LINEAR AND NONLINEAR EFFECTS ON THE NEWTONIAN GRAVITATIONAL CONSTANT AS DEDUCED FROM THE TORSION BALANCE

Transcription

1 International Journal of Modern Physics A Vol., No. 9 (007) c World Scientific Publishing Company Int. J. Mod. Phys. A 007.: Downloaded from by UNIVERSITY OF TORINO DEPARTMENT OF MATHEMATICS on 09/6/1. For personal use only. LINEAR AND NONLINEAR EFFECTS ON THE NEWTONIAN GRAVITATIONAL CONSTANT AS DEDUCED FROM THE TORSION BALANCE M. ROSSI Dipartimento di Matematica, Università deglistudiditorino, Via Carlo Alberto 10, 1013 Torino, Italy L. ZANINETTI Dipartimento di Fisica Generale, Università degli Studi di Torino, Via Pietro Giuria 1, I-1015 Torino, Italy Received 4 June 007 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. PACS numbers: y, 06.0.Jr 1. Introduction After many years of measurements, begun by H. Cavendish, the Newtonian gravitational constant value, said G, is still affected by a large error. 1 The CODATA recommends G =(6.674 ± 0.001) m3, meaning a relative standard uncertainty kg s of 150 parts per million (ppm). This value has had recent confirmation by means, on one hand, of a superconducting 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 fiber. 6 8 Moreover recent studies point out that the period of a torsion pendulum might vary under disturbances of environmental noise factors, see Ref. 9. Other authors suggest a possible deviation from Newton s law specified as an additional contribution of Yukawa potential type. 10 This paper first analyzes the linear and nonlinear equations governing the torsional balance rotational dynamics (Sec. ). By means of a gravitational torsion balance, same values of G are obtained and summarized (Sec. 3). The variation of oscillation period, due to a nonlinear effect, is then discussed (Sec. 4). 5391

2 539 M. Rossi & L. Zaninetti Torsion Balance r Int. J. Mod. Phys. A 007.: Downloaded from by UNIVERSITY OF TORINO DEPARTMENT OF MATHEMATICS on 09/6/1. For personal use only. Fig. 1. luminous source. The Basic Equations The form of Newton s law of gravitation is distant scale Schematic view of the torsional balance. F = G mm r, (1) where G is the gravitational constant, M the great mass, m the small mass and r their relative distance. The Leybold balance represents a widespread instrument to determine the constant G, see Fig. 1, and is constructed with the following components. (1) A freely oscillating horizontal bar, of length d, holding two small lead balls of mass m as in Fig. 1 supported by a torsion fiber that has a torsional constant τ. () Two larger balls of mass M that can be positioned next to the small balls as in Fig. 1. The center of mass of the two m and M are supposed to be all on a plane perpendicular to the fiber. (3) A luminous source directed toward the center of mass of the bar where is reflected by a mirror. (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. Figure shows a plot of the motion. The moment of inertia of the bar, I is The fundamental equation of rotational dynamics is I md. () I ϕ = M g + M v + M t, (3)

3 Linear and Nonlinear Effects on the Newtonian Gravitational Constant 5393 Forces Int. J. Mod. Phys. A 007.: Downloaded from by UNIVERSITY OF TORINO DEPARTMENT OF MATHEMATICS on 09/6/1. For personal use only. where u Fig.. GF 1 GF u 1 =r+s Top view of the Cavendish balance. M v = β ϕ, (4) M t = τϕ. (5) Here β is the coefficient of viscosity of air, ϕ the angle between bar and bar itself when the torque is zero. This angle is measured in the anticlockwise direction. The term M g represents the torque of the gravitational forces. From Eq. (1), we obtain M g =dgf (ϕ), (6) where F is a function of the angle ϕ. The law of dependence of F with ϕ is complex and will here be analyzed. When the motion starts the resulting force is where F 1 (ϕ) =mm cos ϕ u 1 F (ϕ) =F 1 (ϕ) F (ϕ), (7), (8) F (ϕ) =mm cos ( π (ϕ + α)) sin(ϕ + α) u = mm u, (9) with α := arcsin ( ) r s u (where r and s are defined as in Fig. 1). With our data (see Sec. 3) the maximum angular excursion of the angle ϕ is ( ) xmax x min =arcsin. (10) l

4 5394 M. Rossi & L. Zaninetti Int. J. Mod. Phys. A 007.: Downloaded from by UNIVERSITY OF TORINO DEPARTMENT OF MATHEMATICS on 09/6/1. For personal use only. The angle ϕ has a low values, see Table 1, and a Taylor series expansion that keep terms to order ϕ will be adopted. This means to forget quantities less than The series representation gives u 1 = ( ( d (r + s) +(d dcos ϕ) r 1+ϕ r + d r ϕ + d 4r ϕ3 )) 1. (11) Developing the last term with a Maclaurin series, we obtain ( u 1 r 1 ϕ ( ) ) d r + d r ϕ d r ϕ + O(ϕ 3 ) r + dϕ. (1) As a consequence F 1 mm 1 ϕ (r + dϕ), (13) and expanding the denominator, we obtain ) (r + dϕ) = (r dr ϕ +3d 3 r 4 ϕ + O(ϕ 3 ), (14) that means F 1 mm ( r 1 d ( 3d r ϕ + r 1 ) )ϕ. (15) An expression for F can be obtained from Eq. (9): sin ϕ cos α +cosϕsin α r + dϕ r F = mm u mm (d + r rdϕ d ϕ ) 3. (16) On Taylor expanding the denominator, (4d + r rdϕ d ϕ ) 3 ( =(4d + r ) dr 4d + r ϕ + 3d (4d +5r ) ) (4d + r ) ϕ + O(ϕ 3 ), (17) and therefore F r + 4d(d +r ) 4d +r ϕ + r(0d4 +13d r r 4 ) (4d +r ) ϕ. (4d + r ) 3 (18) Now F from Eq. (7) can be expressed as a Taylor expansion truncated at O(ϕ 3 ): F A 0 + A 1 ϕ + A ϕ, (19) where ( ) 1 A 0 := mm r r, (0) (4d + r ) 3 ( d A 1 = mm r 3 + d(d + r ) ), (4d + r ) 5 (1) A =3 mmd r 4 1 mm r + 1 mmr(4d 4 10r d + r 4 ). () (4d + r ) 7/

5 Linear and Nonlinear Effects on the Newtonian Gravitational Constant 5395 Three methods that allow to obtain an expression for G in terms of measurable quantities are now introduced. Further on the well-known formula for G extracted from the Leybold manual is reviewed. Int. J. Mod. Phys. A 007.: Downloaded from by UNIVERSITY OF TORINO DEPARTMENT OF MATHEMATICS on 09/6/1. For personal use only..1. Averaged G Let F (ϕ) = F for all the experience; in first approximation we may assume that F is given by Eq. (19) to first order F = A 0 + A 1 ϕ, (3) where ϕ is the average of the values that ϕ assumes between the first position, P 1, and the last position, P, of the balance; A 0, A 1 are given by Eqs. (0) and (1). The differential equation that describes the motion is and its solution is I ϕ + β ϕ + τϕ =dg F, (4) ϕ(t) =ce δt cos(ωt + φ)+ where c represents the amplitude and dg F τ, (5) δ := β I, (6) 4Iτ β ω :=. (7) I The angle ϕ that represents the bar position at the end of the phenomena P can be determined as follows: ϕ x x o = x x 1, (8) l 4l and should be the same as predicted by the theory F lim ϕ(t) =dg ; (9) t + τ therefore G = τϕ. (30) d F In order to continue a value for τ should be derived. This can be obtained from the period of oscillation of the bar T = π ω = 4πI 4Iτ β. (31) We continue by identifying T with the empirical value T. We continue on assuming that β 4I is small; therefore from Eqs. (), (30) and (3), the following is obtained: G = π Iϕ. (3) d(a 0 + A 1 ϕ) T

6 5396 M. Rossi & L. Zaninetti.. G with air viscosity From formula (4) is possible to deduce the viscosity of the air once the coefficient of damping δ is known, see Sec. 3 on data analysis. From (30) and (6) we should add to the value of G reported in Eq. (3): Int. J. Mod. Phys. A 007.: Downloaded from by UNIVERSITY OF TORINO DEPARTMENT OF MATHEMATICS on 09/6/1. For personal use only. obtaining G = G β := β ϕ, (33) 8Id F π Iϕ d(a 0 + A 1 ϕ) T + β ϕ 8Id(A 0 + A 1 ϕ). (34).3. G to the first order Let us assume that F (ϕ) is not constant. We can assume at the order O(ϕ )with the aid of formula (19): F A 0 + A 1 ϕ, (35) where A 0 and A 1 are defined in Eqs. (0) and (1), respectively. In this case the law of motion is still Eq. (3): and the solution is I ϕ + β ϕ +(τ dga 1 )ϕ =dg A 0, (36) ϕ(t) =ce δt cos(ω t + φ)+ dga 0, (37) τ dga 1 where the angular velocity ω has now the following expression: 4I(τ ω dga1 ) β :=. (38) I As a consequence and therefore τ = 4π I T ϕ = +dga 1 + β 4I, (39) lim ϕ(t) = dga 0, (40) t + τ dga 1 τϕ G = d(a 0 + A 1 ϕ ). (41) Once Eq. (39) is substituted in this relationship we obtain G = π Iϕ da 0 T + β ϕ 8dIA 0. (4)

7 Linear and Nonlinear Effects on the Newtonian Gravitational Constant 5397 Table 1. Parameters of the torsion balance. Parameter Value Unit Int. J. Mod. Phys. A 007.: Downloaded from by UNIVERSITY OF TORINO DEPARTMENT OF MATHEMATICS on 09/6/1. For personal use only. M (1.5 ± 10 3 ) kg m ( ± 10 3 ) kg r ( ± 10 3 ) m d ( ± 10 3 ) m l (5.475 ± 10 3 ) m ϕ ( ± ) rad β ( ± ) kg m s.4. G from Leybold manual The deduction of G trough the Leybold torsional balance is widely known, see Ref. 11. We simply report the final expression G = π b d S MT (1 + β), (43) l where b 3 β = (b +4d ) b +4d. (44) The meaning of the symbols is: b: distance between the centers of the great mass and small mass; S: total deflection of the light spot; d: the length of the lever arm; l: distance between mirror and screen; M: great mass; T : period of the oscillations. 3. Analysis of the Data The physical parameters and their uncertainties are reported in Table 1. The data were analyzed through the following fitting function: ( ) ( πt y(t) =A 0 + A 1 cos exp t ). (45) T τ The data has been processed through the Levenberg Marquardt method (subroutine MRQMIN in Ref. 1) in order to find the parameters A 0, A 1, T and τ. The results are reported in Table together with the derived quantities. The value of G can be derived, coupling the basic parameters of the torsion balance, see Table 1, and the measured parameters of the damped oscillations, see Table. Table 3 reports the four values of G here considered with the uncertainties expressed in absolute value and in ppm; the precision of the measure in respect

8 5398 M. Rossi & L. Zaninetti Table. Parameters of the nonlinear fit. Parameter Value Unit A 0 ( ± ) m A 1 ( ± ) m T (55.98 ± 0.16) s τ ( ± 4.8) s Int. J. Mod. Phys. A 007.: Downloaded from by UNIVERSITY OF TORINO DEPARTMENT OF MATHEMATICS on 09/6/1. For personal use only. Table 3. Values of G. Method Equation Value Uncertainty Accuracy (ppm) (%) G averaged (3) (6.67 ± 0.34) m3 kg s G with air viscosity (34) (6.7 ± 0.35) m3 kg s G to the first order (4) (6.80 ± 0.34) m3 kg s G from Leybold manual (43) (6.71 ± 0.33) m3 kg s 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 10 3 m. Adopting a rotating gauge method 13 the uncertainty in the determination of the span between the two spheres is m; this is the way to lower the uncertainty in Table Nonlinear Effects in the Vacuum Starting from the equation of rotational dynamics up to the second order the case of β = 0 is analyzed, I ϕ + β ϕ +(τ dga 1 )ϕ dga 0 =dga ϕ, (46) I ϕ +(τ dga 1 )ϕ dga 0 =dga ϕ, (47) which corresponds to perform the experiment in the vacuum. On dropping the constant term and dividing by I, we obtain On imposing ϕ + (τ dga 1) I ϕ = dga ϕ. (48) I ω0 = (τ dga 1), (49) I the nonlinear ordinary differential equation, in the following ODE, has the form ϕ + ω0 ϕ = dga ϕ. (50) I

9 Linear and Nonlinear Effects on the Newtonian Gravitational Constant 5399 On adopting the transformation T = t ω 0, the nonlinear ODE is where ϕ + ϕ ɛϕ =0, (51) ɛ = dga (τ dga 1 ). (5) Int. J. Mod. Phys. A 007.: Downloaded from by UNIVERSITY OF TORINO DEPARTMENT OF MATHEMATICS on 09/6/1. For personal use only. The solution of Eq. (51) is given in App. A and in our case ɛ = We now have a measured period, T MS, that is equalized to the nonlinear value, T NL.The period of the linear case, T L, can be written as and therefore T NL = T MS = T L, (53) T L = T MS (54) In the various formulae of G without damping, for example Eqs. (3) and (43), periods T L and T MS are raised to the square TL = T MS , (55) and in the denominator, making the nonlinear G NL greater than the linear G L : G NL =1.0009G L. (56) The value of this correction, δg, can be evaluated as a difference between 1 and the multiplicative factor of G L : δg =( ) m 3 kg 1 s = m 3 kg 1 s. (57) The official error on G is m 3 kg 1 s and therefore the nonlinear correction can be expressed as the double of the official error on G. Appendix A. The Eardrum Equation The equation ẍ + x + ɛx =0, (A.1) is well known under the name eardrum equation. It can be solved 14 transforming it in Ω d dt X(T )+X(T)+ɛ(X(T)) =0, (A.) and adopting the method of Poisson that imposes the following solution to X: x(t )=x 0 (T )+x 1 (T )ɛ + x (T )ɛ, (A.3)

10 5400 M. Rossi & L. Zaninetti and to Ω: Ω=1+ω 1 ɛ + ω ɛ. The computer algebra system (CAS) gives the followings: (A.4) ω 1 =0, ω := 5 1. (A.5) Int. J. Mod. Phys. A 007.: Downloaded from by UNIVERSITY OF TORINO DEPARTMENT OF MATHEMATICS on 09/6/1. For personal use only. Acknowledgments We thank Richard Enns who provided us the Maple routine example 04-S08 extracted from Ref. 14. References 1. G. T. Gillies, Rep. Prog. Phys. 60, 151 (1997).. P. J. Mohr and B. N. Taylor, Rev. Mod. Phys. 77, 1 (005). 3. P. Baldi, E. G. Campari, G. Casula, S. Focardi, G. Levi and F. Palmonari, Phys. Rev. D 71, 000 (005). 4. S. Schlamminger, E. Holzschuh, W. Kündig, F. Nolting, R. E. Pixley, J. Schurr and U. Straumann, Phys. Rev. D 74, (006). 5. J. Schurr, F. Nolting and W. Kündig, Phys. Lett. A 48, 95 (1998). 6. C. H. Bagley and G. G. Luther, Phys. Rev. Lett. 78, 3047 (1997). 7. K. Kuroda, Phys. Rev. Lett. 75, 796 (1995). 8. S. Matsumura, N. Kanda, T. Tomaru, H. Ishizuka and K. Kuroda, Phys. Lett. A 44, 4 (1998). 9. J. Luo, D. Wang, Q. Liu and C. Shao, Chin. Phys. Lett., 169 (005). 10. S. Kononogov and V. Mel nikov, Measurement Techniques 48, 51 (005). 11. Leybold Physics Leaflets: Determining the gravitational constant, Leybold Didactic GMBH, Cologne, W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes in FORTRAN. The Art of Scientific Computing (Cambridge University Press, Cambridge, 199). 13. J. Luo, D. Wang, Z. Hu and X. Wang, Chin. Phys. Lett. 18, 101 (001). 14. R. H. Enns and G. C. McGuire, Computer Algebra Recipes for Classical Mechanics (Birkhäuser, Boston, 00).