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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: Q22E (page 210)

In problems 21-24the given figure represents the graph of an equation of motion for a damped spring/mass system. Use the graph to determine
(a) Whether the initial displacement is above or below the equilibrium position and
(b) Whether the mass is initially released from rest, heading downward, or heading upward.

Short Answer

Expert verified

(a) The mass is below the equilibrium position.

(b) The mass is released from rest.

Step by step solution

01

Definition:

When all the forces that act upon an object are balanced, then the object is said to be in a state of equilibrium.

02

(a) Initial displacement:

The graph is position $x$ versus time $t$ . So since the initial point on the axis is in the positive part of the plane, the initial displacement is below the equilibrium point. Recall that we will regard the equilibrium point as the negative aspect of the system.

The mass is below the equilibrium position.

03

(b) Mass position:

The velocity of the mass will be the derivative of the position function.

Looking at the graph, the initial condition appears to be on a maximum of the graph. So there is no initial velocity since the derivative of a function at a maximum is zero.

The mass is released from rest.

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Most popular questions from this chapter

(a) Show that $x (t)$ given in part (a) of Problem 43 can be written in the form

role="math" localid="1664195280056" $x (t) = \frac{- 2 F_{0}}{ω^{2} - γ^{2}} \sin \frac{1}{2} (γ - ω) t s i n \frac{1}{2} (γ + ω) t$

(b) If we define $ε = \frac{1}{2} (γ - ω)$ show that when is small an approximate solution is

$x (t) = \frac{F_{0}}{2 ε γ} s i n ε t s i n γ t$

When $ε$ is small, the frequency $\frac{γ}{2 π}$ of the impressed force is close to the frequency $\frac{ω}{2 π}$ of free vibrations. When this occurs, the motion is as indicated in Figure 5.1.23. Oscillations of this kind are called beats and are due to the fact that the frequency of $s i n ε t$ is quite small in comparison to the frequency of $s i n γ t$ . The dashed curves, or envelope of the graph of $x (t)$ , are obtained from the graphs of $\pm (\frac{F_{0}}{2 ε γ}) s i n ε t$ .Use a graphing utility with various values of $F_{0}, ε$ and $γ$ to verify the graph in Figure5.1.23.

Discussion problems.

21.Discuss why the damping term in equation(3) is written as

$β |\frac{d x}{d t}| \frac{d x}{d t}$ instead of $β {(\frac{d x}{d t})}^{2}$

Find the effective spring constant of the parallel-spring system shown in Figure 5.1.5when both springs have the spring constant. Give a physical interpretation of this result.

Consider the boundary-value problem

$γ^{''} + λ y = 0, y (- π) = y (π), y^{'} (- π) = y^{'} (π)$

(a) The type of boundary conditions specified are called periodic boundary conditions. Give a geometric interpretation of these conditions.

(b) Find the eigenvalues and eigenfunctions of the problem.

(c) Use a graphing utility to graph some of the eigenfunctions. Verify your geometric interpretation of the boundary conditions given in part (a).

The critical loads of thin columns depend on the end conditions of the column. The value of the Euler load $P_{1}$ in Example 4 was derived under the assumption that the column was hinged at both ends. Suppose that a thin vertical homogeneous column is embedded at its base $(x = 0)$ and free at its top $(x = L)$ and that a constant axial load P is applied to its free end. This load either causes a small deflection as shown in Figure 5.2.9 or does not cause such a detection $δ$ . In either case the differential equation for the detection $y (x)$ is

$E I \frac{d^{2} y}{d x^{2}} + p y = p δ$

FIGURE 5.2.9 Deflection of vertical column in Problem 24

(a) What is the predicted deflection when $δ = 0$ ?

(b) When $δ \pm 0$ , show that the Euler load for this column is

one-fourth of the Euler load for the hinged column in

Example 4.

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