ is displacement from equilibrium. The minus sign means force opposes displacement; larger $k$ = stiffer spring = stronger snap-back. Any system whose net force is proportional to displacement and opposes it executes SHM — the spring-mass system and, for small angles, the simple pendulum.

Period and Frequency of Common SHM Systems

Must know

The period $T$ is the time for one complete oscillation.

For a spring-mass system:

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

For a simple pendulum (small angle, $\theta < 15°$):

$T = 2\pi\sqrt{\frac{L}{g}}$

where $L$ is pendulum length and $g \approx 10 \text{ m/s}^2$.

The spring period depends on mass and stiffness but not amplitude; the pendulum period depends on length and gravity but not mass or amplitude. These independence relationships are favorite MCAT traps.

Energy in SHM

Must know

KE and PE trade off continuously, but their sum—total mechanical energy—stays constant (no friction):

$E_{total} = \frac{1}{2}kA^2$

where $A$ is amplitude, so total energy is set by amplitude. At equilibrium, all energy is kinetic (maximum speed); at the turning points ($x = \pm A$), all energy is potential and the mass is momentarily at rest.

Velocity and Acceleration in SHM

Must know

For $x(t) = A\cos(\omega t)$, the maxima are $v_{\max} = A\omega$ and $a_{\max} = A\omega^2$. The phase relationships the MCAT tests:

• Velocity is maximum at equilibrium ($x = 0$) and zero at the turning points.
• Acceleration is maximum at the turning points and zero at equilibrium — since $a = -\omega^2 x$, $|a|$ is largest where $|x|$ is largest.

So velocity and acceleration are exactly out of step: where one peaks, the other vanishes.

Quick check: A spring-mass system oscillates with amplitude 0.2 m and spring constant 50 N/m. What is its total mechanical energy?

$E = \frac{1}{2}kA^2 = \frac{1}{2}(50)(0.2)^2 = 1.0 \text{ J}$

Note: If amplitude doubles, energy quadruples—a quadratic relationship to remember.

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Amplitude, Frequency, and Phase

Amplitude

Must know

Amplitude ($A$) is the maximum displacement from equilibrium (always positive). It does not affect period or frequency in ideal SHM, but it does set total energy ($E \propto A^2$).

Passage-level

For a sound wave, amplitude sets loudness (pressure variation); for light, it sets intensity (brightness).

Frequency and Period

Must know

Frequency ($f$) is oscillations per second, in hertz ($1\text{ Hz} = 1\text{ s}^{-1}$). Period and frequency are reciprocals:

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

Frequency is set by the source and does not change when a wave moves into a new medium—only wavelength and speed change.

Angular frequency $\omega$ (rad/s) is how fast the phase rotates around a circle:

$\omega = 2\pi f$

Phase

Must know

At $t = 0$ a mass on a spring could start at the turning point, or at equilibrium moving rightward, or leftward—same oscillation, just shifted in time. Phase quantifies that starting offset:

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

where $\phi$ is the initial phase. If $\phi = 0$, the oscillator starts at $x = A$; if $\phi = -\pi/2$, it starts at $x = 0$ moving positive.

Phase difference drives interference: waves in phase ($\Delta\phi = 0$ or $2\pi$) add constructively; out of phase ($\Delta\phi = \pi$, i.e. 180°) they cancel destructively.

Quick check: A mass on a spring has period $T = 0.5$ s. What is its frequency? Its angular frequency?

$f = 1/T = 2$ Hz. $\omega = 2\pi f = 4\pi \approx 12.6$ rad/s.

Biological relevance: A resting heart rate of 75 bpm ≈ 1.25 Hz.

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Transverse and Longitudinal Waves

Must know

A wave transfers energy through a medium (or vacuum, for EM waves) without permanently transporting matter. The key question: in what direction does the medium oscillate relative to the wave's travel?

Transverse Waves

Must know

In a transverse wave, the medium oscillates perpendicular to propagation (shake a rope up and down while the wave travels horizontally). Examples: light and all EM waves, waves on a string. Because displacement is perpendicular to travel, transverse waves can show polarization—EM waves are polarizable, sound is not.

Longitudinal Waves

Must know

In a longitudinal wave, the medium oscillates parallel to propagation (a pushed-and-pulled Slinky). The key example is sound (including ultrasound), where compressions (higher pressure) and rarefactions (lower pressure) alternate. The eardrum and cochlear hair cells respond to these pressure oscillations.

Quick check: Is an ultrasound pulse used in echocardiography a transverse or longitudinal wave?

Longitudinal. Sound is always longitudinal—tissue oscillates parallel to the travel direction. Light (used in ophthalmoscopy) would be transverse.

Wavelength

Must know

Wavelength ($\lambda$) is the distance between two consecutive in-phase points—crest to crest, or compression to compression (units: meters).

Shorter wavelength → more cycles per unit length → higher frequency (at fixed speed), and vice versa.

Wave Speed and the Fundamental Wave Equation

Must know

The speed at which the wave pattern travels is the propagation speed ($v$):

$v = f\lambda$

This is the single most tested wave equation—know it cold ($v$ in m/s, $f$ in Hz, $\lambda$ in m).

Wave speed depends on the medium, not the source. When a wave enters a new medium (different $v$), its frequency stays constant and its wavelength changes:

$\lambda = \frac{v}{f}$

This is why light refracts entering glass: $v$ and $\lambda$ decrease, $f$ stays the same.

Know the logic

Stiffer/more elastic media transmit mechanical waves faster (sound: solids > liquids > gases). Quantitative speed-vs-medium and $v = c/n$ for light are developed in the Sound and Light guides; here the takeaway is that crossing media changes speed, fixes frequency, and adjusts wavelength via $\lambda = v/f$.

Quick check: A 440 Hz sound (speed in air 343 m/s) has $\lambda_{air} = 343/440 \approx 0.78$ m. Entering water ($v = 1500$ m/s), frequency is unchanged (set by the source), so $\lambda_{water} = 1500/440 \approx 3.4$ m—scaled up by the same ratio as the speed. ✓

Quick check: If you double the frequency of a sound wave at constant wave speed, what happens to wavelength?

Since $\lambda = v/f$ with $v$ constant, doubling $f$ halves $\lambda$—inversely proportional at fixed speed.

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Damping and Resonance

Must know

A real oscillator loses energy to friction or drag, so its amplitude decays over time—damped harmonic motion (a pendulum coming to rest, a car's shock absorbers).

Every system has a natural (resonant) frequency $f_0$ at which it "wants" to oscillate ($f_0 = \frac{1}{2\pi}\sqrt{k/m}$ for a spring-mass system). Driving it with a periodic force produces a small response unless the driving frequency matches $f_0$—then energy transfers most efficiently and amplitude peaks. This is resonance.

Know the logic

Resonance is high-yield and biological—the cochlea's basilar membrane resonates at different positions to sort sound by frequency; MRI is nuclear magnetic resonance; a singer can shatter a glass at its natural frequency. Damping sets the peak: heavy damping gives a broad, low resonance; light damping a sharp, tall one.

Passage-level

Standing waves (nodes, antinodes, harmonics), beats, superposition/interference, sound properties (pitch, intensity, decibels, Doppler), and the EM spectrum are developed in the Sound and Light guides. This unit's job is the underlying periodic-motion and general-wave framework.

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Common Confusions & Tricks

1. Period of a pendulum depends on $L$, not mass. A heavier pendulum does not swing faster—$T = 2\pi\sqrt{L/g}$ has no mass term. Likewise the spring-mass period depends on $m/k$, not amplitude.

2. Frequency does not change when a wave changes medium. Only $v$ and $\lambda$ change. If sound enters bone from tissue, frequency stays the same and wavelength shortens because speed drops.

3. Energy and amplitude. Total energy scales as $A^2$, not $A$: double the amplitude → four times the energy ($E = \tfrac{1}{2}kA^2$).

4. Longitudinal waves DO have a wavelength. It's the distance between successive compressions—wavelength isn't exclusive to the peak-to-trough transverse picture.

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Key Equations

Equation	Variables & When to Use
$F = -kx$	Hooke's Law: $k$ = spring constant (N/m), $x$ = displacement. Restoring force in SHM.
$T = \dfrac{1}{f}$	Period ($T$, s) and frequency ($f$, Hz) are reciprocals.
$\omega = 2\pi f$	Angular frequency ($\omega$, rad/s). Used when SHM is written $x = A\cos(\omega t + \phi)$.
$T_{spring} = 2\pi\sqrt{\dfrac{m}{k}}$	Period of spring-mass; independent of amplitude.
$T_{pendulum} = 2\pi\sqrt{\dfrac{L}{g}}$	Period of simple pendulum (small angles); independent of mass and amplitude.
$E = \dfrac{1}{2}kA^2$	Total mechanical energy in SHM; scales as amplitude squared.
$v = f\lambda$	The fundamental wave equation. $v$ = speed (m/s), $f$ = frequency (Hz), $\lambda$ = wavelength (m).
$\lambda = \dfrac{v}{f}$	Rearranged wave equation; wavelength changes when a wave enters a new medium while $f$ stays fixed.