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📘 Oscillations: Classical Mechanics

Intermediate classical mechanics of oscillations from SHM to resonance

11
lessons
~30 min
to learn
Adults
level
Start the course →

What you’ll learn

  1. Defining Oscillatory MotionIdentify the conditions that produce periodic motion and distinguish oscillations from other trajectories.Periodic return to a neighborhood of an equilibrium position defines oscillation. A restoring force proportional to displacement initiates simple cases. Students quantify period, amplitude, and frequency from position-time data.
  2. Energy in Undamped OscillatorsApply mechanical energy conservation to relate amplitude, maximum speed, and total energy.Potential and kinetic terms exchange while their sum remains constant. Students derive maximum velocity from total energy and plot energy versus time.
  3. The Simple PendulumObtain the period of a simple pendulum under the small-angle limit and state its range of validity.Torque balance produces θ'' + (g/L)θ = 0 for small angles. Period equals 2π√(L/g). Students compare exact elliptic-integral periods with the approximation.
  4. Damped Harmonic MotionClassify underdamped, critically damped, and overdamped solutions and sketch their time series.The characteristic equation yields three root cases. Students match initial conditions to each regime and compute the quality factor Q.
  5. Driven OscillatorsWrite the particular solution for a driven damped oscillator and obtain its amplitude expression.The steady-state solution is x(t) = D cos(ωt − δ). Amplitude D peaks when driving frequency approaches natural frequency modified by damping.
  6. ResonanceLocate the resonance frequency for amplitude and for power, and calculate power at resonance.Velocity resonance occurs at ω = ω0. Average power is maximum when driving matches the damped natural frequency. Students evaluate bandwidth from the full-width at half-maximum.
  7. Coupled OscillatorsFormulate the equations for two coupled masses and find the normal frequencies and mode shapes.Symmetric and antisymmetric modes decouple the system. Students solve the eigenvalue problem and predict energy transfer periods.
  8. Normal Modes and BeatsDerive the beat frequency from superposition and connect it to normal-mode splitting.Sum of two cosines yields a carrier modulated by cos(Δω t / 2). Students measure beat frequency from recorded waveforms.
  9. Fourier Analysis of Periodic MotionExpress a periodic driving force as a Fourier series and predict which terms produce large amplitudes.Any periodic function decomposes into harmonic sinusoids. Students compute coefficients for common waveforms and filter by the resonance curve.
  10. Continuous Systems and WavesModel longitudinal or transverse waves on a continuous string or rod and obtain standing-wave frequencies.Wave equation solutions with fixed-end conditions produce discrete modes. Students calculate wave speed from material properties.
  11. Applications and Measurement LimitsConnect oscillator concepts to precision instruments and identify dominant noise sources.Resonant sensors, seismometers, and atomic clocks rely on controlled oscillations. Students estimate thermal noise via equipartition and discuss Q optimization.

Questions this course answers

Which of the following motions satisfies the conditions for oscillation?

Only the mass-spring system has both an equilibrium position and a restoring force that returns it repeatedly, meeting the definition of oscillatory motion.

Match each quantity to the feature it describes in an oscillation

Period measures repetition rate, amplitude measures excursion size, and the restoring force is the physical agent that forces repeated returns to equilibrium.

A 0.2 kg mass on a 200 N/m spring oscillates with amplitude 0.1 m. Estimate its maximum speed.

Total energy is ½kA² = 1 J. At equilibrium this equals ½mv_max², so v_max = √(2×1/0.2) ≈ 3.16 m/s.

A second identical mass is launched from the same amplitude but on a stiffer spring. How does its maximum speed compare with the original system?

Stiffer k raises ½kA², so total energy and therefore maximum kinetic energy both increase, producing a higher v_max.

A simple pendulum of length 0.25 m is released from 6°. Estimate its period under the small-angle approximation.

T = 2π √(L/g) = 2π √(0.25/9.81) ≈ 1.00 s exactly under the approximation.

A lab pendulum of length 0.50 m is released from 35°. Which statement best describes the measured period relative to 2π √(L/g)?

At 35° the elliptic-integral period exceeds the small-angle value by about 3–4 %, so the measured period is noticeably longer while remaining periodic.

Grounded in trusted sources

  • Massachusetts Institute of Technology
  • California Institute of Technology
  • National Institute of Standards and Technology

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