Kater Pendulum Theoretical Considerations

Randall D. Peters

Department of Physics
Mercer University
Macon, Georgia 31207

1  Idealized Kater Pendulum

Consider a uniform rectangular pendulum as illustrated in Figure 1. One axis of rotation is an end, and we wish to find another axis for which the periods of oscillation about the two parallel axes are the same. Note that the pendulum must be turned upside down to oscillate about axis 2, the adjustable knife edge.

Figure 1: Illustration of a simple Kater pendulum.

For small amplitudes of free decay, the equations of motion are given by Ii d2q/dt2 + MgLiq  =  0; and the moments of inertia are

Ii  =  Ic + MLi2,  i  =  1,2
by use of the parallel axis theorem. If the pendulum length L is considerably greater than its width or thickness, then the center of mass moment is given by Ic  =  ML2/12. Additionally, L1  =  L/2, and L2 is to be determined.

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

Ti  =  2p[Ii/(MgLi)]1/2
It is convenient to define the radius of gyration, k, whereby Ic  =  Mk2. From equations 1 and 2, the condition T1  = T2 requires that
k2  =  L1L2  for period matching
Additionally, since k  =  L/121/2, we see that L2  =  k2/L1  =  L/6. Thus, for the matched condition,
T1  =  T2  =  T  =  2p[(L1 + L2)/g]1/2  =  2p[2L/(3g)]1/2

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

1.1  Variation of Period with Axis Position

It is instructive to consider the pendulum as outfitted with a hypothetical single axis, rather than the two required for operation, and look at the variation of period with the position y of this single axis relative to the center (of mass) of the pendulum, in the range from 0 < y < L/2, as shown in Figure 2.
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/s2.

2  Non-Ideal Features

2.1  Effect of Period Difference

Ultimately, there will always be a measurable difference between the periods,  T1 and T2, but attempts to match them is an exercise that is unnecesary, since the effect of small difference can be factored into one's estimate for the acceleration of gravity  g .

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 T2 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

T2  =  2p(
[ L2
 + ( L

[( L
+D) g]

which is seen to equal T1 when D  =  0.

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

lm  =   2
L + D

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

T2  =  T1 (1 -  3

which combines with (6) to yield

gest  =   4p2lm

The usefulness of (8) derives from the manner in which lm changes with the ratio T2/T1. If T2/T1 > 1 then lm < 2/3L. On the other hand if T2/T1 < 1 then lm > 2/3L. Moreover, Equation (8) provides automatic compensation over a surprisingly large range of deviations in DT = T2-T1, 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 T2-T1 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

dg/g  =  [(dlm/lm)2 + (2dT2/T2)2 + (3dT1/T1)2)]1/2
Weighting by the factor of 3 in the T1 errors (due to the cubic term) requires more careful period measurement as compared to the slider method.

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 T1 and T2 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

3  Practical pendulum

The idealized rectangular pendulum treated above is not practical, since axis engineering has been ignored. Some things that were tried unsuccessfully involved (i) bearings mounted in the pendulum for sharpened pins to ride against, and (ii) small cylinders inserted through the pendulum. The only practical means found for providing axes is the scheme next described, involving a pair of holes drilled through the pendulum.

3.1  Configuration with a pair of holes for a knife edge

Let each hole be of radius r, their centers being located at y1 (positive) above the center of the pendulum, and y2 (negative) below the center. Since y2 is in magnitude less than y1, there is a small negative shift of the center of mass of the pendulum with respect to the center of the rectangle (-0.8 mm for the TEL-Atomic design). The position of the center of mass is given by
yc  =  -p r2(y1 + y2)/(Lw - 2pr2)
where L is the overall length of the pendulum and w is its width. If there were no holes in the pendulum, its mass normalized moment of inertia, with respect to the center of mass (c.m. = geometric center), would be
Ic, ideal  =  (L2 + w2 )/12
where t is the thickness. Although the term involving t  =  0.32 cm is presently ignorable, the one involving w  =  1.3 cm is not (as compared to L  =  37.4 cm). To match the periods about the two axes to a few parts in 10,000 requires the computer. The easiest way to accommodate the holes is to treat them as negative masses. By using this technique and the parallel axis theorem, one obtains the following expression for the nonideal mass normalized c.m. moment:
Ic  =  [(L2+w2)Lw/12+Lwyc2-pr2[(y1-yc)2+(y2-yc)2]-pr4]/(Lw - 2pr2)
To estimate the period for each of the axes, it is necessary to specify the distance of each axis from the center of mass:
d1  =  y1 + r - yc
d2  =  y2 - r - yc
in which care must be taken with regard to algebraic signs. Using then the moments of inertia
I1  =  Ic + d12
I2  =  Ic + d22
the periods are found to be
T1  =  2p[I1/(gd1)]1/2
T2  =  2p[I2/(g(-d2))]1/2

3.2  Design of the TEL-Atomic pendulum

The holes were selected to be 0.25 in diameter, and w  =  1.3 cm so that the amount of side material at the axis-2 position would be adequate for structural integrity. Similar considerations figured into the placement of the axis-1 hole; i.e., the distance from the center of the pendulum was chosen at 18.18 cm to give a 2 mm piece of metal where the knife edge rides.

The length L  =  37.4 cm and the axis-2 hole position of y2  =  -5.995 cm were obtained by programming the computer with the equations given above, and allowing L and y2 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), y2 was allowed to vary quasi-smoothly through a range which permits T1  =  T2. In the calculations, g was set to 9.797, and the matched period condition is one which yields for the estimated acceleration of gravity

g  =  4p2 (d1 - d2)/T2,   T  =  T1  =  T2
with d1  =  18.58 cm, d2  =  -6.232 cm
Note that the nominal simple pendulum equivalent length is d1 - d2  =  lm (Eq.8)  =  24.81 cm. For the indicated parameters, the nominal period values were T1  =  0.9998 s and T2  =  0.9999 s. Attempts to get closer to unity were found to be futile for reasons that probably derive from the brass of the pendulum being less than perfectly homogeneous in terms of mass density. For example, one pendulum to the next typically shows deviations from the nominal periods on the order of 5 parts per 10,000.

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On 19 Nov 1999, 11:23.