文档介绍：Section
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Applications: Pendulums
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In this section we will investigate two applications of our work in Section . First, we
will consider the motion of a pendulum, a problem originally mentioned in Section
in connection with the trigonometric functions. Second, we will discuss the motion of an
object vibrating at the end of a spring.
The motion of a pendulum
Consider a pendulum consisting of a bob of mass m at the end of a rigid rod of length
b. We will assume that the mass of the rod is negligible parison with the mass of
the bob. Let x(t) be the angle between the rod and the vertical at time t, with x(t) > 0
for angles measured in the counterclockwise direction and x(t) < 0 for angles measured
in the clockwise direction. See Figure . Suppose the bob is pulled through an angle
α and then released. That is, suppose our initial conditions are x(0) = α andx ˙(0) = 0.
If we view the motion of the pendulum in plex plane, with the real axis vertical,
positive direction downward, and the imaginary axis horizontal, positive direction to the
right, then the position of the bob at time t is given by
z(t) = beix(t). ()
b
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Figure A pendulum
Then we have
z˙= ibxe˙ ix ()
and
z¨ = −bx˙ 2eix + ibxe¨ ix
= −bx˙ 2(cos(x) + i sin(x)) + ibx¨(cos(x) + i sin(x)) ()
= (−bx˙ 2 cos(x) − bx¨ sin(x)) + i(−bx˙ 2 sin(x) + bx¨ cos(x)).
1
2 Applications: Pendulums and Mass-Spring Systems Section
Nowz ¨ is the acceleration of the pendulum, and so mz¨ must be equal to the force of gravity
acting on the bob, namely, a force of magnitude mg acting in the downward direction, the
direction of the positive real axis. Hence we must have g =z ¨, that is,
g = (−bx˙ 2 cos(x) − bx¨ sin(x)) + i(−bx˙ 2 sin(x) + bx¨ cos(x)). ()
Equating the real and imaginary parts of the tw