Chapter 5: Problem 31
Consider a cart on a spring with natural frequency
Short Answer
Expert verified
Plot position for each provided ; use formulas based on damping level.
Step by step solution
01
Understand the Problem
We have a damped harmonic oscillator with a natural frequency of . We are asked to plot the position for different damping constants ( ) and for a specific time interval from to . We should apply the appropriate formula for based on whether , , or .
02
Formulas for Position x(t)
The position for a damped harmonic oscillator is given by different formulas based on the damping constant:- For (underdamped), .- For (critically damped), .- For (overdamped), .
03
Determine the Damping Scenarios
Calculate using the given value , which gives us approximately 6.28. We'll evaluate different values: 0, 1, 2, 4, 6, 2\pi (approx. 6.28), 10, and 20. Some of these will correspond to underdamped, critically damped, and overdamped cases.
04
Apply the Formulas
For each damping constant :- If (0, 1, 2, 4, 6): use the formula for underdamped motion.- If : use the critical damping formula.- If (10, 20): use the formula for overdamped motion.Calculate values from to .
05
Graph the Results
Use graphing software to plot each scenario. Create a graph showing on the y-axis and on the x-axis, with separate curves for each value. Ensure that each curve is clearly labeled to show the different damping constants.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Natural Frequency
The natural frequency, often denoted as , is a fundamental characteristic of oscillatory systems like springs. It represents the frequency at which a system oscillates when not subjected to any external force or damping. In our exercise, this frequency is given as , which simplifies calculations since it can convert easily between cycles and angular measurements.
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 . Whether the system is underdamped, critically damped, or overdamped will depend on the relationship between and .
It's crucial because it helps us use the right formula for calculating position , which changes based on what type of damping occurs.
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
It's crucial because it helps us use the right formula for calculating position
Damping Constant
The damping constant, symbolized by , measures how quickly an oscillating system loses energy over time. It's a key player in defining the behavior of damped harmonic oscillators. In the given exercise, we're asked to consider various values of like 0, 1, 2, 4, 6, 10, and 20.
A higher value indicates stronger damping, meaning the system will return to equilibrium more quickly. Conversely, a lower 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. accurately over time.
A higher
- 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
These terms describe how a damped harmonic oscillator behaves based on the relationship between the damping constant and the natural frequency .
Underdamped:
When , the system is underdamped. It will oscillate, but the amplitude decreases over time. The position function is . This reflects oscillatory motion that gradually diminishes.
Critically Damped:
When , the system is critically damped. Here, the system returns to equilibrium as fast as possible without oscillations. The mathematical expression is .
Overdamped:
When , the system is overdamped. It returns to equilibrium without oscillating. The relevant formula is . This signifies a slower response compared to the critically damped case.
Underdamped:
When
Critically Damped:
When
Overdamped:
When
Graphing of Position Over Time
Graphing the position over time is essential to visualize how a damped harmonic oscillator behaves under different damping scenarios. In our exercise, we plot from to for various values.
Using a graph helps us compare how quickly the oscillations decay depending on the damping constant. Here's what to expect:
Using a graph helps us compare how quickly the oscillations decay depending on the damping constant. Here's what to expect:
- For a small
, 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.