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

Three ways a swing can die

Science · Forces & Motion · ages 19-20
Name ______________________   Date ____________

  1. A door closer has damping coefficient b = 15 kg/s. For its mass and spring, 2*sqrt(m*k) = 10 kg/s. This door closer is overdamped.

    Circle one:   True   False

  2. A mass on a spring follows m x'' + b x' + k x = 0, with m = 2 kg, k = 8 N/m, and b = 8 kg/s. The critical damping value is b_c = 2*sqrt(m*k). Which regime is this system in?

    • Underdamped
    • Critically damped
    • Overdamped
    • Undamped

  3. A mass on a spring follows m x double prime + b x prime + k x = 0, with m = 2, k = 8, b = 8. Here b_c = 8. Which regime is it in?

    • Underdamped
    • Critically damped
    • Overdamped

  4. A door closer has b = 15. For its mass and spring, 2 root(m k) = 10. This door closer is overdamped.

    Circle one:   True   False

  5. Rank these three door closer regimes from fastest to slowest return to the closed position: underdamped, critically damped, overdamped.

    • Critically damped, then overdamped, then underdamped
    • Underdamped, then critically damped, then overdamped
    • Overdamped, then critically damped, then underdamped
    • All three return at the same speed

  6. An overdamped door closer reaches the closed position faster than a critically damped one.

    Circle one:   True   False

  7. A spring-mass-damper has m = 2 kg and k = 18 N/m. What value of b (in kg/s) puts the system exactly at critical damping?

    Answer: ______________

  8. An overdamped door closer reaches closed faster than a critically damped one.

    Circle one:   True   False

  9. A system has omega_n = 10 and zeta = 0.6. Find omega_d = omega_n root(1 minus zeta squared).

    Answer: ______________

  10. A system has natural frequency omega_n = 20 rad/s and damping ratio zeta = 0.6. Find the damped oscillation frequency omega_d, in rad/s.

    Answer: ______________

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Answer key

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

Three ways a swing can die W1-mt_JCpYpFVn6s-s1

  1. True · b (15) is bigger than the critical value (10), so the damping term wins and the door never swings past closed, just slower than it could.
  2. Critically damped · b matches b_c exactly here, so the system sits right at the boundary: critically damped.
  3. Critically damped · b matches b_c exactly, so the system sits on the boundary.
  4. True · b sits above the critical value, so damping wins.
  5. Critically damped, then overdamped, then underdamped · Critically damped wins the race to rest. Overdamped comes next since the extra drag only costs a little time. Underdamped often swings back and forth past the resting point several times before it truly settles, which costs the most time.
  6. False · Critically damped is the fastest way back to rest without crossing over. Extra damping beyond that just adds drag and slows things down.
  7. 12 · Critical damping happens at the exact boundary b_c = 2*sqrt(m*k).
  8. False · Critical is the fastest way home without crossing; extra damping drags.
  9. 8 · Root(0.64) = 0.8, and 10 times 0.8 is 8.
  10. 16 · Same formula as before, just with a bigger natural frequency, so the answer scales up too.
Worksheet · LightMySky