the geometry of which is shown in Fig. 1; where the hole nearest an end provides for rotation about axis-1, and the other hole (approximately

Figure 1: Geometry of the TEL-Atomic Kater pendulum.

A photograph of the pendulum mounted to swing about axis-1 is provided
in Fig. 2. Similarly, Fig. 3 is a photograph corresponding to motion about
axis-2. Also shown and identified with labels in Figures 2 and 3
are various support items necessary to do the experiment, such as the Timer.
A variety of timers may be used, depending
on the mode of operation. For example, software of the **KIS Interface**
provides computer generated mean and standard deviation for a data set.
In choosing a timer, it is *important* to insure that it be
accurate (stable) to about 0.01%, which requires that the timer operate
with a quartz crystal oscillator.

Figure 2: Pendulum mounted to swing about axis-1.

Figure 3: Pendulum mounted to swing about axis-2.

Check for acceptability of the knife edge orientation as follows. With a finger, initiate motion by displacing the pendulum from equilibrium by an amount that is large compared to the amount used for data collection (discussed later). When so doing, purposely include a small out of plane component. If the resulting out-of-plane rocking motion is fairly symmetric between right and left turnings, and if this motion dampens quickly compared to the desired planar motion; then the knife edge should be sufficiently level.

Figure 4: Positioning the photogate relative to the pendulum.

Once the photogate is properly aligned, no further adjustments are necessary-except for slider positioning- as the pendulum is swung first about one axis and then the other.

The quick method uses the pendulum without a slider. Thus there is only
a single mass configuration of the instrument. In turn, there is a single
estimated period, T_{1} associated with axis-1 and likewise T_{2} for axis-2.
One must carefully estimate each of these periods by collecting enough
data to reduce the influence of random errors.

The preferred method collects a different T_{1} and T_{2} for each position
of the slider. If every possible position of the slider that does not
interfere with the photogate is investigated, then about 30 different data
sets result for each of T_{1} and T_{2} (total of 60 sets). Because a
regression analysis is performed with the mean of each set, fewer period
measurements per slider position are required, as compared to the
quick method. *Although a full 60 data sets provide better appreciation
for the physics involved, the minimum number of slider positions necessary
for an accurate estimate of g can be as few as four or five. Thus the
total number of individual period measurements can be less than 200.*

To estimate the acceleration of gravity, use the formula (see Appendix Eq. 8)

| (1) |

| (2) |

If measurements over a large range of
slider positions are performed, a graph of the two periods
versus slider position should
look like that provided in Figure 5.

Figure 5: Variation of axis-1 and axis-2 Period over the full
range of Slider Positions.

Of course, such a wide range of slider positions is not necessary to
estimate g. As previously mentioned, only four or five positions are
really required. They must be chosen, however, to include the
position for which T_{1} = T_{2}. One does not actually attempt to
place the slider at the exact position of equality; rather a graph
such as that of Figure 6 is produced.

Figure 6: Period versus Slider position in the vicinity of period
matching.

Here, using Excel, a linear regression has
been separately performed on each of the T_{1} data and the T_{2} data.
Individual points correspond to the mean value of the period at a
given slider position.
From the regression analyses, the period T corresponding to
T_{1} = T_{2} = T is calculated (using the value of Slider position
which yields equality). Then the estimate for g, along
with its uncertainty is obtained from the following equations.

| (3) |

| (4) |

File translated from T

On 19 Oct 1999, 15:41.