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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

Expert verified
Plot position \( x(t) \) for each provided \( \beta \); 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 \( \omega_{\mathrm{o}} = 2\pi \). We are asked to plot the position \( x(t) \) for different damping constants (\( \beta \)) and for a specific time interval from \( t = 0 \) to \( t = 2 \). We should apply the appropriate formula for \( x(t) \) based on whether \( \beta < \omega_{\mathrm{o}} \), \( \beta = \omega_{\mathrm{o}} \), or \( \beta > \omega_{\mathrm{o}} \).
02

Formulas for Position x(t)

The position \( x(t) \) for a damped harmonic oscillator is given by different formulas based on the damping constant:- For \( \beta < \omega_{\mathrm{o}} \) (underdamped), \( x(t) = x_{\mathrm{o}} e^{-\beta t} \cos(\sqrt{\omega_{\mathrm{o}}^2 - \beta^2} \cdot t) \).- For \( \beta = \omega_{\mathrm{o}} \) (critically damped), \( x(t) = x_{\mathrm{o}} (1 + \omega_{\mathrm{o}} t) e^{-\omega_{\mathrm{o}} t} \).- For \( \beta > \omega_{\mathrm{o}} \) (overdamped), \( x(t) = x_{\mathrm{o}} e^{-\beta t} \cosh(\sqrt{\beta^2 - \omega_{\mathrm{o}}^2} \cdot t) \).
03

Determine the Damping Scenarios

Calculate \( \omega_{\mathrm{o}} \) using the given value \( \omega_{\mathrm{o}} = 2\pi \), which gives us approximately 6.28. We'll evaluate different \( \beta \) 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 \( \beta \):- If \( \beta < 6.28 \) (0, 1, 2, 4, 6): use the formula for underdamped motion.- If \( \beta = 6.28 \): use the critical damping formula.- If \( \beta > 6.28 \) (10, 20): use the formula for overdamped motion.Calculate \( x(t) \) values from \( t = 0 \) to \( t = 2 \).
05

Graph the Results

Use graphing software to plot each scenario. Create a graph showing \( x(t) \) on the y-axis and \( t \) on the x-axis, with separate curves for each \( \beta \) 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 \( \omega_{\mathrm{o}} \), 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 \( \omega_{\mathrm{o}} = 2\pi \), 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 \( \beta \). 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
The damping constant, symbolized by \( \beta \), 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 \( \beta \) like 0, 1, 2, 4, 6, 10, and 20.
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.
Each of these behaviors requires a specific formula to measure \( x(t) \) accurately over time.
Underdamped, Critically Damped, and Overdamped Cases
These terms describe how a damped harmonic oscillator behaves based on the relationship between the damping constant \( \beta \) and the natural frequency \( \omega_{\mathrm{o}} \).
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
Graphing the position \( x(t) \) over time is essential to visualize how a damped harmonic oscillator behaves under different damping scenarios. In our exercise, we plot \( x(t) \) from \( t = 0 \) to \( t = 2 \) for various \( \beta \) values.
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.
This graphical representation simplifies understanding how different damping constants affect motion. By visually interpreting the data, students can grasp how energy dissipation varies in each scenario. It’s a powerful way of connecting theoretical formulas with real-world behavior.

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Most popular questions from this chapter

An unusual pendulum is made by fixing a string to a horizontal cylinder of radius \(R\), wrapping the string several times around the cylinder, and then tying a mass \(m\) to the loose end. In equilibrium the mass hangs a distance \(l_{\mathrm{o}}\) vertically below the edge of the cylinder. Find the potential energy if the pendulum has swung to an angle \(\phi\) from the vertical. Show that for small angles, it can be written in the Hooke's law form \(U=\frac{1}{2} k \phi^{2} .\) Comment on the value of \(k\).

Write down the potential energy \(U(\phi)\) of a simple pendulum (mass \(m,\) length \(l\) ) in terms of the angle \(\phi\) between the pendulum and the vertical. (Choose the zero of \(U\) at the bottom.) Show that, for small angles, \(U\) has the Hooke's law form \(U(\phi)=\frac{1}{2} k \phi^{2},\) in terms of the coordinate \(\phi .\) What is \(k ?\)

(a) Consider a cart on a spring which is critically damped. At time \(t=0\), it is sitting at its equilibrium position and is kicked in the positive direction with velocity \(v_{\mathrm{o}} .\) Find its position \(x(t)\) for all subsequent times and sketch your answer. (b) Do the same for the case that it is released from rest at position \(x=x_{\mathrm{o}} .\) In this latter case, how far is the cart from equilibrium after a time equal to \(\tau_{\mathrm{o}}=2 \pi / \omega_{\mathrm{o}}\) the period in the absence of any damping?

The force on a mass \(m\) at position \(x\) on the \(x\) axis is \(F=-F_{0} \sinh \alpha x,\) where \(F_{0}\) and \(\alpha\) are constants. Find the potential energy \(U(x),\) and give an approximation for \(U(x)\) suitable for small oscillations. What is the angular frequency of such oscillations?

The potential energy of a one-dimensional mass \(m\) at a distance \(r\) from the origin is $$U(r)=U_{0}\left(\frac{r}{R}+\lambda^{2} \frac{R}{r}\right)$$ for \(0 < r < \infty,\) with \(U_{\mathrm{o}}, R,\) and \(\lambda\) all positive constants. Find the equilibrium position \(r_{\mathrm{o}} .\) Let \(x\) be the distance from equilibrium and show that, for small \(x\), the PE has the form \(U=\) const \(+\frac{1}{2} k x^{2}\). What is the angular frequency of small oscillations?

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