Department of Physics

Mercer University

Macon, Georgia 31207

Figure 1: Illustration of a simple Kater pendulum.

For small amplitudes of free decay,
the equations of motion are given by I_{i} d^{2}q/dt^{2} + MgL_{i}q = 0;
and the moments of inertia are

| (1) |

From the solution to the equation of motion, the periods are found to be

| (2) |

| (3) |

| (4) |

We thus see that the pendulum, for the matched period condition, is equivalent to a simple pendulum of length 2L/3.

Figure 2: Period vs axis position for a pendulum of Figure 1 type.

The first thing one should note from Figure 2 is that there are two cardinal
axis positions-one at ^{L}/_{2} and the one which yields the same period
at ^{L}/_{6}. The length of the pendulum for Figure 2 was chosen at
L = 37.25 cm to yield a 1 s period when g = 9.803 m/s^{2}.

Imagine that the position of pivot 2
could be changed by a small amount, D, from
its nominal ^{L}/_{6} from the center of the pendulum.
(In our initial consideration of the influence of departures from ideal
geometry, the offset distance of pivot 1 from the end of
the pendulum will be ignored-a complication which will be addressed later.)
The dependence of T_{2} on D is determined by two factors: (i) the
moment arm of the gravitational force and (ii) the change in the moment of
inertia about the pivot (readily calculated using the parallel axis theorem).
The period is found to obey

| (5) |

which is seen to equal T_{1} when D = 0.

Assume that one makes an accurate measurement of the separation distance between the two pivots

| (6) |

By expanding the square root term in (5) with the binomial theorem and retaining terms only to first order in D, one obtains

| (7) |

which combines with (6) to yield

| (8) |

The usefulness of (8) derives from the manner in which *l*_{m} changes with
the ratio T_{2}/T_{1}.
If T_{2}/T_{1} > 1 then *l*_{m} < ^{2}/_{3}L. On the other hand if
T_{2}/T_{1} < 1 then *l*_{m} > ^{2}/_{3}L. Moreover,
Equation (8) provides automatic compensation over a surprisingly large
range of deviations in DT = T_{2}-T_{1}, for reasons that can be
appreciated qualitatively
by studying Figure 2.
A quantitative indicator of the compensation of Eq. 8 is
provided in Figure 3.

Figure 3: Errors of Eq. 8

To generate the experimental results, a very crude
pendulum was cut out of aluminum with a hacksaw to the approximate size of
the TEL-Atomic instrument. An end 1/4 in hole was drilled for axis-1, and
7 axis-2 holes were drilled at various positions around the 2L/3 point. The
total range of distances between the axes was 5.7 cm. In this range, the
period difference
T_{2}-T_{1} varied from -58 ms at 28.2 cm separation to 97 ms at 22.5 cm
separation of the axes.
Eq. 8 is not capable of fully describing this
system because it corresponds to a 2-hole pendulum rather than the 7-hole
pendulum tested. Nevertheless the equation fits the data reasonably well
as shown in Figure 3. Interestingly, over the full 5.7 cm range, the largest
error in the g estimate does not exceed 8%. Moreover, for a certain 3 cm
range (favoring smaller as opposed to larger than nominal separations), the
error does not exceed 1%. For tolerances on hole placement held to
approximately 1 mm, Eq. 8 results in errors not to exceed 1 part per 10,000.
It is worth repeating-errors in the placement
of the axis-2 hole are less significant if its placement is shy of
(as opposed to greater than) the
L/6 nominal distance from the center.

One must recognize a shortcoming to the utility of Eq. 8 as compared to the more elaborate method of using a small mass slider as described in the operational part of this manual. The relative uncertainty in the estimate of g, when using eq. 8 is given by

| (9) |

In spite of the extra contribution to the uncertainty in g, Eq. 8 is
still quite useful. For example, it was used with the data obtained
from the TEL-Atomic pendulum employing a 2.6 g slider. The errors of
Fig. 4 derive from the calculation of g by means of quadratic
fits to the T_{1} and T_{2} data. They are seen to be less than one
part per thousand for every position of the slider between 0 and 30 cm.

Figure 4: Errors in g estimate using Eq. 8 on TEL-Atomic pend. with slider

| (10) |

| (11) |

| (12) |

| (13) |

| (14) |

| (15) |

The length L = 37.4 cm and the axis-2 hole position of y_{2} = -5.995 cm
were obtained by programming the computer
with the equations given above, and allowing L and y_{2} to vary.
The objective was to select for these values a matched
condition corresponding to a period of 1 s.
For a small set of L values (expected from idealized pendulum
considerations to be in the neighborhood of 37 1/2 cm), y_{2} was
allowed to vary quasi-smoothly through a range which permits T_{1} = T_{2}.
In the calculations, g was set to 9.797, and
the matched period condition is one which yields for the estimated
acceleration of gravity

| (16) |

File translated from T

On 19 Nov 1999, 11:23.