Resonance - UNSOLVED PRACTICE SET

Q7. Define resonance. Under what conditions does resonance occur in a forced oscillator?

Q8. Explain why the amplitude at resonance is finite even though the driving frequency equals the natural frequency.

Q9. What is meant by the "sharpness of resonance"? How does damping affect it?

Q10. Give two examples of useful resonance and two examples of destructive resonance from everyday life.

Q11. Why is it dangerous for soldiers to march in step across a suspension bridge? Explain using the concept of resonance.

Q12. A radio receiver uses resonance to select a particular station. Explain how this works.

Q13. Explain the phenomenon of resonance in detail. Using the expression for amplitude of forced oscillations, show that:

(i) The amplitude is maximum when the driving frequency equals the natural frequency (for small damping)

(ii) The maximum amplitude depends inversely on the damping constant

(iii) The resonance curve becomes broader as damping increases

Draw resonance curves for light damping, moderate damping, and heavy damping, and explain the practical significance of each.

Q14. Discuss the Tacoma Narrows Bridge collapse and explain how resonance contributed to this disaster. Also discuss:

(i) How modern bridge designs prevent resonance

(ii) How resonance is used beneficially in musical instruments

(iii) How MRI machines use resonance

(iv) How quartz crystals in watches use resonance

Q15. A student sets up an experiment with a tuning fork of frequency 512 Hz and a hollow tube partially filled with water. By adjusting the water level, loud sound is heard at certain positions.

(i) Explain why loud sound is heard at certain water levels.

(ii) Calculate the length of the air column for the first resonance. (Speed of sound = 340 m/s)

(iii) Calculate the length for the second resonance.

(iv) How would the resonance positions change if a tuning fork of higher frequency were used?

Q16. A driven harmonic oscillator has natural frequency f₀ = 2 Hz, mass m = 0.5 kg, and damping constant b = 0.2 s⁻¹. The driving force has amplitude F₀ = 1 N.

(i) Calculate the amplitude of oscillation when the driving frequency is 1 Hz.

(ii) Calculate the amplitude at resonance.

(iii) Calculate the amplitude when the driving frequency is 3 Hz.

(iv) Calculate the quality factor Q of the oscillator.

(v) How many oscillations occur before the energy of a free oscillator decays to 1/e² of its initial value?

Q17. A bridge has a natural frequency of 1.5 Hz. Soldiers marching across it produce a periodic force. The damping in the bridge is such that the amplitude at resonance would be 10 cm if the soldiers marched at the natural frequency.

(i) At what marching frequency would resonance occur?

(ii) If the soldiers march at 120 steps per minute, is this dangerous? Calculate the driving frequency.

(iii) If the damping is increased by a factor of 2, what happens to the amplitude at resonance?

(iv) Modern bridges are designed with damping systems. Explain how this prevents catastrophic resonance.

Q18. In a school science exhibition, a student builds a resonance tube apparatus. A tuning fork of frequency 440 Hz is held above a tube filled with water. The water level is gradually lowered.

(i) Calculate the length of the air column for the first three resonance positions. (Speed of sound = 340 m/s, end correction can be neglected)

(ii) If the student uses a tuning fork of 880 Hz, calculate the new resonance positions.

(iii) Explain why the resonance positions are not exactly at λ/4, 3λ/4, 5λ/4, etc., in a real experiment.

(iv) How can the student use this apparatus to determine the speed of sound in air?