tre were 39.14 in. below its point of support, it
would vibrate once per second. This, however, is not the case. For as
the pendulum swings, the ball also tends to turn in space to and fro
round a horizontal axis perpendicular to the direction of its motion.
Hence the force stored up in the pendulum is expended, not only in
making it swing, but also in causing the ball to oscillate to and fro
through a small angle about a horizontal axis. We have therefore to
consider not merely the vibrations of the rod, but the oscillations of
the bob. The moment of the momentum of the system round the point of
suspension, called its moment of inertia, is composed of the sum of
the mass of each particle multiplied into the square of its distance
from the axis of rotation. Hence the moment of inertia of the body I
= [Sigma](ma^2). If k be defined by the relation [Sigma](ma^2) =
[Sigma](m) X k^2, then k is called the radius of gyration. If k be the
radius of gyration of a bob round a horizontal axis through its centre
of gravity, h the distance of its centre of gravity below its point of
suspension, and k' the radius of gyration of the bob round the centre
of suspension, then k'^2 = h^2 + k^2. If l be the length of a simple
pendulum that oscillates in the same time, then lh = k'^2 = h^2 + k^2.
Now k can be calculated if we know the form of the bob, and l is the
length of the simple pendulum = 39.14 in.; hence h, the distance of
the centre of gravity of the bob below the point of suspension, can be
found.

In an ordinary pendulum, with a thin rod and a bob, this distance h is
not very different from the theoretical length, l = 39.14 in., of a
simple theoretical pendulum in which the rod has no weight and the bob
is only a single heavy point. For the effect of the weight of the rod
is to throw the centre of oscillation a little above the centre of
gravity of the bob, while the effect of the size of the bob is to
throw the centre of oscillation a little down. In ordinary practice it
is usual to make the pendulum so that the centre of gravity is about
39 in. below the upper free end of the suspension spring and leave the
exact length to be determined by trial.

[Illustration: FIG. 7.--Section of Westminster Clock Pendulum.]

Regulation.

Since T = [pi]sqrt(L/g), we have, by differentiating, dL/L = 2dT/T,
that is, any small percentage of increase in L will corresp