Damped and Driven Oscillations · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Swings that fade and swings that build

Mathematics · Differential Equations · ages 20-21
Name ______________________   Date ____________

  1. A lightly damped oscillator is driven at various frequencies. When is the steady response largest?

    • When the driving frequency equals the natural frequency
    • When the driving frequency is zero
    • When the driving frequency is twice the natural frequency
    • When the driving frequency is half the natural frequency

  2. Characteristic roots are minus 2 and minus 3, both real and negative. What is the motion?

    • Overdamped
    • Underdamped
    • Critically damped
    • Undamped

  3. A driven solution is the sum of a transient and a steady state. Which part is still moving long after the start?

    • The steady state at the driving frequency
    • The transient at the natural frequency
    • Both die out together
    • Neither survives friction

  4. Characteristic roots are minus 1 plus 2 i and minus 1 minus 2 i. What is the motion?

    • Overdamped
    • Underdamped
    • Critically damped

  5. Which damping returns to rest fastest without overshooting?

    • Underdamped
    • Overdamped
    • Critically damped

  6. The steady state oscillates at the driving frequency, not the natural one.

    Circle one:   True   False

  7. A mass on a spring and a series circuit share one equation. What does that buy you?

    • Results transfer: solve once, read both as motion or charge
    • Springs and circuits behave oppositely
    • Only the spring can resonate

  8. In a driven answer, which part survives long after the start?

    • The transient, which grows forever
    • The steady state at the driving frequency
    • Neither part survives

  9. A lightly damped system is driven exactly at its natural frequency. What happens and why?

    • The motion dies out instantly
    • Nothing special happens at that frequency
    • Resonance builds a huge steady arc, limited only by damping

  10. A student watches early motion and calls the dying wobble the steady state. What is the error?

    • Early on both parts mix; the wobble that fades is the transient
    • Transients never appear in real data
    • The steady state always dies first
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Swings that fade and swings that build W1-mt_Y8K1njb1PH-s1

  1. When the driving frequency equals the natural frequency · Matching frequencies is resonance, where each push lands in step with the motion.
  2. Overdamped · Two distinct real roots mean heavy damping with no oscillation.
  3. The steady state at the driving frequency · Damping kills the transient, while the driver keeps feeding the steady state.
  4. Underdamped · A complex pair means oscillation, and the negative real part means it decays.
  5. Critically damped · Critical damping sits exactly between swing and creep, the fastest quiet return.
  6. True · The persistent wave follows the driver beating it, not the old natural song.
  7. Results transfer: solve once, read both as motion or charge · Mass, damping, and stiffness trade roles with inductance, resistance, and capacitance.
  8. The steady state at the driving frequency · Damping kills the transient, leaving only the persistent forced wave.
  9. Resonance builds a huge steady arc, limited only by damping · Matched rhythms stack push on push, and only light damping curbs the growth.
  10. Early on both parts mix; the wobble that fades is the transient · The start shows transient plus steady together, and only the transient fades.
Worksheet · LightMySky