Resonance and Forced Vibrations Quiz
This quiz tests Lesson 5: Resonance and Forced Vibrations from the course Sound Waves: Principles, Superposition, and Applications for IIT JEE. It is designed for IIT JEE aspirants at an intermediate level.
You will be examined on:
- The definition of resonance and the condition for maximum amplitude
- Natural frequency versus driving frequency in the steady state
- Resonance in air columns, strings, and the resonance tube experiment
- Sharpness of resonance and the quality factor ##Q##
- Damped and undamped oscillations, and the phase relationship at resonance
- Everyday and engineering examples, including the Tacoma Narrows bridge collapse
- The electrical analogy of resonance and energy transfer
All questions are answerable from the lesson material alone. Use the explanations after each question to consolidate your understanding of the forced harmonic oscillator and its applications.
Q1. Resonance is best defined as the condition in which a driven oscillatory system responds with maximum amplitude because:
Resonance occurs when the driving frequency approaches a natural frequency of the system, producing maximum steady-state amplitude. A large driving force or large damping does not define resonance, and release from rest describes a free oscillation, not a driven one.
Q2. For a damped harmonic oscillator of mass ##m## and stiffness ##k##, the natural angular frequency is given by:
The lesson states that the natural angular frequency is ##\omega_0 = \sqrt{k/m}##. The reciprocal form ##\sqrt{m/k}## would correspond to a time period scale, not the angular frequency.
Q3. The damping ratio of the oscillator model is defined as:
The lesson defines the damping ratio as ##\zeta = b/(2\sqrt{km})##, where ##b## is the damping constant. This dimensionless ratio classifies the system as underdamped, critically damped, or overdamped.
Q4. Acoustic resonators of practical interest, such as air columns and strings, are almost always:
The lesson states that acoustic resonators of practical interest are almost always underdamped. This is why they can sustain vibration and show a pronounced resonant peak.
Q5. At ##\omega = \omega_0##, the steady-state displacement amplitude of the driven oscillator reduces to:
At resonance the term ##(\omega_0^2-\omega^2)^2## vanishes, leaving ##A = F_0/(2\zeta k)##. This grows without bound as the damping approaches zero, explaining the large response of lightly damped systems.
Q6. The steady-state amplitude of the driven oscillator is governed mainly by:
The amplitude expression shows that ##A(\omega)## depends on how close ##\omega## is to ##\omega_0## and on the damping term in the denominator. Initial conditions affect only the transient, which damping removes.
Q7. The quality factor ##Q## is used in this lesson primarily to quantify:
The lesson states that sharpness of resonance, quantified by the quality factor ##Q##, separates a lightly damped tuning fork from a heavily damped loudspeaker enclosure. It is a measure of how sharp the resonant peak is.
Q8. A lightly damped tuning fork and a heavily damped loudspeaker enclosure are contrasted in the lesson mainly through:
The lesson uses the quality factor ##Q## to distinguish the lightly damped tuning fork, which has a sharp resonance, from the heavily damped loudspeaker enclosure, which has a broad response.
Q9. In the driven oscillator, the transient response at the natural frequencies is removed by:
The lesson states that damping removes the transient, leaving a driven oscillation whose amplitude peaks near resonance. The steady state then oscillates at the driving frequency.
Q10. The natural frequencies of an acoustic resonator are determined by:
The lesson states that every acoustic resonator possesses natural frequencies determined by its geometry, boundary conditions, and the speed of the wave within it. The driving force does not set the natural frequencies.
Q11. The collapse of the Tacoma Narrows bridge is used in the lesson as an example of:
The lesson cites the failure of a bridge under wind loading as an example of resonance that must be suppressed. It illustrates why engineers avoid unwanted resonance in structures.
Q12. In the electrical analogy of resonance, the mechanical mass ##m##, stiffness ##k##, and damping ##b## correspond respectively to:
In the standard electrical analogy, mass corresponds to inductance, stiffness to the inverse of capacitance, and damping to resistance. The resonant condition then mirrors the matching of driving frequency to natural frequency.
Q13. Resonance in a musical instrument body, such as the body of a violin, is exploited to:
The lesson states that resonance converts modest periodic forces into large pressure oscillations and governs the tuning of a violin body. The instrument body is designed to respond strongly at selected frequencies.
Q14. In the resonance tube experiment, resonance is detected when the air column responds with maximum amplitude because:
The resonance tube experiment relies on matching the driving frequency to a natural frequency of the air column, giving maximum response. The material and the medium outside the tube do not by themselves create resonance.
Q15. For a driven damped harmonic oscillator, which statement about the phase of the steady-state response relative to the driving force is correct?
For a driven damped oscillator, the steady-state displacement is approximately in phase with the drive well below resonance, lags by exactly ##\pi/2## at resonance, and lags by nearly ##\pi## well above resonance. Thus the phase varies with frequency and is not constant, so only B is correct.