Chapter 4: Q3E (page 156)

Question: Show that if $F_{ext} (t) = 0, m = 1, k = 9$ , and , then the equation has the "critically damped" solutions $y_{1} (t) = e^{- 3 t}$ and $y_{2} (t) = t e^{- 3 t}$ . What is the limit of these solutions as $t \to \infty$ ?

Short Answer

Expert verified

Answer

The $y (t) = c_{1} e^{- 3 t} + c_{2} {te}^{- 3 t}$ solution is and the limit of the solution is .

Step by step solution

Finding the differential equation and the general solution.

The differential equation for the mass-spring oscillator is

my'' + by' + ky = F_{ext} .

Given $F_{ext} = 0, m = 1, k = 9$ and, then the differential equation is $y'' + 6 y' + 9 y = 0 .$

The auxiliary equation for the given differential equation is, $m^{2} + 6 m + 9 = 0 {(m + 3)}^{2} = 0$

Check for critically damped.

If , then the system is critically damped. Here and , then;

$\begin{array}{rcl} b^{2} - 4 a c & = & 6^{2} - 9 \times 4 \\ = & 0 \end{array}$

So, the system is critically damped and, then the solutions are;

$y (t) = c_{1} e^{- 3 t} + c_{2} {te}^{- 3 t}$

If $t \to \infty$ , then $e^{- 3 t} = 0$ . Therefore,

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Question: Show that if $F_{ext} (t) = 0, m = 1, k = 9$ , and , then the equation has the "critically damped" solutions $y_{1} (t) = e^{- 3 t}$ and $y_{2} (t) = t e^{- 3 t}$ . What is the limit of these solutions as $t \to \infty$ ?

Short Answer

Step by step solution

Finding the differential equation and the general solution.

Check for critically damped.

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Question: Show that if Fext(t)=0,m=1,k=9, and , then the equation has the "critically damped" solutionsy1(t)=e-3t andy2(t)=te-3t . What is the limit of these solutions as t→∞?

Short Answer

Step by step solution

Finding the differential equation and the general solution.

Check for critically damped.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Applied Mathematics

Discrete Mathematics

Statistics

Theoretical and Mathematical Physics

Decision Maths

Study anywhere. Anytime. Across all devices.

Question: Show that if $F_{ext} (t) = 0, m = 1, k = 9$ , and , then the equation has the "critically damped" solutions $y_{1} (t) = e^{- 3 t}$ and $y_{2} (t) = t e^{- 3 t}$ . What is the limit of these solutions as $t \to \infty$ ?