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Linear and Non Linear Effects on the Newtonian Gravitational Constant as deduced from the Torsion Balance

M. Rossi Dipartimento di Matematica,
Università degli Studi di Torino
Via Carlo Alberto 10,
10123 Torino, Italy
   L. Zaninetti Dipartimento di Fisica Generale,
Università degli Studi di Torino
Via Pietro Giuria 1,
I-10125 Torino, Italy
phone +390116707460, fax +390116699579
(Day Month Year; Day Month Year)
Abstract

The Newtonian gravitational constant has still 150 parts per million of uncertainty. This paper examines the linear and nonlinear equations governing the rotational dynamics of the torsion gravitational balance. A nonlinear effect modifying the oscillation period of the torsion gravitational balance is carefully explored.

keywords:
Experimental studies of gravity ; Determination of fundamental constants;
{history}
\ccode

PACS numbers: 04.80.-y ; 06.20.Jr;

1 Introduction

After many years of measurements, begun by H. Cavendish, the Newtonian gravitational constant value, said GG , is still affected by a large error [1]. The CODATA recommends G=(6.6742±0.001)×1011m3kgs2G=(6.6742\pm 0.001)\times 10^{-11}\frac{m^{3}}{kgs^{2}}, meaning a relative standard uncertainty of 150 parts per million (in the following ppm) [2]. This value has had recent confirmation by means, on one hand, of a super-conducting gravimeter [3] and, on the other hand, of a careful analysis of the possible beam balance nonlinearity [4]. The original Cavendish method of measure, employing the torsion balance, still reveals a large discrepancy from the recommended value: about 500 ppm [5] . This is probably due to imperfections of the crystalline structure of the torsion fibre [6, 7, 8]. Moreover recent studies point out that the period of a torsion pendulum might vary under disturbances of environmental noise factors, see see [9]. Other authors suggest a possible deviation from Newton’s law specified as an additional contribution of Yukawa potential type  [10]. This paper first analyses the linear and non linear equations governing the torsional balance rotational dynamics (Section 2). By means of a gravitational torsion balance, same values of GG are obtained and summarised (Section 3). The oscillation period’s variation, due to a non linear effect, is then discussed (Section 4).

2 The basic equations

The form of Newton’s law of gravitation is

F=GmMr2,F=G\frac{mM}{r^{2}}\quad, (1)

where GG is the gravitational constant , MM the great mass , mm the small mass and rr their relative distance. The Leybold balance represents a widespread instrument to determine the constant GG, see Figure 1,

Refer to caption
Figure 1: Schematic view of the torsional balance

and is constructed with the following components.

  1. 1.

    A freely oscillating horizontal bar , of length 2d2d, holding two small lead balls of mass mm as in Figure 1 supported by a torsion fibre that has a torsional constant τ\tau .

  2. 2.

    Two larger balls of mass MM that can be positioned next to the small balls as in Figure 1. The center of mass of the two mm and MM are supposed to be all on a plane perpendicular to the fibre.

  3. 3.

    A luminous source directed toward the center of mass of the bar where is reflected by a mirror.

  4. 4.

    A scale at distance l where the reflected light beam is measured.

Thus the equilibrium position about which the pendulum oscillates is different for the two positions and it is this difference which we use to determine G. The Figure 2 reports a plot of the motion.

Refer to caption
Figure 2: Top view of the Cavendish balance

The moment of inertia of the bar,II is

I2md2.I\sim 2md^{2}\quad. (2)

The fundamental equation of rotational dynamics is

Iφ..=Mg+Mv+Mt,I\stackrel{{\scriptstyle..}}{{\varphi}}=M_{g}+M_{v}+M_{t}\quad, (3)

where

Mv=βφ.,M_{v}=-\beta\stackrel{{\scriptstyle.}}{{\varphi}}\quad, (4)
Mt=τφ,M_{t}=-\tau\varphi\quad, (5)

here β\beta is the coefficient of viscosity of air , φ\varphi the angle between bar and bar itself when the torque is zero. This angle is measured in the anti clockwise direction. The term MgM_{g} represents the torque of the gravitational forces. From equation (1)({\ref{newton}}) , we obtain

Mg=2dGF(φ),M_{g}=2dGF(\varphi)\quad, (6)

where FF is a function of the angle φ\varphi. The law of dependence of FF with φ\varphi is complex and will here be analysed. When the motion starts the resulting force is

F(φ)=F1(φ)F2(φ),F(\varphi)=F_{1}(\varphi)-F_{2}(\varphi)\quad, (7)

where

F1(φ)=mMcosφu12,F_{1}(\varphi)=mM\frac{\cos\varphi}{u_{1}^{2}}\quad, (8)
F2(φ)=mMcos(π2(φ+α))u2=mMsin(φ+α)u2,F_{2}(\varphi)=mM\frac{\cos(\frac{\pi}{2}-(\varphi+\alpha))}{u^{2}}=mM\frac{\sin(\varphi+\alpha)}{u^{2}}\quad, (9)

now α:=arcsin(rsu)\alpha:=\arcsin(\frac{r-s}{u}) (where rr e ss are defined as in Figure 1). With our data , see Section 3 the maximum angular excursion of the angle φ\varphi is

Δ=arcsin(xmaxxmin2l).\Delta=\arcsin(\frac{x_{\max}-x_{\min}}{2l})\quad. (10)

The angle φ\varphi has a low values, see Table 1, and a Taylor series expansion that keep terms to order φ2\varphi^{2} will be adopted. This means to forget quantities less than 3 10510^{-5}. The series representation gives

u1=(r+s)2+(ddcosφ)2r(1+φ(2dr+d2r2φ+d24r2φ3))12.u_{1}=\sqrt{(r+s)^{2}+(d-d\cos\varphi)^{2}}\sim r(1+\varphi(\frac{2d}{r}+\frac{d^{2}}{r^{2}}\varphi+\frac{d^{2}}{4r^{2}}\varphi^{3}))^{\frac{1}{2}}\quad. (11)

Developing the last term with a Maclaurin series we obtain

u1r(1φ2(2dr+d2r2φ)d22r2φ2+O(φ3))r+dφ.u_{1}\sim r(1-\frac{\varphi}{2}(\frac{2d}{r}+\frac{d^{2}}{r^{2}}\varphi)-\frac{d^{2}}{2r^{2}}\varphi^{2}+O(\varphi^{3}))\sim r+d\varphi\quad. (12)

As a consequence

F1mM1φ22(r+dφ)2,F_{1}\sim mM\frac{1-\frac{\varphi^{2}}{2}}{(r+d\varphi)^{2}}\quad, (13)

and expanding the denominator we obtain

(r+dφ)2=(r22dr3φ+3d2r4φ2+O(φ3)),\left(r+d\varphi\right)^{-2}=({r}^{-2}-2\,{\frac{d}{{r}^{3}}}\varphi+3\,{\frac{{d}^{2}}{{r}^{4}}}{\varphi}^{2}+O\left({\varphi}^{3}\right))\quad, (14)

that means

F1mMr2(12drφ+(3d2r212)φ2).F_{1}\sim\frac{mM}{r^{2}}(1-\frac{2d}{r}\varphi+(\frac{3d^{2}}{r^{2}}-\frac{1}{2})\varphi^{2})\quad. (15)

An expression for F2F_{2} can be obtained from equation (9)({\ref{F2}})

F2=mMsinφcosα+cosφsinαu2mMr+dφr2φ2(d2+r22rdφd2φ2)32.F_{2}=mM\frac{\sin\varphi\cos\alpha+\cos\varphi\sin\alpha}{u^{2}}\sim mM\frac{r+d\varphi-\frac{r}{2}\varphi^{2}}{(d^{2}+r^{2}-2rd\varphi-d^{2}\varphi^{2})^{\frac{3}{2}}}\quad. (16)

On Taylor expanding the denominator

(4d2+r22rdφd2φ2)32=(4d2+r2)32(1+3dr4d2+r2φ+3d2(4d2+5r2)2(4d2+r2)2φ2+O(φ3)),(4d^{2}+r^{2}-2rd\varphi-d^{2}\varphi^{2})^{-\frac{3}{2}}=(4d^{2}+r^{2})^{-\frac{3}{2}}(1+\frac{3dr}{4d^{2}+r^{2}}\varphi+\frac{3d^{2}(4d^{2}+5r^{2})}{2(4d^{2}+r^{2})^{2}}\varphi^{2}+O(\varphi^{3}))\quad, (17)

and therefore

F2r+4d(d2+r2)4d2+r2φ+r(20d4+13d2r2r4)2(4d2+r2)2φ2(4d2+r2)32.F_{2}\sim\frac{r+\frac{4d(d^{2}+r^{2})}{4d^{2}+r^{2}}\varphi+\frac{r(20d^{4}+13d^{2}r^{2}-r^{4})}{2(4d^{2}+r^{2})^{2}}\varphi^{2}}{(4d^{2}+r^{2})^{\frac{3}{2}}}\quad. (18)

Now FF from equation (7)({\ref{F}}) can be expressed like a Taylor expansion truncated at O(φ3)O(\varphi^{3})

FA0+A1φ+A2φ2,F\sim A_{0}+A_{1}\varphi+A_{2}\varphi^{2}\quad, (19)

where

A0:=mM(1r2r(4d2+r2)32),A_{0}:=mM(\frac{1}{r^{2}}-\frac{r}{(4d^{2}+r^{2})^{\frac{3}{2}}})\quad, (20)
A1=2mM(dr3+2d(d2+r2)(4d2+r2)52),A_{1}=-2mM(\frac{d}{r^{3}}+\frac{2d(d^{2}+r^{2})}{(4d^{2}+r^{2})^{\frac{5}{2}}})\quad, (21)
A2=3mMd2r41/2mMr2+1/2mMr(4d410r2d2+r4)(4d2+r2)7/2.A_{2}=3\,{\frac{mM{d}^{2}}{{r}^{4}}}-1/2\,{\frac{mM}{{r}^{2}}}+1/2\,{\frac{mMr\left(4\,{d}^{4}-10\,{r}^{2}{d}^{2}+{r}^{4}\right)}{\left(4\,{d}^{2}+{r}^{2}\right)^{7/2}}}\quad. (22)

Three methods that allow to obtain an expression for GG in terms of measurable quantities are now introduced. Further on the well known formula for G extracted from the Leybold manual is reviewed.

2.1 Averaged G

Let F(φ)=F¯F(\varphi)=\overline{F} for all the experience ; in first approximation we may assume that FF is given by equation (19)({\ref{svF}}) to first order

F¯=A0+A1φ¯,\overline{F}=A_{0}+A_{1}\overline{\varphi}\quad, (23)

where φ¯\overline{\varphi} is the average of the values that φ\varphi assumes between the first position , P1P_{1}, and the last position ,PP_{\infty}, of the balance ; A0,A1A_{0},A_{1} are given by equations ( 20) and (21) . The differential equation that describes the motion is

Iφ..+βφ.+τφ=2dGF¯,I\stackrel{{\scriptstyle..}}{{\varphi}}+\beta\stackrel{{\scriptstyle.}}{{\varphi}}+\tau\varphi=2dG\overline{F}\quad, (24)

and it’s solution is

φ(t)=ceδtcos(ωt+ϕ)+2dGF¯τ,\varphi(t)=ce^{-\delta t}\cos(\omega t+\phi)+\frac{2dG\overline{F}}{\tau}\quad, (25)

where cc represents the amplitude and

δ:=β2I,\delta:=-\frac{\beta}{2I}\quad, (26)
ω:=4Iτβ22I.\omega:=\frac{\sqrt{4I\tau-\beta^{2}}}{2I}\quad. (27)

The angle φ\varphi_{\infty}, that represents the bar position at the end of the phenomena PP_{\infty} can be determined as follows

φxxo2l=xx14l,\varphi_{\infty}\sim\frac{x_{\infty}-x^{o}}{2l}=\frac{x_{\infty}-x_{1}}{4l}\quad, (28)

and should be the same as predicted by the theory

limt+φ(t)=2dGF¯τ;\lim_{t\rightarrow+\infty}\varphi(t)=\frac{2dG\overline{F}}{\tau}\quad; (29)

therefore

G=τφ2dF¯.G=\frac{\tau\varphi_{\infty}}{2d\overline{F}}\quad. (30)

In order to continue a value for τ\tau should be derived. This can be obtained from the period of oscillation of the bar

T=2πω=4πI4Iτβ2.T=\frac{2\pi}{\omega}=\frac{4\pi I}{\sqrt{4I\tau-\beta^{2}}}\quad. (31)

We continue by identifying TT with the empirical value T¯\overline{T}. We continue on assuming that β24I\frac{\beta^{2}}{4I} is small ; therefore from equations (2)({\ref{In}}), (30)({\ref{Gprimo}}) and (23)({\ref{Fmedio}}), the following is obtained

G=2π2Iφd(A0+A1φ¯)T¯2.G=\frac{2\pi^{2}I\varphi_{\infty}}{d(A_{0}+A_{1}\overline{\varphi}){\overline{T}}^{2}}\quad. (32)

2.2 G with air viscosity

From formula (4) is possible to deduce the viscosity of the air once the coefficient of damping δ\delta is known, see Section 3 on data analysis. From formula (30) and (26) we should add to the value of GG reported in equation (32)

Gβ:=β2φ8IdF¯,G_{\beta}:=\frac{\beta^{2}\varphi_{\infty}}{8Id\overline{F}}\quad, (33)

obtaining

G=2π2Iφd(A0+A1φ¯)T¯2+β2φ8Id(A0+A1φ¯).G=\frac{2\pi^{2}I\varphi_{\infty}}{d(A_{0}+A_{1}\overline{\varphi}){\overline{T}}^{2}}+\frac{\beta^{2}\varphi_{\infty}}{8Id(A_{0}+A_{1}\overline{\varphi})}\quad. (34)

2.3 G to the first order

Let assume that F(φ)F(\varphi) is not constant, we can assume at the order O(φ2)O(\varphi^{2}) with the aid of formula (19)

FA0+A1φ,F\sim A_{0}+A_{1}\varphi\quad, (35)

where A0A_{0} e A1A_{1} are defined in equations (20)({\ref{A0}}) and (21)({\ref{A1}}) respectively. In this case the law of motion is still equation (3)({\ref{eqcard}})

Iφ..+βφ.+(τ2dGA1)φ=2dGA0,I\stackrel{{\scriptstyle..}}{{\varphi}}+\beta\stackrel{{\scriptstyle.}}{{\varphi}}+(\tau-2dGA_{1})\varphi=2dGA_{0}\quad, (36)

and the solution is

φ(t)=ceδtcos(ωt+ϕ)+2dGA0τ2dGA1,\varphi(t)=ce^{-\delta t}\cos(\omega^{\prime}t+\phi)+\frac{2dGA_{0}}{\tau-2dGA_{1}}\quad, (37)

where the angular velocity ω\omega^{\prime} has now the following expression

ω:=4I(τ2dGA1)β22I.\omega^{\prime}:=\frac{\sqrt{4I(\tau-2dGA_{1})-\beta^{2}}}{2I}\quad. (38)

As a consequence

τ=4π2IT¯2+2dGA1+β24I,\tau=\frac{4\pi^{2}I}{\overline{T}^{2}}+2dGA_{1}+\frac{\beta^{2}}{4I}\quad, (39)
φ=limt+φ(t)=2dGA0τ2dGA1,\varphi_{\infty}=\lim_{t\rightarrow+\infty}\varphi(t)=\frac{2dGA_{0}}{\tau-2dGA_{1}}\quad, (40)

and therefore

G=τφ2d(A0+A1φ).G=\frac{\tau\varphi_{\infty}}{2d(A_{0}+A_{1}\varphi_{\infty})}\quad. (41)

Once equation (39)({\ref{tors2}}) is substituted in this relationship we obtain

G=2π2IφdA0T¯2+β2φ8dIA0.G=\frac{2\pi^{2}I\varphi_{\infty}}{dA_{0}{\overline{T}}^{2}}+\frac{\beta^{2}\varphi_{\infty}}{8dIA_{0}}\quad. (42)

2.4 G from Leybold manual

The deduction of G through the Leybold torsional balance is widely known , see [11]. We simply report the final expression

G=π2b2dΔSMT2l×(1+β),G={\pi^{2}b^{2}d\Delta S\over{MT^{2}l}}\times(1+\beta)\quad, (43)

where

β=b3(b2+4d2)b2+4d2.\beta={\displaystyle b^{3}\over\displaystyle{(b^{2}+4d^{2})\sqrt{b^{2}+4d^{2}}}}\quad. (44)

The meaning of the symbols is

  • bb:Distance between center of the great mass and small mass

  • Δ\Delta S:Total deflection of the light spot

  • dd:The length of the lever arm

  • ll:Distance between mirror and screen

  • MM: Great mass

  • T Period of the oscillations

3 Analysis of the data

The physical parameters as well their uncertainties are reported in Table 1.

Table 1: Parameters of the torsion balance.
parameter value unit
MM (1.5 ±\pm 10310^{-3}) KgKg
mm (1.5102\cdot 10^{-2} ±\pm 10310^{-3}) KgKg
rr (4.65102\cdot 10^{-2} ±\pm 10310^{-3}) mm
dd (5.0102\cdot 10^{-2} ±\pm 10310^{-3}) mm
ll (5.475 ±\pm 10310^{-3}) mm
φ\varphi (-6.715103\cdot 10^{-3} ±\pm 1.3105\cdot 10^{-5}) radrad
β\beta ( 1.432107\cdot 10^{-7} ±\pm 1.11108\cdot 10^{-8}) kgm2s\frac{kg\cdot m^{2}}{s}

The data were analysed through the following fitting function

y(t)=A0+A1cos(2πtT)exp(tτ).y(t)=A_{0}+A_{1}cos(\frac{2\pi t}{T})\exp(-\frac{t}{\tau})\quad. (45)

The data has been processed through the Levenberg–Marquardt method ( subroutine MRQMIN in [12]) in order to find the parameters A0A_{0},A1A_{1} ,TT and τ\tau. The results are reported in Table 2 together with the derived quantities.

Table 2: Parameters of the nonlinear fit.
parameter value unit
A0A_{0} (8.188102\cdot 10^{-2} ±\pm1.11104\cdot 10^{-4}) mm
A1A_{1} ( 0.1470 ±\pm 2.851042.85\cdot 10^{-4}) mm
TT (552.98 ±\pm 0.16) ss
τ\tau (1047.0 ±\pm 4.8) ss

The value of GG can be derived coupling the basic parameters of the torsion balance , see Table 1, and the measured parameters of the damped oscillations , see Table 2. Table 3 reports the four values of GG here considered with the uncertainties expressed in absolute value and in ppm ; the precision of the measure in respect of the so called ”true” value is also reported. A considerable source of error is the uncertainty in the determination of the span between the two spheres that in our case is 103m\approx 10^{-3}m . Adopting a rotating gauge method [13] the uncertainty in the determination of the span between the two spheres is 0.5×106m\approx 0.5\times 10^{-6}m; this is the way to lower the uncertainty in Table 3.

Table 3: Values of GG
method equation value uncertainty [ppm] accuracy %\%]
GG averaged (32) (6.67 ±\pm 0.34)1011m3kgs2\cdot 10^{-11}\frac{m^{3}}{kg\cdot s^{2}} 52433 0.0161
GG with air viscosity (34) (6.72 ±\pm 0.35)1011m3kgs2\cdot 10^{-11}\frac{m^{3}}{kg\cdot s^{2}} 52433 0.69
GG to the first order (42) (6.80 ±\pm 0.34)1011m3kgs2\cdot 10^{-11}\frac{m^{3}}{kg\cdot s^{2}} 49989 1.92
GG from Leybold manual (43) (6.71 ±\pm 0.33) 1011m3kgs2\cdot 10^{-11}\frac{m^{3}}{kg\cdot s^{2}} 49600 0.64

4 Non linear effects in the vacuum

By starting from the equation of rotational dynamics up to the second order

Iφ..+βφ.+(τ2dGA1)φ2dGA0=2dGA2φ2,I\stackrel{{\scriptstyle..}}{{\varphi}}+\beta\stackrel{{\scriptstyle.}}{{\varphi}}+(\tau-2dGA_{1})\varphi-2dGA_{0}=2dGA_{2}\varphi^{2}\quad, (46)

the case of β\beta=0 is analysed ,

Iφ..+(τ2dGA1)φ2dGA0=2dGA2φ2,I\stackrel{{\scriptstyle..}}{{\varphi}}+(\tau-2dGA_{1})\varphi-2dGA_{0}=2dGA_{2}\varphi^{2}\quad, (47)

that corresponds to perform the experiment in the vacuum. On dropping the constant term and dividing by II we obtain

φ..+(τ2dGA1)Iφ=2dGA2Iφ2.\stackrel{{\scriptstyle..}}{{\varphi}}+\frac{(\tau-2dGA_{1})}{I}\varphi=\frac{2dGA_{2}}{I}\varphi^{2}\quad. (48)

On imposing

ω02=(τ2dGA1)I,\omega_{0}^{2}=\frac{(\tau-2dGA_{1})}{I}\quad, (49)

the nonlinear ordinary differential equation , in the following ODE , has the form

φ..+ω02φ=2dGA2Iφ2.\stackrel{{\scriptstyle..}}{{\varphi}}+\omega_{0}^{2}\varphi=\frac{2dGA_{2}}{I}\varphi^{2}\quad. (50)

On adopting the transformation T=tω0T=t*\omega_{0} the nonlinear ODE is

φ..+φϵφ2=0,\stackrel{{\scriptstyle..}}{{\varphi}}+\varphi-\epsilon\varphi^{2}=0\quad, (51)

where

ϵ=2dGA2(τ2dGA1).\epsilon=\frac{2dGA_{2}}{(\tau-2dGA_{1})}\quad. (52)

The solution of equation (51) is reported in the Appendix A and in our case ϵ\epsilon=0.0187. We now have a measured period ,TMST_{MS}, that is equalised to the non linear value , TNLT_{NL} . The period of the linear case , TLT_{L} , can be written as

TNL=TMS=1.00014TL,T_{NL}=T_{MS}=1.00014T_{L}\quad, (53)

and therefore

TL=TMS1.00014.T_{L}=\frac{T_{MS}}{1.00014}\quad. (54)

In the various formulae of GG without damping , for example equations (32) and (43), the periods TLT_{L} and TMST_{MS} are raised to the square

TL2=TMS21.00029,T_{L}^{2}=\frac{T_{MS}^{2}}{1.00029}\quad, (55)

and in the denominator, making the non linear GNLG_{NL} greater than the linear GLG_{L}

GNL=1.00029GL.G_{NL}=1.00029\,G_{L}\quad. (56)

The value of this correction , δG\delta G, can be evaluated as a difference between 1 and the multiplicative factor of GLG_{L}

δG=(1.000291)×6.6742×1011m3kg1s2=0.19×1013m3kg1s2.\delta G=(1.00029-1)\times 6.6742\times 10^{-11}m^{3}kg^{-1}s^{-2}=0.19\times 10^{-13}~m^{3}kg^{-1}s^{-2}\quad. (57)

The official error on GG is 0.1×1013m3kg1s20.1\times 10^{-13}~m^{3}kg^{-1}s^{-2} and therefore the nonlinear correction can be expressed as the double of the official error on GG.

Appendix A The eardrum equation

The equation

x¨+x+ϵx2=0,\ddot{x}+x+\epsilon x^{2}=0\quad, (58)

is well known under the name ”eardrum equation”. It can be solved , see [14], transforming it in

Ω2d2dT2X(T)+X(T)+ϵ(X(T))2=0,{\Omega}^{2}{\frac{d^{2}}{d{T}^{2}}}X\left(T\right)+X\left(T\right)+\epsilon\,\left(X\left(T\right)\right)^{2}=0\quad, (59)

and adopting the method of Poisson that imposes the following solution to XX

x(T)=x0(T)+x1(T)ϵ+x2(T)ϵ2,{\it x}\left(T\right)={\it x_{0}}\left(T\right)+{\it x_{1}}\left(T\right)\epsilon+{\it x_{2}}\left(T\right){\epsilon}^{2}\quad, (60)

and to Ω\Omega

Ω=1+ω1ϵ+ω2ϵ2.\Omega=1+{\it\omega_{1}}\,\epsilon+{\it\omega_{2}}\,{\epsilon}^{2}\quad. (61)

The computer algebra system (CAS) gives

ω1=0ω2:=5/12.\omega_{1}=0\quad\omega_{2}:=-5/12\quad. (62)

Acknowledgements

We thank Richard Enns who has provided us the Maple routine Example 04-S08 extracted from [14].

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