   Chapter 17.3, Problem 12E

Chapter
Section
Textbook Problem

Consider a spring subject to a frictional or damping force.(a) In the critically damped case, the motion is given by x = c1er1 + c2tert. Show that the graph of x crosses the t-axis whenever c1 and c2 have opposite signs.(b) In the overdamped case, the motion is given by x = x = c 1 e r 1 t + c 2 e r 2 t where r1 > r2. Determine a condition on the relative magnitudes of c1and c2 under which the graph of x crosses the t-axis at a positive value of t.

To determine

To show: The graph of x crosses the t-axis whenever c1 and c2 have opposite signs.

Explanation

Given data:

The motion is given as follows,

x=c1ert+c2tert

Consider the motion expression as follows.

x=c1ert+c2tert (1)

If x crosses t-axis, then x=0 .

Substitute 0 for x in equation (1),

0=c1ert+c2tertert(c

(b)

To determine

A condition on the relative magnitudes of c1 and c2 under which the graph of x crosses the t-axis at a positive value of t.

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