1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

530 CHAPTER 12 • FOURIER SERIES AND THE LAPLACE TRANSFORM

--------------------------r------

s = U(t)

-~--- ---·-------~·

(a) System in static equilibrium. (b) System in motion.

Figure 12 .17 The spring- mass system.

by

F1 = mg,

where g is the acceleration of gravity. The next force to be considered is the

spring force acting on the mass and directed upward if the spring is stretched

and downward if it is compressed. It obeys Hooke's law,

F2 = ks,

where s is the amount the spring is stretched when s > 0 and is the amount it

is compressed wheu s < O.

When the system is in static equilibrium and the spring is stretched by the

amount so, the resultant of the spring force and the gravitational force is zero,

which is expressed by the equation

mg-kso = 0.

We let s = U (t) denote the displacement from static equilibrium with the pos-

itive s direction pointed downward, as indicated in Figure 12 .17(b), and write

the spring force as

F2 = -k (so+ U (t)] = -ks 0 - kU (t).

The resultant force FR is

FR= Fr + F2 =mg - kso - kU (t) = -kU (t) - (12-16)