1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Linear Equations with Constant Coefficients 285

Since r 1 < 0 and r2 < 0, both er,t and er^2 t are monotone decreasing functions.

Therefore y(t) --> 0 as t --> oo and no oscillation occurs. The three graphs

shown in Figure 6.6 are also typical of the graph of equation (22)- overdamped

motion.

Figure 6.6 Typical Critically Damped and Overdamped Motion

EXAMPLE 1 Equation of Motion of an Undamped Pendulum

An undamped pendulum of length .9 meters (m) with a bob of mass
.2 kilograms (kg) is moved to the left of vertical so that the pendulum m akes
an angle of -.6 radians (rads) with the vertical and is released imparting a
velocity of .3 radians/second (rads/s) to the pendulum.

a. Write the equation of motion for the pendulum. (The expressing "write
the equation of motion for the pendulum" is another way of saying "write the
general solution of the initial value problem for the pendulum.")

b. What is the amplitude, period, and frequency of oscillation?

c. What is the phase angle?

SOLUTION

a. On the earth 's surface g = 9.8 meters/second^2 (m/s^2 ) is the mean gravi-

tational constant. So the pendulum constant is

k =mg/£= (.2 kg)(9.8 m/s^2 )/.9 m = 2.178 kg/s^2.

Since the pendulum is undamped and there is no forcing function, the

initial value problem which we need to solve is

(23) my"+ ky = O; y(O) = - .6 rads, y'(O) = .3 rads/s.