Page 132 - Schaum's Outline of Differential Equations

CHAP. 14] SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS 115

Hooke's law: The restoring force F of a spring is equal and opposite to the forces applied to the spring and
is proportional to the extension (contraction) I of the spring as a result of the applied force; that is, F = —kl,
where k denotes the constant of proportionality, generally called the spring constant.

Example 14.1. A steel ball weighing 128 Ib is suspended from a spring, whereupon the spring is stretched 2 ft from its
natural length. The applied force responsible for the 2-ft displacement is the weight of the ball, 128 Ib. Thus, F = -128 Ib.
Hooke's law then gives -128 = -k(2), or k = 64 Ib/ft.

For convenience, we choose the downward direction as the positive direction and take the origin to be the
center of gravity of the mass in the equilibrium position. We assume that the mass of the spring is negligible
and can be neglected and that air resistance, when present, is proportional to the velocity of the mass. Thus, at
any time t, there are three forces acting on the system: (1) F(t), measured in the positive direction; (2) a restoring
force given by Hooke's law as F s = -kx, k > 0; and (3) a force due to air resistance given by F a=— ax, a > 0,
where a is the constant of proportionality. Note that the restoring force F s always acts in a direction that will
tend to return the system to the equilibrium position: if the mass is below the equilibrium position, then x is
positive and -kx is negative; whereas if the mass is above the equilibrium position, then x is negative and -kx
is positive. Also note that because a > 0 the force F a due to air resistance acts in the opposite direction of the
velocity and thus tends to retard, or damp, the motion of the mass.
It now follows from Newton's second law (see Chapter 7) that mx = - kx - ax + F(t), or

If the system starts at t = 0 with an initial velocity v 0 and from an initial position x 0, we also have the initial
conditions

(See Problems 14.1-14.10.)
The force of gravity does not explicitly appear in (14.1), but it is present nonetheless. We automatically
compensated for this force by measuring distance from the equilibrium position of the spring. If one wishes to
exhibit gravity explicitly, then distance must be measured from the bottom end of the natural length of the
spring. That is, the motion of a vibrating spring can be given by

if the origin, x = 0, is the terminal point of the unstretched spring before the mass m is attached.

ELECTRICAL CIRCUIT PROBLEMS

The simple electrical circuit shown in Fig. 14-2 consists of a resistor R in ohms; a capacitor C in farads;
an inductor L in henries; and an electromotive force (emf) E(t) in volts, usually a battery or a generator, all

Fig. 14.2