1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Systems of Equations 485

function. Dividing by mf(t) and letting y 1 = y and y 2 = y', we can rewrite

(9) as the following equivalent first-order system

(10)

I f'(t) C g , f(t)

Y2 = -(2 f(t) + m )y2 - f(t) sm Y1 + mf(t).

s

I

I

I

I

Vertical 1

I

y

e (t)

~

bob of

massm

Figure 10.24 Variable Length Pendulum

Foucault Pendulum The French astronomer and physicist, Jean Bernard

Leon Foucault is, perhaps, best remembered as the first person to demonstrate

the rotation of the earth without using a point of reference outside the earth-

such as stars or the sun. In 1851, Foucault suspended a 62 pound ball on a

220 feet long steel wire from the dome of the Pantheon in Paris. A pin

protruded from the bottom of the ball and was adjusted to draw a mark

through a circle of wet sand beneath the ball. Foucault pulled the ball to

one side of the circle of sand and released it. With each swing the pendulum

made a mark in the sand and appeared to rotate in a clockwise direction.

(The pendulum only appears to change its plane of oscillation while swinging.

It is actually the floor under the pendulum that moves counterclockwise due

to the rotation of the earth.) Thus, Foucault demonstrated his prediction

that the pendulum would revolve approximately 270° in a 24 hour period and

proved that the earth revolved. Foucault presented an intuitive explanation

for the motion of his pendulum.

When damping is absent or compensated for, the equations of motion of a

Foucault pendulum are

x^11 = 2wy' sin </> - g:

(11)

y II = -^2 WX /·,1, Slll<y--gy

f

where w = 7.29 x 10-^5 rads/sec is the angular velocity of the earth's rotation,

</>is the latitude of the pendulum, g = 9.8 m/sec^2 is the gravitational constant,