Measurement of the Gravitational Constant

Transcription

1 Measurement of the Gravitational Constant Dylan L. Renaud and Albert Fraser Department of Physics, New Jersey Institute of Technology, Newark, New Jersey, 070 USA (Dated: May, 07) The gravitational constant, G, is determined by using a compact variation of the Cavendish apparatus. The influence of calibrating the device s torsion balance is investigated experimentally by analyzing smooth (calibrated) and disturbed (un-calibrated) oscillation data. Analysis of smooth oscillations yields a value for G with an error of approximately 33.4% relative to the current standard established by the Committee on Data for Science and Technology (CODATA). Similar analysis for the case of disturbed oscillations produces a value of G with a relative error of nearly 54%. I. INTRODUCTION Gravity is one of the most extensively studied phenomena in nature. At the macroscopic level, the observation of the interplay of the gravitational force and finite mass systems is commonplace. The trajectory of a falling object, orbits of celestial bodies, and motion of a pendulum all exhibit explicit dependencies on gravity. In lay terminology, gravity is generally described as a force by which systems with mass attract one another. However, as elucidated by Einstein in his seminal works on general relativity, it is now known that gravity s influence extends to massless systems, such as light, and even space itself []. Needless to say, a proper understanding of this ever-present phenomenon is vital for the development of a complete description of the observable universe and nature itself. The first documented effort towards a quantitative description of gravity was the product of Sir Isaac Newton and Robert Hooke in 666. Newton, spurred on by the suggestion of Hooke that gravity was an attractive force who s magnitude depended on the inverse square of the distance between interacting systems [], began by investigating planetary motion. Through observations of bodies in the solar system, Newton developed and empirically confirmed a number of mathematical relations that ultimately served to validate the previously a-priori inverse square law [3]. One of the first experimental ventures into measuring the gravitational constant was performed by Henry Cavendish in 798 [4]. Following after the work of geologist John Michell and Francis John Hyde Wollaston, Cavendish developed an apparatus to measure the nearly imperceptible gravitational attraction between a pair of small and large lead spheres. Cavendish s device consisted of a torsion balance - a wooden rod suspended by a thin wire - with two small lead spheres at the ends. A significantly larger spherical lead mass was then positioned near each small sphere. The attraction between each large and corresponding small sphere caused the balance to rotate, thereby precipitating the twisting of the thin wire. After twisting through a particular angle, the torque induced on the balance by the gravitational attraction was balanced by the oppositely-directed torsion torque. Measurement of this angle, along with the imparted torque, allowed for the calculation of G. The present work replicates Cavendish s original experiment while utilizing a more compact apparatus. In general, there are three methods which may be utilized to obtain G using Cavendish s system. These methods are known as the measurement by deflection, measurement by equilibrium, and measurement by acceleration techniques [5]. In the present work, only the final deflection method is utilized. Here, the complete oscillations of the torsion-balance are measured until they have died out in order to obtain the period of oscillation. This period, in combination with the equilibrium position (angle about which the oscillations occur) are used to then determine G. II. THEORY In the Cavendish experiment, two large masses (M) are rotated into extreme positions (I and II - shown in FIG. ). Each large mass then exhibits a gravitational force on its corresponding neighboring small mass (m) of magnitude F = GmM b () where G is the gravitational constant and b is the distance between the centers of mass of the neighboring large and small masses. This force imparts a torque on the small mass-dumbbell system with net magnitude τ = F d () with d being the distance between either small mass and the point of rotation. A torsion band - a thin piece of wire - provides an equal and opposite torque to maintain the system s equilibrium. This balancing torque takes the form τ tor = κθ (3) where κ is the torsion constant of the band and θ is the total angle through which it is rotated [5]. Utilizing eq. (), (), and (3) and solving for G yields the following Correspondence dlr6@njit.edu G = κθb dmm (4)

2 which contains two parameters, κ and θ, who s values are to be experimentally obtained. The torsion constant κ is related to the period of the measured motion by κ = 8π m d + 5 r T (5) where T is the period of oscillation of small massdumbbell system. The solid angle θ is obtained by θ = (6) L denotes the distance from mirror in FIG. to the region of measurement, while denotes the difference in equilibrium points for the two extreme positions of the large masses. Through careful measurement of both and T, the gravitational constant may be determined. Finally, through inclusion of an additional correction factor [5], the gravitational constant becomes where b is defined as G 0 = G b (7) III. b 3 b = (b + 4d ) 3 EXPERIMENTAL METHODS (8) FIG. : Experimental set-up of Cavendish apparatus. The parameters L, S, and S indicate the distance from the mirror to the -meter stick, first equilibrium position, and second equilibrium position, respectively. A. Experimental Set-up Klinger Educational Products Gravitation Torsion Balance, shown in FIG., is utilized for experimentation. The device s two small masses are spatially calibrated to be parallel to the device s plastic faces (not shown). This is done in order to ensure that neither of the small masses collides with either of the plastic faces during oscillations. A HeNe laser (633 nm) is directed to the internal mirror of the Gravitation Torsion Balance. The reflected beam then travels to a -meter measuring stick to determine the oscillatory behaviour of the smallmass dumbbell system. B. Measurements Measurement of linear oscillations is performed in conjunction with an audible stopwatch with beep cycles every 5 seconds. The position of the laser on the -meter stick is noted at each beep and subsequently recorded. Equilibrium positions are determined by observing the position of the laser after oscillations have been sufficiently damped (i.e. no visible motion of laser spot). To ensure that oscillations have been adequately damped, the system is left alone for approximately 4 hrs before measurement of each equilibrium position. IV. RESULTS AND DISCUSSION A. Undisturbed Oscillations Angular oscillation data for the small-mass dumbbell system is provided in FIG.. The measured linear oscillations, (x(t) [cm]), are converted into angular oscillations, (φ(t) [ ]), by way of the relation φ(t) = tan ( x(t) S L ) (9) Due to the apparent damped behaviour of the oscillations, the angular data is fitted with a damped sinusoidal function in order to obtain the period ( ) π φ(t) = ae bt sin c t d + e (0) where c is the desired period of oscillation. In implementing the fit, the period is bounded such that c [600, 60] to guide the Levenberg-Marquardt fitting algorithm. The coefficient of determination, R, and parameter values are included in FIG.. The torsion constant, κ, is obtained by using the fitted value for the period with eq. (5). Calculation of the torsion angle, θ, is achieved using the equilibrium positions,

3 3 Measured Fit - eq. (0) φ [rad] 0 R = a =.9 b = 8.55e 4 c = 604 d =.04 e = ,000,00,400,600,800,000,00 t [s] G κ θ T S S dg dκ dθ [m 3 kg s ] [N m rad ] [rad] [± s] [±.5 cm] [±.5 cm] [m 3 kg s ] [N m rad ] [rad] 4.44e- 8.4e e e-.7e-09 9.e-04 L b r d M [±.3 cm] [mm] [mm] [mm] [kg] FIG. : Undisturbed oscillations. Table shows measured and provided (m, r, d and M) parameters. S and S, in conjunction with eq. (6). In calculating these values, theoretical error may is determined by considering the propagation of error caused by uncertainty in the obtained values for T, S, and S. For κ, the propagation relation is. While determination of uncertainty in the calculated value of G can be obtained from the uncertainties yielded by eq. () and (), a more direct method entails combining eq. (4), (5), and (6). The explicit uncertainty relation then takes the form ( ) δκ dκ = (dt ) δt ( 6π m d + = 5 r T 3 ) (dt ) () where dt is the uncertainty in the period. While propagation of error in θ yields ( ) ( ) ( ) δθ δθ δθ dθ = (ds ) δs + (ds ) δs + (dl) δl ( ) ( ) ( ) = (ds ) + (ds ) + (dl) = ( ) (ds ) + (ds ) + (dl) () Here, ds and ds are the uncertainties in the measured equilibrium positions. Both observed and calculated uncertainties are provided in the table accompanying FIG. ( ) δg dg = (ds ) δs + δg ) (ds ) δs +... ( ) ( ) δg δg (dt ) + (dl) δt δl ( = π b d + ) 5 r T (ds ) +... (π b d + ) 5 r T (ds ) +... ( π b d + ) 5 r T 3 (dt ) +... ( π b d + ) 5 r (dl) T ML d = π b d + 5 r (ds T ) + (ds ) +... ( ) ( ) T (dt ) + L (dl) (3)

4 4 B. Disturbed Oscillations In the case that the two small masses are not spatially calibrated as outlined in sec. III A, the oscillations shown in FIG. will no longer be smooth. As the dumbbell system oscillates, the masses collide with the device s plastic faces, resulting in apparent disturbed oscillations. An example of this behavior is provided in FIG Measured Fit - eq. (4) this occurs proves difficult, as shown by the small number of data points prior to the first bounce in FIG. 3. In the present study, this method is not implemented due to the inability to measure the oscillations in the required manner. Method requires fitting the data with a damped sine squared term. If the system is left to oscillate, collisions will cause the appearance of several sharp bounces. The resulting behavior resembles that of a squared sinusoidal waveform with an exponentially decaying envelope. For this reason, in fitting the data, a function of the form R = φ(t) = ae bt sin ( π c t d ) + e (4) φ [rad] 0 4 bounce is considered. Eq. (4) differs in form from that of eq. (0) only by the squared sine term. The fitting period, c, may then be used in a manner identical to that proposed in sec. IV A in order to obtain G. The obtained values for T and G are provided in the table accompanying FIG. 3. The respective uncertainties are approximately the same as those calculated in sec. IV A t [s] C. Comparison of Disturbed and Undisturbed Oscillations G [m 3 kg s ] [± s] 3.0e- 73 FIG. 3: Disturbed oscillations. Shaded regions indicate data points excluded during fitting. Table shows calculated values for G and T obtained from disturbed oscillation data. Under such circumstances, it may still be possible to determine the period of the oscillations by one of two methods:. Fitting of sinusoidal waveform through the first half of the initial oscillation. Fitting using a damped squared sinusoidal waveform Concerning case, prior to a collision between an internal dumbbell mass and a plastic face, the movement of the internal mass is identical to that found in an undisturbed system. Therefore, fitting the curve at a time period prior to the first sharp bounce - shown in FIG. 3 - should result in an approximate period similar to that of an undisturbed system. The primary challenge that accompanies this method resides in the fact that measurements must commence at the exact moment the large external masses are rotated, or equivalently, before one of the small internal masses collides with a plastic face. Obtaining a significant number of measurements before T Calculated values of G for both disturbed and undisturbed oscillations can be compared with the consensus value for G established by the Committee on Data for Science and Technology (CODATA) [6]. Here, the precision of the CODATA value is limited to agree with that of our measurements. The relative errors for the two calculations are then (.73) 6.67 for undisturbed oscillations, and (.73) 6.67 = 33.4% (5) = 53.5% (6) for disturbed oscillations. Considering the uncertainty, the relative error of the undisturbed oscillation value for G comes to % that of the CODATA value. While a relatively large discrepancy exists between our measured values, the latter falls within the uncertainty of the former. Nonetheless, discrepancies can be attributed to a non-trivial influence of the plastic face during collisions. For this reason, the squared-sine fitting method is suggested to be a poorer method for accurately obtaining the period necessary to calculate G. V. CONCLUSION The gravitational constant, G, has been measured using a compact version of the Cavendish apparatus. Analysis of data for smooth oscillations yields a value for G with a relative error of approximately 33%. Similar

5 5 analysis for the case of disturbed oscillations using the damped squared sinusoidal waveform method produces a value of G with a relative error of nearly 54%. This indicates the significance of proper calibration of the devices internal masses prior to measurement. Regardless, it is shown that it is possible to utilize oscillations influenced by un-desirable external factors to approximate G. [] Albert Einstein. Relativity: The special and general theory. [] Cohen I. Bernard and George Edwin Smith. The Cambridge Companion to Newton. Cambridge University Press, 00. [3] Isaac Newton. The principia: mathematical principles of natural philosophy, 999. [4] Henry Cavendish. Experiments to determine the density of the earth. by henry cavendish, esq. frs and as. Philosophical Transactions of the Royal Society of London, 88:469 56, 798. [5] Cavendish methods. [6] NIST. Newtonian constant of gravitation, 04.