Question

In case of a forced vibration, the resonance wave becomes very sharp when the:

• A. Damping force is small
• B. Restoring force is small
• C. Applied periodic force is small
• D. Quality factor is small

Answer 1

The Correct Option is A

To solve this problem, we need to understand the concept of resonance in forced vibrations. When an object is subject to forced vibrations, resonance occurs if the frequency of the applied force matches the natural frequency of the system, leading to maximum amplitude of oscillation.

1. Role of Damping: Damping is a force that opposes the motion and dissipates the energy of the system, reducing the amplitude of oscillations over time. In the context of resonance, a small damping force allows the system to reach higher amplitudes because energy is not quickly lost to the environment. Therefore, a sharp resonance curve indicates high amplitudes and occurs when damping is minimal. The quality of resonance is described by the sharpness of the resonance peak.
2. Effect of Small Damping: With small damping, the energy losses are minimized, allowing the amplitude at resonance to increase significantly. This results in a sharp and distinct resonance peak in the amplitude versus frequency graph, as the amplitude rapidly changes around the natural frequency.
3. Analyzing the Options:
   • Damping force is small: Correct. As explained, small damping leads to a sharp and distinct resonance curve.
   • Restoring force is small: Incorrect. The restoring force is related to the system's stiffness and affects the natural frequency rather than the sharpness of resonance.
   • Applied periodic force is small: Incorrect. The amplitude of vibrations depends on the magnitude of the applied force, but small applied force doesn't enhance the sharpness of resonance.
   • Quality factor is small: Incorrect. A high-quality factor indicates low damping and sharper resonance, not a small quality factor.

Based on the above explanation and analysis, the correct answer is that in a forced vibration scenario, the resonance wave becomes very sharp when the damping force is small.

In quantitative terms, the sharpness of the resonance is measured by the Quality Factor (Q), which is quantitatively given by:

Q = \frac{\text{Natural Frequency}}{\text{Bandwidth}}

A small damping force implies a higher Q factor, leading to a sharper resonance curve.