Computerized Cavendish Balance Experiment Computerized Cavendish Balance Experiment

Randall D. Peters
Physics Department
Mercer University
Macon, Georgia 31207

## Abstract

This document summarizes the basic physics involved in estimating the Newtonian constant G. It emphasizes the dynamic method in which the balance is driven at resonance by `swinging the masses' at turning points of the motion.

## 1  Background

As shown in the complete manual for the computerized Cavendish instrument, sold by Tel-Atomic, Inc.; G is estimated (before corrections to account for non-ideal factors like boom mass) using the expression:
 G  =  KqD R2/(2 M m d)
(1)
where M is the mass of each large (perturbing) spherical mass, m is the mass of each small (boom) mass, d is the distance from the rotation axis to the center of the small sphere, R is the distance between the centers of the large and small spheres, and K is the torsion constant of the 25 mm dia tungsten fiber that supports the boom with its pair of small masses (refer to Figure 1 of the manual).

The angle qD is determined by the amount of static twist imparted to the boom, when the large perturbing masses are `swung' from their `zero' (centered) position to their maximum displacement (touching the glass plates of the housing). In moving from zero, the otherwise balanced boom is disturbed from its static equilibrium starting position qe. Because the masses are typically swung in roughly 2 s, and because the period of oscillation of the boom is roughly 250 s, the disturbance represents essentially a Heaviside step function. The resulting response of the boom is a simple harmonic free-decay, in which the mean position of the decay is displaced from the starting position by the angle qD. If the masses had been swung the opposite direction by the same amount, then the final displacement (after transient decay) would be -qD.

## 2  Free-decay

Shown in fig. 1 is an example of free-decay around a mean position of zero. In practice, oscillation about qe » 0 is almost impossible to achieve, within the measurement precision attainable with the instrument. Fortunately, the influence of non-zero qe can be eliminated from the equations used to determine qD and thus estimate G. Details are provided in the manual. Because of the simpler resulting expressions, we here assume it to be zero; and nothing of conceptual significance is lost in so doing.

The example of fig. 1 was generated by modeling the well known case from classical mechanics, of a viscous damped simple harmonic oscillator. Parameters were chosen to approximate the manual case illustrated in Figure B1. In both that document and the present document, all position data are expressed in terms of sensor voltage rather than angle in radians. Because the sensor is essentially linear, the two are proportional through a calibration constant, with a representative value being 45 V/rad.

Fig. 1 Simulation of a viscous damped simple harmonic oscillator.

The only output from the sensor is a voltage corresponding to boom position. Also shown in Fig. 1 is a ``velocity'' curve that has been labeled with quotation marks to draw attention to the fact that it was scaled in magnitude to yield a curve of comparable magnitude to the position curve. It is provided to illustrate phase differences that are important for understanding how to drive the boom at its resonance frequency (later discussion).

### 2.1  Damping coefficient

It is necessary to quantify the amount of damping in the instrument, if the preferred method of measurement is employed. (The reason for preferring a `dynamic' method instead of a `static' method is discussed later.)

The damping coefficient b can be determined from any three adjacent turning points, such as q1, q2, and q3 shown in fig. 1 as follows:

 b  =  - 2 T ln (- q3 - q2 q2 - q1 )
(2)
It is worthy of note that eq. (2) is still valid even if qe is not equal to zero!

Because of measurement errors, accuracy is improved by working with more than three turning points. The total number N employed must be odd, and a near-optimum case for the Cavendish balance has been shown to be N = 11. (Details along with the expression required for using arbitrary N to determine x  =  e(-bT/2) are provided in the manual.)

## 3  Driving at Resonance

Shown in Fig. 2 is the response of the boom that results for one of the two possible cases (resonance and antiresonance)-in which the large masses are swung at each turning point of the motion. The square wave (displacement of position) drive is of amplitude qD, and the oscillatory amplitude is seen to quickly rise above this static value. If the mass swings continue without interruption until steady state is acquired (roughly 20 cycles), then the final amplitude of the boom motion is 13.3 times qD for the particular case shown in which bT  =  0.30. To drive in this manner is useful for `magnifying' the influence of the small gravitational force between the small and large masses; even if only a small number of swings are employed. The amount of magnification is determined by the quality factor Q of the oscillator, which is given by
 Q  = p bT
(3)

Fig. 2 Illustration of `magnification' that results from driving at the resonance frequency.

### 3.1  Amplitude change per half-cycle

From one half-cycle to the next, the amplitude A builds (for Q >> 0.5) in accord with the relationship
 DA A =  - bT 2 + 2qD A
(4)
If there were no qD drive, the fractional change in the amplitude per half cycle would just be -(bT/2), consistent with the exponential decay of fig. 1.

At steady state the energy gained from the drive is just balanced by the energy lost to friction, so that the amplitude for this condition is given by
 Amax  = 4 bT qD  = 4 Q p qD
(5)

### 3.3  Phase relationships

It is seen from Fig.2 that resonance results when the fundamental Fourier component of the square wave drive is in phase with the velocity, which is equivalent to the addition of a `negative' damping term to the equation of motion. This condition is the same as the usual resonance requirement for a harmonic drive; i.e., the drive frequency is the same as the oscillator's natural frequency and leads the oscillator motion by 90 degrees. It should be noted that we are here ignoring the damping `redshift', in which frequencies are altered by the viscous friction. For example, textbooks routinely give the free decay frequency as
 w1  = æÖ w02 - b2)
(6)
where w0® w in the present document. In actuality the difference between w0 and w1 is for virtually all oscillators too small to be measured because of noise. Eq.(1) of the manual retains this formal difference for the sake of theoretical completeness; however, one need not employ the difference in estimating G. For the representative present case of b  =  0.3/(250 s), the damping redshift is only 0.1%, which is much smaller than the 2% precision possible with the instrument.

If the phase were instead lagging the motion by 90 degrees, it would correspond to an antiresonance drive. This would cause the sign of the 2qD/A term in eq. (4) to be negative, with a resulting motion that dies out faster than the free decay case. Accomplished by swinging the masses 180 degrees different in phase from the case shown in Fig. 2, it can be an effective means for bringing the balance more quickly to an operational status.

### 3.4  Observations related to Fourier series

For quality factors significantly greater than critical damping (Q = 0.5) harmonics present in the drive are inconsequential to the instrument's response when driven at resonance.

Since the fundamental component of a square wave is 4/p times the amplitude of the square wave, one sees that equation (5) is consistent with the better known case of a harmonic drive; i.e., the steady state amplitude at resonance in that case is simply Q times the amplitude of the drive.

## 4  Practical Consideration

In practice it is not necessary to drive all the way to steady state to take advantage of the magnification that results from resonant `excitation'. If one has the patience to drive for such a long time, then there is the advantage of a very simple result ; i.e., qD  =  (bT/4) Amax, where Amax is the steady state amplitude in radians. For those with less time and/or patience, the manual provides an equation (eq. 15) that permits one to determine qD using any odd number N > 3 of turning points of the buildup curve corresponding to Fig. 2. As with the free-decay case, this expression is valid for non-zero qe, as long as the equilibrium position is constant.

## 5  Final comment

It should be noted that the resonance drive method for estimating G has advantages beyond just `magnification'. The tungsten fiber that supports the boom with its masses is not an ideal Hooke's law torsional element. In fact, Hooke's law is not to be realized with any real mechanical oscillator. The `jerky' behavior of real systems, because of the Portevin LeChatelier effect; causes a variety of difficulties. The mean position can consequently move about (to include creep). This is cause for greater complications to a static measurement than to a dynamic measurement. The relevant mesoanelastic complexity is too involved to be presently discussed in detail. It is one of the primary reasons that G remains the poorest known of the fundamental constants. For detailed information on how these factors influence damping, the reader is referred to the author's contributions (ch. 20 and 21) of CRC's Vibration and Shock Handbook, ed. Clarence deSilva, 2005.

File translated from TEX by TTH, version 1.95.
On 9 Aug 2005, 08:28.