Problem 12 Consider a spring subject to a f... [FREE SOLUTION]

Chapter 17: Problem 12

Consider a spring subject to a frictional or damping force. (a) In the critically damped case, the motion is given by \(x=c_{1} e^{r t}+c_{2} t e^{r t}\). Show that the graph of \(x\) crosses the \(t\) -axis whenever \(c_{1}\) and \(c_{2}\) have opposite signs. (b) In the overdamped case, the motion is given by \(x=c_{1} e^{r_{i} t}+c_{2} e^{r_{2} t},\) where \(r_{1}>r_{2}\). Determine a condition on the relative magnitudes of \(c_{1}\) and \(c_{2}\) under which the graph of \(x\) crosses the \(t\) -axis at a positive value of \(t .\)

Short Answer

Expert verified

(a) Opposite signs of \(c_1\) and \(c_2\) make \(x\) cross the \(t\)-axis. (b) In the overdamped case, \(|c_2| > |c_1|\) allows \(x\) to cross at positive \(t\).

Step by step solution

Understand Critically Damped Motion

In the critically damped case, the motion is described by the function \(x=c_{1} e^{r t}+c_{2} t e^{r t}\). A root (crossing the \(t\)-axis) occurs when the expression equals zero, \(x = 0\).

Setup Equation for Crossing the t-axis

For the critically damped case, solve \( c_{1} e^{r t} + c_{2} t e^{r t} = 0\). Factor out \(e^{r t}\) to get \(e^{r t} (c_{1} + c_{2} t) = 0\). Since \(e^{r t}eq 0\), the condition for crossing is \(c_{1} + c_{2} t = 0\).

Analyze the Sign Condition

Solve \(c_{1} + c_{2} t = 0\) for \(t\), leading to \(t = -\frac{c_{1}}{c_{2}}\). If \(c_{1}\) and \(c_{2}\) have opposite signs, \(-\frac{c_{1}}{c_{2}}\) is positive, meaning \(x\) can cross the \(t\)-axis.

Understand Overdamped Motion

In the overdamped case, the motion is given by \(x = c_{1} e^{r_{1} t} + c_{2} e^{r_{2} t}\), with \(r_{1} > r_{2}\). The function crosses the \(t\)-axis when \(x = 0\).

Setup Equation for Overdamped Crossing

Set \(c_{1} e^{r_{1} t} + c_{2} e^{r_{2} t} = 0\). Rewrite to \(c_{1} e^{r_{1} t} = -c_{2} e^{r_{2} t}\) and solve for \(t\): \(t = \frac{1}{r_{1} - r_{2}} \ln\left(-\frac{c_{2}}{c_{1}}\right)\).

Determine Condition on Magnitudes

For \(t\) to be positive, \(-\frac{c_{2}}{c_{1}} > 1\) (since \(r_{1} > r_{2}\)). Thus, \(|c_{2}| > |c_{1}|\) is the condition for \(t\) to be positive.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Critically Damped Motion

Critically damped motion occurs when a system returns to equilibrium without oscillating. Imagine a spring system with just enough damping to prevent it from bouncing back and forth. The mathematical expression for this motion is given as \[ x = c_{1} e^{r t} + c_{2} t e^{r t} \].The graph of this function crosses the \(t\)-axis (meaning it has a root) when the function equals zero. To find when this crossing happens, we factor out \(e^{r t}\):

Factor the equation: \( e^{r t} (c_{1} + c_{2} t) = 0 \)
Since \( e^{r t} eq 0 \), the crossing occurs at \( c_{1} + c_{2} t = 0 \)

Solving for \(t\), we obtain \( t = -\frac{c_{1}}{c_{2}} \). This value is positive when \(c_{1}\) and \(c_{2}\) have opposite signs, indicating the motion's path crosses the \(t\)-axis after a positive time has elapsed.

Overdamped Motion

In overdamped motion, the system returns to equilibrium more slowly than in the critically damped case. The motion is described by the function \[ x = c_{1} e^{r_{1} t} + c_{2} e^{r_{2} t} \],where \( r_{1} > r_{2} \).The graph crosses the \(t\)-axis if the expression equals zero:

Set the equation: \( c_{1} e^{r_{1} t} + c_{2} e^{r_{2} t} = 0 \)
Rearrange to find \( c_{1} e^{r_{1} t} = -c_{2} e^{r_{2} t} \)

Solving for \(t\) yields:\[ t = \frac{1}{r_{1} - r_{2}} \ln \left( -\frac{c_{2}}{c_{1}} \right) \].For \(t\) to be positive, the ratio \(-\frac{c_{2}}{c_{1}} > 1\). This implies that for the graph to cross the \(t\)-axis at a positive time, the magnitude of \(c_{2}\) must be greater than that of \(c_{1}\), i.e., \(|c_{2}| > |c_{1}|\).

Crossing the t-axis

For any motion described by a function to cross the \(t\)-axis, the function itself must equal zero at that point. This is a universal check for when a system's motion transitions from a positive to a negative value or vice versa. Both critically damped and overdamped systems have conditions under which their expressions become zero, leading to a crossing.In critically damped systems:

The crossing occurs when \( c_{1} + c_{2} t = 0 \)
Thus, \( t = -\frac{c_{1}}{c_{2}} \) gives the crossing point, provided \(c_{1}\) and \(c_{2}\) have opposite signs.

In overdamped systems:

The equation \( c_{1} e^{r_{1} t} + c_{2} e^{r_{2} t} = 0 \) must be satisfied.
This simplifies to \( t = \frac{1}{r_{1} - r_{2}} \ln \left( -\frac{c_{2}}{c_{1}} \right) \), ensuring crossing if \(|c_{2}| > |c_{1}|\).

Understanding these conditions is key in predicting how and where the system's graph will pass the \(t\)-axis, offering insight into the behavior of damped harmonic systems.

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Short Answer

Step by step solution

Understand Critically Damped Motion

Setup Equation for Crossing the t-axis

Analyze the Sign Condition

Understand Overdamped Motion

Setup Equation for Overdamped Crossing

Determine Condition on Magnitudes

Key Concepts

Critically Damped Motion

Overdamped Motion

Crossing the t-axis

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