A radio receiver circuit has a very high quality factor Q. Compared to a circuit with low Q tuned to the same frequency, how does it behave when exposed to multiple nearby radio signals?
AIt responds equally strongly to all frequencies near its resonance — the high Q makes it more sensitive overall
BIt has a broad, low resonance peak, providing equal amplification across a wide frequency band
CIt has a sharp, tall resonance peak, responding strongly to its target frequency and weakly to nearby frequencies — enabling selective tuning
DIt resonates at a much lower frequency than a low-Q circuit, because high Q reduces the resonance frequency
High Q means low damping: the oscillator rings for a long time when struck and has a narrow, tall resonance peak. For a radio receiver, this means strong response at the target frequency and rapid falloff at adjacent frequencies — exactly the selectivity needed to tune to one station while ignoring others. A low-Q circuit (heavily damped) would have a broad, flat peak, responding to many nearby frequencies with roughly equal strength — good for shock absorption but terrible for frequency selection. High Q = high selectivity = sharp tuning.
Question 2 Multiple Choice
A child is being pushed on a swing. Each push is timed to coincide with the swing reaching its maximum backward height. Over time, the swing's amplitude grows. Why does timing the pushes this way produce growth, while random timing does not?
APushes at the maximum height add maximum potential energy, which accumulates regardless of the swing's natural frequency
BRandom timing averages to zero net energy, while synchronized pushing consistently adds energy in phase with the natural motion, allowing amplitude to build
CThe swing's natural frequency changes as amplitude grows, and timed pushes track this changing frequency
DRandom pushes cancel the swing's motion by sometimes pushing against it, while timed pushes prevent any cancellation at all
Resonance is about consistent, in-phase energy transfer. When pushes are synchronized with the natural period (pushing when the swing moves away), each push adds a fixed amount of energy to the oscillation. Over many cycles, energy accumulates and amplitude grows. With random timing, pushes sometimes add energy (in phase) and sometimes remove it (out of phase) — these effects average out, leaving amplitude roughly constant. The natural frequency ω₀ is not the frequency where you push hardest; it is the frequency at which energy addition is consistently constructive rather than partially canceling.
Question 3 True / False
In an underdamped oscillator, the steady-state amplitude at the resonance peak increases as damping decreases.
TTrue
FFalse
Answer: True
This is the defining behavior of resonance in damped systems. The steady-state amplitude at driving frequency ω is proportional to F₀/m divided by |ω₀² − ω² + iγω|. At resonance (ω ≈ ω₀), the first term in the denominator vanishes and the amplitude is governed by the damping term γω₀. Smaller γ (less damping) means smaller denominator, which means larger amplitude. In the limit of zero damping, the amplitude at resonance grows without bound. Damping limits the resonance peak — it is both what keeps resonance from being infinite and what determines the Q factor.
Question 4 True / False
The resonance frequency of a damped oscillator is exactly equal to its natural frequency ω₀, regardless of the amount of damping present.
TTrue
FFalse
Answer: False
For a damped driven oscillator, the frequency of maximum steady-state amplitude — the resonance frequency — is not exactly ω₀, but ω_res = √(ω₀² − γ²/2), which is slightly below ω₀. Damping shifts the peak downward. For light damping (high Q), this shift is negligible, and ω_res ≈ ω₀ to excellent approximation. But as damping increases, the shift becomes significant and the peak also broadens and lowers. In the limit of very heavy damping, the resonance peak disappears entirely. The approximation ω_res ≈ ω₀ is useful but not exact.
Question 5 Short Answer
Explain why resonance can be both deliberately exploited and deliberately avoided in engineering, and give an example of each use case.
Think about your answer, then reveal below.
Model answer: Resonance is exploited when large-amplitude response to a small driving force is desired. In a radio receiver, a high-Q circuit resonates at one frequency to select that carrier signal from background noise. In musical instruments, air columns and strings are designed to resonate at specific frequencies to amplify sound. Resonance is avoided when large oscillations would be destructive. Bridges, buildings, and mechanical structures must have natural frequencies far from typical driving frequencies (traffic, wind, seismic activity) to prevent runaway oscillation — the Tacoma Narrows Bridge collapsed when wind drove oscillations near a structural resonance. The same principle (driving frequency matching natural frequency → large amplitude) is a tool when controlled and a hazard when not.
The key is that resonance itself is neither good nor bad — it is a physical phenomenon whose consequences depend on context. Whether you want to amplify a signal (radio), sustain a tone (instrument), or prevent catastrophic vibration (structure) determines whether resonance is the goal or the danger.