A 3.0 -kg mass is vibrating on a spring. It has a resonant...

A 3.0 -kg mass is vibrating on a spring. It has a resonant angular speed of $2.4 \mathrm{rad} / \mathrm{s}$ and a damping angular speed of $0.14 \mathrm{rad} / \mathrm{s}$. If the driving force is $2.0 \mathrm{~N}$, find the maximum amplitude if the driving angular speed is (a) $1.2 \mathrm{rad} / \mathrm{s},$ (b) $2.4 \mathrm{rad} / \mathrm{s}$, and $($ c) $4.8 \mathrm{rad} / \mathrm{s}$.

A 3.0 -kg mass is vibrating on a spring. It has a resonant angular speed of $2.4 \mathrm{rad} / \mathrm{s}$ and a damping angular speed of $0.14 \mathrm{rad} / \mathrm{s}$. If the driving force is $2.0 \mathrm{~N}$, find the maximum amplitude if the driving angular speed is
(a) $1.2 \mathrm{rad} / \mathrm{s},$ (b) $2.4 \mathrm{rad} / \mathrm{s}$, and $($ c) $4.8 \mathrm{rad} / \mathrm{s}$.

Key Concepts

Damped Harmonic Oscillator

This concept describes systems where oscillations diminish over time due to the presence of a damping force, such as friction or air resistance. The damping is typically modeled by a term proportional to velocity in the differential equation, and it affects both the transient response and the steady-state behavior of the system. Understanding damping is crucial to predicting how the amplitude of oscillations decreases over time and influences the resonance peak.

Forced Oscillations

Forced oscillations occur when an external periodic force drives a system, which responds by oscillating at the driving frequency. The system’s response depends on the relation between the driving frequency and the system’s natural (or resonant) frequency. Analyzing forced oscillations is essential for situations where continuous energy input counteracts the energy loss due to damping, leading to a steady-state oscillatory motion.

Resonance

Resonance happens when the driving frequency of an external force matches the natural frequency of a system, causing the amplitude of oscillations to peak. In damped systems, resonance is moderated by the damping factor, which broadens the resonance peak and limits the maximum amplitude. Recognizing resonance is key in both avoiding destructive oscillations in engineering structures and optimizing the response in systems like musical instruments or sensors.

Amplitude Response

The amplitude response of a system under forced oscillation quantifies how the maximum displacement (amplitude) depends on the driving frequency, the system parameters (such as mass and stiffness), and the damping. This response is described by a function that shows sharp peaks near resonance and lower amplitudes when the driving frequency deviates from the natural frequency. It is a fundamental tool for analyzing and predicting the behavior of oscillatory systems under various driving conditions.

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