*Project Gutenberg*. Public domain.

34. *Q.*—By what circumstance is the
velocity of vibration of a pendulous body determined?

*A.*—By the length of the suspending
rod only, or, more correctly, by the distance between
the centre of suspension and the centre of oscillation.
The length of the arc described does not signify, as
the times of vibration will be the same, whether the
arc be the fourth or the four hundredth of a circle,
or at least they will be nearly so, and would be so
exactly, if the curve described were a portion of
a cycloid. In the pendulum of clocks, therefore,
a small arc is preferred, as there is, in that case,
no sensible deviation from the cycloidal curve, but
in other respects the size of the arc does not signify.

35. *Q.*—If then the length of a pendulum
be given, can the number of vibrations in a given
time be determined?

*A.*—Yes; the time of vibration bears
the same relation to the time in which a body would
fall through a space equal to half the length of the
pendulum, that the circumference of a circle bears
to its diameter. The number of vibrations made
in a given time by pendulums of different lengths,
is inversely as the square roots of their lengths.

36. *Q.*—Then when the length of the
second’s pendulum is known the proper length
of a pendulum to make any given number of vibrations
in the minute can readily be computed?

*A.*—Yes; the length of the second’s
pendulum being known, the length of another pendulum,
required to perform any given number of vibrations
in the minute, may be obtained by the following rule:
multiply the square root of the given length by 60,
and divide the product by the given number of vibrations
per minute; the square of the quotient is the length
of pendulum required. Thus if the length of a
pendulum were required that would make 70 vibrations
per minute in the latitude of London, then SQRT(39.1393)
x 60/70 = (5.363)^2 = 28.75 in. which is the length
required.

37. *Q.*—Can you explain how it comes
that the length of a pendulum determines the number
of vibrations it makes in a given time?

*A.*—Because the length of the pendulum
determines the steepness of the circle in which the
body moves, and it is obvious, that a body will descend
more rapidly over a steep inclined plane, or a steep
arc of a circle, than over one in which there is but
a slight inclination. The impelling force is
gravity, which urges the body with a force proportionate
to the distance descended, and if the velocity due
to the descent of a body through a given height be
spread over a great horizontal distance, the speed
of the body must be slow in proportion to the greatness
of that distance. It is clear, therefore, that
as the length of the pendulum determines the steepness
of the arc, it must also determine the velocity of
vibration.