**1. Introduction**

The presently official value of the Newtonian constant *G* is listed in the most recent CODATA report (2014) as 6.674 08 × 10−11 m3 kg−1 s−2, with a quoted relative uncertainty of 4.7 × 10−5, which still makes it the least well known of all constants of nature, despite improvements derived from a flurry of efforts undertaken in the last decades.

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Several different approaches have been followed in the realization of experiments aimed at its determination. A short summary can be found in the introduction of [1], where the experiment illustrated in this chapter was proposed, and for a deeper insight, a well-done recent comprehensive review [2] can be used for reference and comparison. It makes metrological sense to devise different experiments for the purpose, so that the set of possible systematic errors be not the same for all and the risk of undetected coherent biases among various *G* determinations be minimized. While refurbished and modernized versions of the original Cavendish torsion balance are still the most commonly adopted sensing device and at least one of them has demonstrated extremely high accuracy [3], experiments based on other configurations have also been developed, and a few of them have yielded some of the best results to date. The latter include a measurement based on a beam balance [4] and one based on a pair of simple pendulums used in the static mode [5]. Both achieved accuracy in the low 10−5 region. These three determinations of *G* agree within their stated uncertainty and are the most influential in the 2014 CODATA value, which, however, was attributed higher uncertainty due to the excessive disagreement of other results. A coordinate effort is being led by the recently established working group WG13 of UIAP, stimulated by a NIST initiative, aimed at improving the status of *G* metrology. The experiments coordinated in this effort are mainly based on the torsion balance approach because of its favorable S/N ratio, hoping to put to fruition the enormous amount of information on systematics affecting it, with the target of improving accuracy by possibly an order of magnitude. However, other approaches are also encouraged, and experiments based differently are monitored or even supported. The free-fall gravimeter [6–8] still appears very promising due to its unique absence of difficult-to-evaluate systematics, but results are still hard to come by, mainly due to the inherently low S/N ratio of these experiments. The experiment presented in this chapter is supported by NIST through its Precision Measurements Grant Program and is based on the adoption of a pair of simple pendulums as a detection device. The target is the determination of *G* with an accuracy of 10−5. The concept of the experiment has evolved from a pilot experiment carried on at Politecnico di Torino from 1998 to 2005, which used a single pendulum in vacuum and yielded preliminary results at 3% accuracy level [9–11].

the two sources. For example, if time resolution is 1 ns, a 1000 s run allows to determine the relative frequency difference to 10−12, provided its stability is adequate. This means that the two frequencies can wander around in parallel by more than that but their difference should not. This is important in considering the use of pendulum oscillators, because the gravity acceleration g is not constant in time due to a variety of causes, and so will be their frequency, which will then show instabilities not much below the 10−7 level [13]. Nevertheless, since such instabilities affect in a similar way all pendulum oscillators, particularly if they are in the same location, it can be expected to be quite possible that the differential instability of two equal pendulums oscillating not far from each other may be adequate for the projected resolution of

Measuring 'Big G', the Newtonian Constant, with a Frequency Metrology Approach

http://dx.doi.org/10.5772/intechopen.75635

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In **Figure 1**, a sketch is shown of the expected evolution of time delay as the active field mass distribution is shifted back and forth between a geometrical configuration in which it increases the frequency of one pendulum and another antisymmetric one in which it increases the frequency of the other one. Suppose one pendulum is slightly slower than the other one (it always will be the case as two exactly equal lengths are very unlikely and even undesirable to avoid coupling). As time goes by, this slower oscillator will show increasing time delay with respect to the other, as indicated in **Figure 1** by the broken trend line. Now, when its frequency is increased by the field masses, it will get closer to that of the faster one, and its time delay rate of change (*DROC*) will decrease. The opposite will happen when the field masses increase the frequency of the other pendulum. The relevant information in this measurement

are the undisturbed frequencies of slow and fast pendulums (*v*f0 − *v*s0 = ∆*v*<sup>0</sup> > 0),

, as illustrated in **Figure 2**.

(1)

(2)

their difference will be modified by field masses as in Eq. (1) below, when the latter are next

The DROC measurement is performed by measuring the time delay accumulated in *n* periods

Therefore, it turns out that the relationship between measured DROC and actual frequencies

**Figure 1.** Time delay slope changes as field masses are moved back and forth between the two pendulums with repetition

the experiment under discussion.

If *v*s0 and *vf*<sup>0</sup>

is the difference between *DROC*s in the two configurations.

of the slow pendulum and dividing it by *nT*<sup>s</sup>

of the two oscillators is.

period *T*R.

to the slow pendulum, and as in Eq. (2) when they are next to the fast one.
