Oscillations
Motion repeats: a pendulum swings back and forth, a mass on a spring bounces up and down, a guitar string vibrates.
Start with the simplest version: this lesson is about Oscillations. If you can explain the core idea to a friend using everyday language, examples, and one clear reason why it matters, you have moved from memorising to understanding.
Motion repeats: a pendulum swings back and forth, a mass on a spring bounces up and down, a guitar string vibrates. Such oscillatory motion is ubiquitous in nature. Understanding it reveals patterns in mechanical systems, predicts behavior of springs, and explains why bridges collapse under certain vibrations. This chapter explores periodic and oscillatory motion, focusing on simple harmonic motion—one of physics' most elegant patterns.
Periodic and Oscillatory Motion
Periodic motion repeats itself at regular time intervals. The period (T) is the time for one complete cycle. The frequency (f) is cycles per unit time, measured in hertz (Hz):
f = 1/T
Not all periodic motion is oscillatory. A planet orbiting the sun is periodic (returns to the same position yearly) but not oscillatory (doesn't return to a central position, then leave again).
Oscillatory motion is periodic motion about a central equilibrium. A pendulum oscillates around the lowest point. A mass on a spring oscillates around the equilibrium position. Oscillatory systems have restoring forces pulling them back toward equilibrium.
Simple Harmonic Motion (SHM)
The most important oscillatory motion is simple harmonic motion (SHM). A particle undergoes SHM if the restoring force is proportional to displacement:
F = −kx
where k is the spring constant and x is displacement from equilibrium. The negative sign shows force opposes displacement.
From F = ma:
a = −(k/m)x = −ω²x
where ω = √(k/m) is the angular frequency.
This differential equation has solutions:
x(t) = A cos(ωt + φ)
where A is amplitude (maximum displacement) and φ is the phase constant.
The particle oscillates between −A and +A with period T = 2π/ω and frequency f = ω/(2π).
Characteristics of SHM
Amplitude (A): Maximum displacement. Larger amplitude means wider swings.
Period (T = 2π/ω): Time for one complete oscillation. For a mass-spring system: T = 2π√(m/k). Heavier masses oscillate more slowly (larger inertia); stiffer springs oscillate faster (stronger restoring force).
Velocity and acceleration vary with position:
- At maximum displacement (x = A): velocity = 0, acceleration = maximum (directed toward equilibrium)
- At equilibrium (x = 0): velocity = maximum, acceleration = 0
- The particle speeds up approaching equilibrium, slows down leaving it
Energy alternates between kinetic and potential:
- Maximum potential energy at maximum displacement (KE = 0)
- Maximum kinetic energy at equilibrium (PE = 0)
- Total mechanical energy E = (1/2)kA² is constant
This energy oscillation is crucial: springs and pendulums exchange kinetic and potential energy, converting motion to stretching and back.
The Pendulum: Gravity as a Restoring Force
A pendulum bob swinging from a string exhibits SHM for small angles. The restoring force comes from gravity's component along the arc:
F = −mg sin(θ) ≈ −mg(θ) (for small θ)
This is proportional to displacement (θ is proportional to arc length), making it SHM!
The period is:
T = 2π√(L/g)
where L is string length. The period is independent of amplitude (for small swings) and independent of mass. This is why Galileo could time a chandelier's swing using his pulse—its period didn't depend on how far it swung.
For large angles, the approximation sin(θ) ≈ θ breaks down, and the motion is no longer perfectly harmonic. The period increases slightly with amplitude.
Damped Oscillations
Real oscillations lose energy to friction and air resistance—they damp. The amplitude decreases exponentially:
x(t) = A₀e^(−t/τ) cos(ωt + φ)
where τ is the time constant—time for amplitude to decrease by a factor of e.
Three regimes exist:
- Underdamped (light damping): Oscillates many times before stopping
- Critically damped: Returns to equilibrium fastest without oscillating
- Overdamped (heavy damping): Slowly returns to equilibrium, oscillating slowly or not at all
Car shock absorbers are nearly critically damped—they stop bouncing without excessive oscillation. If underdamped, the car would bounce for a long time. If overdamped, it would slowly settle.
Friction converts mechanical energy to heat, explaining why oscillations stop.
Forced Oscillations and Resonance
Apply a periodic driving force to an oscillating system, and it oscillates at the driving frequency—forced oscillation. The amplitude depends on whether the driving frequency matches the system's natural frequency.
Resonance occurs when the driving frequency equals the natural frequency. The amplitude reaches a maximum and can be extremely large even with modest driving force.
When glasses shatter from loud sound, resonance is at work. A precisely timed frequency excites the natural vibration mode, and amplitude grows until the stress exceeds material strength.
This principle is dangerous for bridges and buildings. Wind or earthquakes might excite resonant frequencies. The Tacoma Narrows Bridge collapse (1940) occurred because wind-induced vibrations matched the bridge's natural frequency, growing until structural failure.
Damping and Resonance
Damping reduces resonant amplitude. A heavily damped system has a lower maximum amplitude at resonance, but the resonant peak is broader (occurs over a wider frequency range).
Underdamped systems have sharp, high resonant peaks. This is useful for radio tuning—the tuned circuit resonates sharply at the desired frequency. But structures must be damped to avoid dangerous resonances.
Coupled Oscillations and Normal Modes
Two coupled pendulums exchange energy. If one is set swinging, it gradually transfers energy to the other. This occurs through a normal mode—a pattern of coordinated motion.
Complex systems (like molecules with multiple bonds) have multiple normal modes. Each mode vibrates at its own frequency. The actual motion is a superposition of all modes.
This is why molecules absorb infrared light at specific frequencies—the frequencies match molecular vibration modes. Different bonds (C=C, C-H, O-H) have different vibrational frequencies, producing characteristic spectroscopic signatures.
Socratic Questions
- Why does a mass on a spring always undergo simple harmonic motion, while a pendulum only does so for small angles? What assumption breaks down for large angles?
- The period of a pendulum is T = 2π√(L/g), independent of mass. Explain physically why a heavier pendulum bob swings with the same period.
- A damped oscillator loses energy to friction. Where does this energy go, and why does the amplitude decrease exponentially rather than linearly?
- At resonance, the amplitude of a driven oscillation can be very large. Yet, if the driving force is weak, why does the amplitude become large? What determines the maximum amplitude at resonance?
- A building's natural vibration frequency can be determined by dropping a heavy object and analyzing oscillations. Why would knowing this frequency matter to a building engineer?