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A First Course In Differential Equations With Modelling Applications

Dennis G. Zill

$Math Studyset Vaia Explanations$ Math

11th Edition · 400 pages

ISBN: 9781305965720

Chapter 5: 4RP (page 196)

Pure resonance cannot take place in the presence of a damping force. _________

Short Answer

Expert verified

The given Statement is “ Pure resonance cannot take place in the presence of a damping force “ that is true.

Step by step solution

01

Definition of pure resonance

Pure resonance occurs whenever the natural interior frequencyequals the natural exterior frequency, resulting in boundless differential equation solution.

The differential equation of pure resonance

$x'' (t) + ω_{0}^{2} x (t) = F_{0} \cos (ωt)$

02

Step 2:

Large mass oscillations would ultimately push the spring over its elastic limit.

Because it overlooks the retarding effects of ever-present damping forces, the resonating model described is entirely impractical.

Although "pure resonance cannot exist when the tiniest quantity of damping is considered," huge and equally damaging amplitudes of vibration (despite being bounded as t-x) can occur.

Therefore, Pure resonance cannot take place in the presence of a damping force statement is true.

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Most popular questions from this chapter

Relief supplies As shown in figurea plane flying horizontally at a constant speed drops _V0 relief supply pack to a person on the ground. Assume the origin is the point where the supply pack is released and that the positive x-axis points forward and that positive y-axis points downward. under the assumption that the horizontal and vertical components of the air resistance are proportional to $(d x / d t)^{2} a n d = (d y / d t)^{2}$ ,respectively, and if the position of the supply pack is given by r(t)=x(i)+y(t)j, then its velocity is $v (t) = (d x / d t) i + (d y / d t) j$ Equating components in the vector form of Newton’s second law of motion.

$m \frac{d y}{d t} = m g - [{(\frac{d x}{d t})}^{2} i + {(\frac{d y}{d t})}^{2} j] m \frac{d^{2} x}{d t^{2}} = m g - k {(\frac{d x}{d t})}^{2}, x (0) = 0, x' (0) =' v_{0} m \frac{d^{2} y}{d t^{2}} = m g - k {(\frac{d y}{d t})}^{2} y (0) = 0, y' (0) =' 0$

a)solve both of the foregoing initial-value problems by means of the substitutions $u = \frac{d x}{d y}, w = \frac{d y}{d t}$ and separation of variable.[Hint: see the Remarks at the end of

section .]

b)suppose the plane files at an altitude of $1000 f t$ ft and that its constant speed $300$ is mi/h. assume that the constant of proportionality for air resistance is $k=0.0053$ and that the supply pack weighs $256$ Ib. use a root-finding application of a CAC or a graphic calculator to determine the horizontal distance the pack travels, measured from its point of release to the point where it hits the ground.

Suppose a pendulum is formed by attaching a massto theend of a string of negligible mass and length l. At $t = 0$ thependulum is released from rest at a small displacement angle $θ_{0} > 0$ to the right of the vertical equilibrium position OP. SeeFigure 5.R.5. At time $t_{1} > 0$ the string hits a nail at a point N onOP a distance $\frac{3}{4} l$ from O, but the mass continues to the left asshown in the figure.

(a) Construct and solve a linear initial-value problem for thedisplacement angleshown in the figure. Find theinterval $[0, t_{1}]$ on which $θ_{1} (t)$ is defined.

(b) Construct and solve a linear initial-value problem for thedisplacement angle $θ_{2} (t)$ shown in the figure. Find theinterval $[t_{1}, t_{2}]$ on which $θ_{2} (t)$ is defined, where $t_{2}$ isthe time that m returns to the vertical line NP.

Find the eigenvalues and eigenfunctions for the given boundary-value problem.

$y^{''} + λ y = 0, y^{'} (0) = 0, y (L) = 0$

In Problem 37 write the equation of motion in the form $x (t) = A s i n (ω t + ϕ) + B e^{- 2 t} s i n (4 t + θ)$ . What is the amplitude of vibrations after a very long time?

Find the eigenvalues and eigenfunctions for the given boundary-value problem.

$y'' + λ y = 0, y' (0) = 0, y' (π) = 0$

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