91Ó°ÊÓ

Resonance in a damped harmonic oscillator occurs when the driving force frequency matches the oscillator's natural frequency. This synchronization leads to maximum energy transfer. For our oscillator with a force constant \(k\) and mass \(m\), the natural frequency is expressed by the angular frequency \(\omega = \sqrt{\frac{k}{m}}\). When a system reaches resonance, the amplitude of oscillation reaches its peak because the input energy from the driving force is most efficiently converted into motion. It's crucial that at this point, the damping effects, which tend to reduce the amplitude over time, are just enough to balance energy input and output. As shown in the problem, when resonance is achieved under initial damping, the amplitude is noted as \(A_1\). Changes in damping can affect the resonant peak size but not its frequency. Thus, adjusting the damping constant shifts the amplitude without changing the frequency.

The damping constant \(b\) describes how resistive forces in the medium, like friction or air resistance, impact the oscillator. A larger \(b\) implies a stronger damping effect that reduces the oscillations and decreases the peak amplitude. In our problem, two scenarios are presented: when the damping constant is increased to \(3b_1\) and decreased to \(b_1/2\).

• For damping \(3b_1\): The amplitude diminishes to one-third of the original, \(A = \frac{A_1}{3}\). The higher resistance caused by increased damping opposes the motion, reducing the effectiveness of the driving force.


• For damping \(b_1/2\): The amplitude grows to double, \(A = 2A_1\). Reduced damping allows the system to oscillate with greater amplitude because less energy is lost in each cycle.

Thus, as damping decreases, the system retains more energy, swaying in grander oscillations.

The driving force is the external influence that keeps the damped harmonic oscillator moving. It is characterized by its strength \(F_{\text{max}}\) and frequency. Without a driving force, the oscillator would gradually cease moving due to damping forces. This force is crucial in maintaining continuous motion. In experiencing the driving force, the system draws energy that is ideally synchronized with its natural frequency to achieve resonance.
The effectiveness of this energy transfer is contingent on both the frequency don the driving force reaches resonance and the magnitude of the force itself. Even if the driving force amplitude remains constant as \(F_{\text{max}}\), changes in damping constant alter the system's response, varying the oscillation amplitude as illustrated in the exercise. Thus, while the driving force operates as the energy supplier, the damping constant modulates how well the supplied energy manifests as oscillation amplitude.

Angular frequency \(\omega\) represents how quickly a system oscillates. In a physical system, this figure is measured in radians per second and is pivotal for understanding the dynamics of the damped harmonic oscillator. It is determined from the relationship \(\omega = \sqrt{\frac{k}{m}}\), where \(k\) is the spring constant and \(m\) is the mass.
Assuming resonance, the driven and natural angular frequencies align, allowing optimal oscillation amplitude. Any disparity would result in less efficient energy transfer and a reduced resonance effect. Calculating this angular frequency helps in setting the system to resonance, ensuring the driving force reinforces each oscillation at the right moment.

• At \(b_1\), resonance maintains, giving maximum amplitude.


• Any change in system parameters, like the damping constant, modifies amplitude response while angular frequency remains defined by mass and force constant.

Understanding this concept is crucial for predicting and manipulating the behavior of systems subjected to external oscillatory forces.