Pushing at the favourite frequency

Science · Forces & Motion · ages 19-20

Name ______________________ Date ____________

For a fixed-amplitude sinusoidal force, what displacement response does light velocity-dependent damping produce?
- A flat response with no change across frequencies
- A tall, narrow peak near the natural frequency
- A broad, low peak with rapid decay of free motion
A damped linear oscillator has an undamped natural frequency of 5 Hz and is driven sinusoidally at 3 Hz. After transients fade, what is its oscillation frequency?
- 3 Hz
- 5 Hz
- 8 Hz
For a linear oscillator with velocity-dependent damping, what does a high quality factor indicate?
- A broad peak and rapidly decaying free motion
- A narrow peak and rapidly decaying free motion
- A narrow peak and slowly decaying free motion
Which pair names a case where resonance is wanted and a case where resonant vibration has to be suppressed?
- An instrument strengthens a note; a bridge suppresses vibration
- An instrument suppresses a note; a bridge strengthens vibration
- A radio rejects its selected station; an engine strengthens vibration
A linear oscillator has light positive velocity-dependent damping. With driving-force amplitude fixed, how does the frequency giving the largest steady displacement compare with the undamped natural frequency?
- Far above the undamped natural frequency
- Exactly at the undamped natural frequency
- Slightly below the undamped natural frequency
For fixed mass and stiffness, tripling a positive velocity-dependent damping coefficient divides the quality factor by three.
Circle one: True False
Resonance needs zero damping to appear at all.
Circle one: True False
A lightly damped linear model of a footbridge is driven near its displacement-amplitude peak. Which change reduces steady displacement if the driving frequency stays fixed?
- Increase the periodic force while keeping damping unchanged
- Increase the damping while keeping the force unchanged
- Decrease the damping while keeping the force unchanged

Answer key

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

Pushing at the favourite frequency W1-mt_FmpOTEZFy4-s1

A tall, narrow peak near the natural frequency · Light damping removes little energy, so the displacement peak is tall and narrow. Its frequency is slightly below the undamped natural frequency.
3 Hz · After transients fade, the steady oscillation follows the 3 Hz driving force. The natural frequency does not replace or add to the driving frequency.
A narrow peak and slowly decaying free motion · High Q indicates weak damping. Weak energy loss gives a narrow driven response and a long free ring-down.
An instrument strengthens a note; a bridge suppresses vibration · "An instrument strengthens a note; a bridge suppresses vibration" matches both conditions. Resonance strengthens selected musical notes; large bridge vibration can cause stress or discomfort.
Slightly below the undamped natural frequency · For light positive velocity-dependent damping, the displacement maximum lies slightly below the undamped natural frequency. Exact equality is only an approximation.
True · Q = m omega_n/b. Fixed mass and stiffness keep the numerator unchanged. Replacing b with 3b gives Q_new = Q/3, so the statement is true.
False · A displacement-amplitude peak exists with small positive damping. Zero damping is not required; sufficiently strong damping can remove this peak.
Increase the damping while keeping the force unchanged · At the same nonzero driving frequency and force amplitude, increasing damping reduces steady displacement in this linear model. A real bridge requires safety instructions and engineering assessment.