Chapter 5: Problem 31
Consider a cart on a spring with natural frequency \(\omega_{\mathrm{o}}=2 \pi,\) which is released from rest at \(x_{\mathrm{o}}=1\) and \(t=0 .\) Using appropriate graphing software, plot the position \(x(t)\) for \(0 < t < 2\) and for damping constants \(\beta=0,1,2,4,6,2 \pi, 10,\) and \(20 .\) [Remember that \(x(t)\) is given by different formulas for \(\left.\beta<\omega_{\mathrm{o}}, \beta=\omega_{\mathrm{o}}, \text { and } \beta > \omega_{\mathrm{o}} .\right]\)
Short Answer
Step by step solution
Understand the Problem
Formulas for Position x(t)
Determine the Damping Scenarios
Apply the Formulas
Graph the Results
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Natural Frequency
Understanding natural frequency helps to identify how fast a system tends to oscillate naturally. In this scenario, it's useful for determining the type of damping effect by comparing it to the damping constant \( \beta \). Whether the system is underdamped, critically damped, or overdamped will depend on the relationship between \( \beta \) and \( \omega_{\mathrm{o}} \).
It's crucial because it helps us use the right formula for calculating position \( x(t) \), which changes based on what type of damping occurs.
Damping Constant
A higher \( \beta \) value indicates stronger damping, meaning the system will return to equilibrium more quickly. Conversely, a lower \( \beta \) leads to a slower dissipation of energy, allowing oscillations to continue longer. This constant guides us to understand whether the system exhibits underdamped, critically damped, or overdamped behavior.
- Underdamped: Oscillates with decreasing amplitude.
- Critically damped: Returns to equilibrium as quickly as possible without oscillating.
- Overdamped: Returns to equilibrium without oscillating, slower than critically damped.
Underdamped, Critically Damped, and Overdamped Cases
Underdamped:
When \( \beta < \omega_{\mathrm{o}} \), the system is underdamped. It will oscillate, but the amplitude decreases over time. The position function is \( x(t) = x_{\mathrm{o}} e^{-\beta t} \cos(\sqrt{\omega_{\mathrm{o}}^2 - \beta^2} \cdot t) \). This reflects oscillatory motion that gradually diminishes.
Critically Damped:
When \( \beta = \omega_{\mathrm{o}} \), the system is critically damped. Here, the system returns to equilibrium as fast as possible without oscillations. The mathematical expression is \( x(t) = x_{\mathrm{o}} (1 + \omega_{\mathrm{o}} t) e^{-\omega_{\mathrm{o}} t} \).
Overdamped:
When \( \beta > \omega_{\mathrm{o}} \), the system is overdamped. It returns to equilibrium without oscillating. The relevant formula is \( x(t) = x_{\mathrm{o}} e^{-\beta t} \cosh(\sqrt{\beta^2 - \omega_{\mathrm{o}}^2} \cdot t) \). This signifies a slower response compared to the critically damped case.
Graphing of Position Over Time
Using a graph helps us compare how quickly the oscillations decay depending on the damping constant. Here's what to expect:
- For a small \( \beta \), the oscillation is more evident, decaying slowly.
- At critical damping, the graph shows a smooth return to equilibrium without oscillating.
- In overdamped conditions, the return to equilibrium is visible but slower, with no oscillations.