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

Mathematics · Differential Equations · ages 20-21
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  1. A lightly damped oscillator is driven at various frequencies. When is the steady response largest?

    • When the driving frequency is zero
    • When the driving frequency is twice the natural frequency
    • When the driving frequency is close to the natural frequency
    • When the driving frequency is half the natural frequency

  2. Characteristic roots are minus 2 and minus 2. What is the motion?

    • Overdamped
    • Underdamped
    • Undamped
    • Critically damped

  3. Characteristic roots are minus 1 and minus 4, both real and negative. What is the motion?

    • Overdamped
    • Underdamped
    • Critically damped

  4. Which damping returns to rest fastest without overshooting?

    • Lightly underdamped
    • Heavily overdamped
    • Critically damped

  5. A driven solution is y = 5e^(-t) cos 2t + 3 sin 4t. Which part is the steady state?

    • 5e^(-t) cos 2t
    • 3 sin 4t
    • The whole expression
    • Neither part

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

    • Solve once, read the answer as motion or charge
    • Springs and circuits always behave in opposite ways
    • Only the spring version can ever resonate

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

    • The transient, which keeps growing without end
    • The steady state at the driving frequency
    • Neither part, since friction stops everything

  8. Characteristic roots are 3i and minus 3i. What is the motion?

    • Underdamped
    • Undamped
    • Overdamped
    • Critically damped

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

    • The motion dies out at once because the pushes cancel
    • Nothing special happens at that frequency
    • Resonance: a huge steady swing, curbed by damping

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

    • The fading wobble is the transient
    • Transients never show up in real measured data
    • The steady state always dies away before the transient
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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 is close to the natural frequency · Resonance: the steady amplitude peaks when the driver is close to the natural frequency, capped only by damping.
  2. Critically damped · A repeated real root is the boundary case: critically damped.
  3. Overdamped · Two distinct real roots mean heavy damping with no oscillation.
  4. Critically damped · Critical damping sits exactly between swing and creep, the fastest quiet return.
  5. 3 sin 4t · 5e^(-t) cos 2t dies away with the exponential, so it is the transient. 3 sin 4t persists, so it is the steady state.
  6. Solve once, read the answer as motion or charge · Mass, damping and stiffness trade roles with inductance, resistance and one over the capacitance, and displacement trades with charge.
  7. The steady state at the driving frequency · Damping kills the transient, leaving only the persistent forced wave.
  8. Undamped · Purely imaginary roots give cos 3t and sin 3t with no decaying factor: a pure sine wave that never fades.
  9. Resonance: a huge steady swing, curbed by damping · Matched rhythms stack push on push, and only light damping curbs the growth.
  10. The fading wobble is the transient · The start shows transient plus steady together, and only the transient fades.
Worksheet · LightMySky