633. It appears that corresponding to each compound pendulum we have a
specific length equal to that of the isochronous simple pendulum. To
take as example the 6' bar already described (Art. 625), this length
is 4'. If I measure off from the edges a distance of 4', and mark this
point upon the bar, the point is called the _centre of oscillation_.
More generally the centre of oscillation is found by drawing a line
equal to the isochronous simple pendulum from the centre of oscillation
through the centre of gravity.
634. In the bar D the centre of oscillation would be at a distance of
20' below the edges; and in general the position will vary with the
position of the edges.
635. In the 6' bar B is the centre of oscillation. I take another pair
of edges and place them on the bar, so that the line of the edges
passes through B. I now lift the bar carefully and turn it upside
down, so that the edges B rest upon the steel plates. In this position
one-third of the bar is above the axis of suspension, and the remaining
two-thirds below. A is of course now at the bottom of the bar, and
is on a level with the ball, C: the pendulum is made to oscillate
about the edges B, and the time of its vibration may be approximately
determined by direct comparison with C, as already explained. I find
that, when I allow C and the bar to swing together, they both vibrate
precisely in the same time. You will remember, that when the ball
was suspended by a string 4' long, its vibrations were isochronous
with those of the bar when suspended from the edges A. Without having
altered C, but having made the bar to vibrate about B, I find that the
time of oscillation of the bar is still equal to that of C. Therefore,
the period of oscillation about A is equal to that about B. Hence, when
the bar is vibrating about B, its centre of oscillation must be 4' from
B, that is, it must be at A: so that when the bar is suspended from A,
the centre of oscillation is B; while, when the bar is suspended from
B, the centre of oscillation is A. This is an interesting dynamical
theorem. It may be more concisely expressed by saying that _the centre
of oscillation and the centre of suspension are reciprocal_.
636. Though the proof that we have given of this curious law applies
only to a uniform bar, yet the law is itself true in general, whatever
be the nature of the compound pendulum.
637. We alluded in the last lecture (Art. 610) to the difficulty of
measuring with accuracy the length of a simple pendulum; but the
reciprocity of the centres of oscillation and suspension, suggested to
the ingenious Captain Kater a method by which this difficulty could be
evaded. We shall explain the principle. Let one pair of edges be at A.
Let the other pair of edges, B, be moved as near as possible to the
centre of oscillation. We can test whether B has been placed correctly:
for the time taken by the pendulum to perform 100 vibrations about A
should be equal to the time taken to perform 100 vibrations about B. If
the times are not quite equal, B must be moved slightly until the times
are properly brought to equality. The length of the isochronous simple
pendulum is then equal to the distance between the edges A and B; and
this distance, from one edge to the other edge, presents none of the
difficulties in its exact measurement which we had before to contend
with: it can be found with precision. Hence, knowing the length of
the pendulum and its time of oscillation, gravity can be found in the
manner already explained (Art. 608).
638. I have adjusted the two edges of the 6' bar as nearly as I could
at the centres of oscillation and suspension, and we shall proceed to
test the correctness of the positions. Mounting the bar first by the
edges at A, I set it vibrating. I take the stop-watch already referred
to (Art. 612), and record the position of its hands. I then place my
finger on the stud, and, just at the moment when the bar is at the
middle of one of its vibrations, I start the watch. I count a hundred
vibrations; and when the pendulum is again at the middle of its stroke,
I stop the watch, and find it records an interval of 110·4 seconds.
Thus the time of one vibration is 1·104 seconds. Reversing the bar, so
that it vibrates about its centre of oscillation B, I now find that
110·0 is the time occupied by one hundred vibrations counted in the
same manner as before; hence 1·100 seconds is the time of one vibration
about B: thus, the periods of the vibrations are very nearly equal, as
they differ only by ¹/₂₅₀th part of a second.
639. It would be difficult to render the times of oscillation exactly
equal by merely altering the position of B. In Kater’s pendulum the
two knife-edges are first placed so that the periods are as nearly
equal as possible. The final adjustments are given by moving a small
sliding-piece on the bar until it is found that the times of vibration
about the two edges are identical. We shall not, however, use this
refinement in a lecture experiment; I shall adopt the mean value of
1·102 seconds. The distance of the knife-edges is about 3'·992; hence
gravity may be found from the expression (Art. 608)
3'·992 × (3·1416 / 1.102)².
The value thus deduced is 32'·4, which is within a small fraction of
the true value.
640. With suitable precautions Kater’s pendulum can be made to give
a very accurate result. It is to be adjusted so that there shall be
no perceptible difference in the number of vibrations in twenty-four
hours, whichever edge be the axis of suspension: the distance between
the edges is then to be measured with the last degree of precision by
comparison with a proper standard.