Classical mechanicsLesson 5 of 6 in the course Oscillations

Damped Spring–Mass Oscillators: Motion, Frequency, and Damping Regimes

5 flashcards · 5 quiz questions · 5 match pairs · 3 fill-in sentences · English

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Summary

Real oscillators lose energy through friction and other nonconservative forces. This lesson develops a mathematical model for that energy loss and explains how it changes the motion of a mass on a spring.

An ideal simple harmonic oscillator (SHO) continues oscillating with constant amplitude because no energy leaves the system. Real systems are different: a plucked guitar string eventually stops, and a playground swing needs repeated pushes to maintain its motion. Friction, air resistance, and fluid viscosity transfer mechanical energy into other forms, commonly thermal energy.

All flashcards

What is damping?
Damping is energy loss caused by nonconservative forces. It makes the amplitude of an oscillator decrease over time.
What does FD=−bvF_D=-bv mean?
The damping force is proportional to velocity, with proportionality constant bb, and acts opposite the direction of motion.
What is the equation of motion for a damped spring–mass system?
It is md2xdt2+bdxdt+kx=0m\frac{d^2x}{dt^2}+b\frac{dx}{dt}+kx=0, where the terms describe inertia, damping, and the spring force.
How does the amplitude change in underdamped motion?
It decreases exponentially according to A(t)=A0e−bt/(2m)A(t)=A_0e^{-bt/(2m)}.
How are damped and natural angular frequency related?
The natural frequency is ω0=k/m\omega_0=\sqrt{k/m}. For an underdamped system, ω=ω02−(b/2m)2\omega=\sqrt{\omega_0^2-(b/2m)^2}, so damping lowers the oscillation frequency.

Quiz questions

  1. For a spring–mass oscillator with mass mm and spring constant kk, what is the critical damping constant?Answer: bc=2mkb_c=2\sqrt{mk}
  2. A spring–mass oscillator has damping constant bb below its critical value bcb_c. Which motion does it exhibit?Answer: It oscillates while its amplitude decreases.
  3. What distinguishes a critically damped spring–mass system?Answer: It returns to equilibrium fastest without oscillating.
  4. How does an overdamped spring–mass system return to equilibrium compared with a critically damped one?Answer: It returns more slowly and does not oscillate.
  5. Why are vehicle shock absorbers designed to provide damping near the critical level?Answer: To settle the suspension promptly without repeated bouncing.

Based on University Physics Volume 1 by William Moebs, Samuel J. Ling, Jeff Sanny (OpenStax), licensed CC BY 4.0. Adapted into a summary, flashcards and quizzes.

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