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Resonance occurs when a system is driven by an external force at a frequency that matches its natural frequency. This results in a significant increase in amplitude, as the system absorbs energy efficiently from the external force. For the damped harmonic oscillator, this natural frequency is given by \( \omega_0 = \sqrt{\frac{k}{m}} \), where \( k \) is the force constant and \( m \) is the mass. At resonance, the energy input from the driving force overcomes energy lost through damping.

When dealing with a damped harmonic oscillator, it's crucial to note that resonance maximizes the system's response. However, the actual peak amplitude at resonance is determined by the balance between the driving force and damping. In practical terms, reduced damping leads to higher peak amplitudes, while increased damping lowers it due to energy dissipation.

The damping constant, \( b \), is a measure of how quickly energy is lost from a system due to resistive forces like friction or air resistance. In a damped harmonic oscillator, the damping affects how the system responds to external forces. As the damping constant increases, it generally slows down the system's response, decreasing the amplitude of oscillations.

In our exercise, the effect of changes in damping constant is clearly illustrated:

• When \( b \) changes from \( b_1 \) to \( 3b_1 \), the amplitude reduces to a third of its initial value. This is because the increased damping dissipates more energy, reducing the system’s ability to oscillate.
• Conversely, if \( b \) is modified to \( \frac{b_1}{2} \), the amplitude doubles. The reduction in damping allows the system to maintain more of its energy, leading to larger oscillations.

Understanding the role of the damping constant is essential in predicting how systems behave under different conditions.

Amplitude variation in a damped harmonic oscillator is directly influenced by the damping constant. The relationship between amplitude (\( A \)) and damping can be understood using the formula \( A = \frac{F_\mathrm{max}}{b \omega_0} \). The formula highlights that changes in the damping constant have an inverse effect on amplitude.

In the given scenario:

• For a damping constant \( 3b_1 \), the amplitude becomes \( \frac{A_1}{3} \). This shows that a threefold increase in damping results in an amplitude that is three times smaller.
• With a damping constant of \( \frac{b_1}{2} \), the amplitude becomes \( 2A_1 \). Halving the damping increases the amplitude twofold, showcasing less energy loss per cycle.

The inverse relationship between amplitude and damping is fundamental to understanding oscillatory systems' behavior. It helps predict how changes in parameters will influence the dynamic response to external stimuli.