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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: Q21E (page 210)

In problemsthe given figure 21-24 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 initial displacement is above the equilibrium position.

(b) The mass is heading upward.

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 $t$ . The point is on the negative x-axis, you can say that the initial displacement is above the equilibrium position.

03

(b) Mass position:

From the graph you can see that the graph continues to head down the negative x-axis before it changes direction, you can say that the mass is heading upward.

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Most popular questions from this chapter

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

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

(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.

A mass weighing 12 pounds stretches a spring 2 feet. The massis initially released from a point 1 foot below the equilibriumposition with an upward velocity of 4 ft/s.

(a) Find the equation of motion.

(b) What are the amplitude, period, and frequency of thesimple harmonic motion?

(c) At what times does the mass return to the point 1 footbelow the equilibrium position?

(d) At what times does the mass pass through the equilibriumposition moving upward? Moving downward?

(e) What is the velocity of the mass at t 5 3y16 s?

(f) At what times is the velocity zero?

(a) Show that the current i(t) in an L R C-series circuit satisfies

$L \frac{d^{2} i}{d t^{2}} + R \frac{d i}{d t} + \frac{1}{C} i = E^{'} (t)$

where $E^{'} (t)$ denotes the derivative of E(t).

(b) Two initial conditions i(0) and $i^{'} (0)$ can be specified for the DE in part (a). If $i (0) = i_{0}$ and, $q (0) = q_{0}$ what is $i^{'} (0)$ ?

Temperature in a Sphere :

Consider two concentric spheres of radius $r = a$ and $r = b, a < b .$ . See Figure 5.2.10. The temperature in the region between the sphere is determined from the boundary – value problem

$r \frac{d^{2} u}{d r^{2}} + 2 \frac{d u}{d r} = 0, u (a) = u_{0}, u (b) = u_{1},$

Where $u_{0}$ and are constants. Solve for .

See all solutions

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