Chapter 5: Problem 26
An undamped oscillator has period \(\tau_{\mathrm{o}}=1.000 \mathrm{s},\) but \(\mathrm{I}\) now add a little damping so that its period changes to \(\tau_{1}=1.001\) s. What is the damping factor \(\beta\) ? By what factor will the amplitude of oscillation decrease after 10 cycles? Which effect of damping would be more noticeable, the change of period or the decrease of the amplitude?
Short Answer
Step by step solution
Understand the Problem
Recall Relevant Equations
Calculate the Damping Factor \(\beta\)
Calculate the Amplitude Decrease Factor
Evaluate Noticeability of Effects
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Damping Factor
To calculate the damping factor \( \beta \), we use the relationship between the damped period \( \tau_1 \) and the natural period \( \tau_0 \). The formula is given by:
- \( \tau_1 = \frac{\tau_0}{\sqrt{1 - (\beta/\omega_0)^2}} \)
By substituting the known periods into the formula and solving for \( \beta \), we gain insight into the damping behavior of the system. A small \( \beta \) indicates low damping, meaning the oscillator will take longer to come to rest, while a large \( \beta \) shows significant damping, leading to faster energy loss.
Period of Oscillation
When damping is present, the period increases slightly, as energy is dissipated with every oscillation. In the specific example given, the period increases from \( \tau_0 = 1.000 \) s to \( \tau_1 = 1.001 \) s. Even a small change in period can significantly affect the timing of oscillations over long durations.
Damping slightly alters the period because the system's energy is reduced, causing it to move slower. This relationship is often negligible for small damping factors, but still a critical aspect in precision-dependent applications.
Amplitude Decay
The amount by which the amplitude decreases follows an exponential decay formula:
- Amplitude decay factor: \( e^{-\beta t} \)
This aspect of damped oscillation is usually more noticeable than the subtle changes in period, especially when the amplitude significantly reduces over time, making it crucial for systems where maintaining motion is essential.
Damped Oscillation
In essence, a damped oscillator's motion becomes more contained or less vigorous as time passes. It is characterized not only by a reduction in amplitude but also a slight modification in the period. The effects of damping are crucial in numerous real-world applications, such as in car shock absorbers, where they help maintain smooth ride quality by reducing the oscillations caused by bumps.
By understanding the interplay between the damping factor, period changes, and amplitude decay, one can predict and control the behavior of oscillating systems effectively. This understanding is vital for designing systems that require precise oscillation control.