endulum swings. To
verify this law we suspend another pendulum beside the first, both being
of the same length. If we draw both pendulums aside and then release
them, they swing together and return together. This might have been
expected. But if we draw one pendulum a great deal to one side, and the
other only a little, the two pendulums still swing sympathetically.
This, perhaps, would not have been expected. Try it again, with even a
still greater difference in the arc of vibration, and still we see the
two weights occupy the same time for the swing.

We can vary the experiment in another way. Let us change the weights on
the pendulums, so that they are of unequal size, though both of iron.
Shall we find any difference in the periods of vibration? We try again:
the period is the same as before; swing them through different arcs,
large or small, the period is still the same. But it may be said that
this is due to the fact that both weights are of the same material. Try
it again, using a leaden weight instead of one of the iron weights; the
result is identical. Even with a ball of wood the period of oscillation
is the same as that of the ball of iron, and this is true no matter what
be the arc through which the vibration takes place.

If, however, we change the _length_ of the wire by which the weight is
supported, then the period will not remain unchanged. This can be very
easily illustrated. Take a short pendulum with a wire only one-fourth of
the length of that of the long one; suspend the two close together, and
compare the periods of vibration of the short pendulum with that of the
long one, and we find that the former has a period only half that of the
latter. We may state the result generally, and say that the time of
vibration of a pendulum is proportional to the square root of its
length. If we quadruple the length of the suspending cord we double the
time of its vibration; if we increase the length of the pendulum
ninefold, we increase its period of vibration threefold.

It is the gravitation of the earth which makes the pendulum swing. The
greater the attraction, the more rapidly will the pendulum oscillate.
This may be easily accounted for. If the earth pulls the weight down
very vigorously, the time will be short; if the power of the earth's
attraction be lessened, then it cannot pull the weight down so quickly,
and the period will be lengthened.

The time of vibration of the pendulum can be determined w