《力学 MECHANICS》课程学习资料：laboratory test of Newton's Second law for small accelerations

PRL98,150801(2007 PHYSICAL REVIEW LETTERS RIL 2007 Laboratory Test of Newtons Second Law for Small accelerations J H. Gundlach, S. Schlamminger, C D. Spitzer, and K -Y. Choi Center for Experimental Nuclear Physics and Astrophysics, University of Washington, Seattle, Washington 98195, USA B.A. Woodall Physics Department, Indiana University-Purdue University, Indianapolis, In 46202.USA J.J. Coy Earth and Space Science Department, Saint Joseph's College, Rensselaer, Indiana 47978, USA Physics Department, Purdue University, West Lafayette, Indiana 47907, USA We have tested the proportionality of force and acceleration in Newtons second law, F=ma, in the limit of small forces and accelerations. Our tests reach well below the acceleration scales relevant to understanding several current astrophysical puzzles such as the flatness of galactic rotation curves, the Pioneer anomaly, and the Hubble acceleration. We find good agreement with Newton's second law at accelerations as small as 5 x 10 4 m/s DOI: 10. 1103/PhysRevLett98150801 PACS numbers: 06.30. Gv. 04.80. Cc Newton's second law is the equation of motion defining standard Newtonian dynamics. The functional form of the the field of dynamics. In its nonrelativistic form, F= transition between the two regimes is not specified. A is perhaps the most famous and most often used equation of smooth transition can be obtained by multiplying the physics. Together with its relativistic and quantum me- right side of F=ma by u(a/ao)=a/ao(1+a2/a3)1/2 chanical variants, this law is implicitly tested in many so that for a > do the function A el and standard applications and experiments, and its validity is simply Newtonian mechanics is recovered. The characteristic ac- assumed at all acceleration scales. Any deviation from celeration ao was determined from fits [4] to galactic F=ma would have profound consequences as it would rotation curves to be ao= 1.2 X 10-10 m/s imply a violation of crucial conservation laws such as Further testing of Newton's second law is motivated by energy and momentum in their conventional definition. the Pioneer anomaly. Doppler-tracking data of the Pioneer At very small accelerations a deviation from Newton's 10 and ll spacecraft shows an unmodeled acceleration at second law could remain hidden in most laboratory scale distances >15 AU of ao =9x 10-10 m/s roughly point experiments, but might appear in astrophysical and cos- ing towards the Sun [5]. Exhaustive efforts have been mological observations undertaken to find a conventional explanation for this One observed fact is the flatness of galactic rotation effect, so far without success. It is also interesting to note curves. The tangential velocity of stars measured as a that the Hubble acceleration aH =cH =7X 10-10 m/s2 function of distance from the galactic center rises first provides a natural acceleration scale, with H being the and flattens for larger distances. Newton's second law Hubble constant. Furthermore, it is also conceivable that together with the gravitational effect of known matter a violation of Newton's second law could play a role in predicts a decrease in the velocities for larger distances, explaining the acceleration of the Universe at large dis- and dark matter has been introduced to resolve this dis- tances, which is usually attributed to dark energy crepancy [1]. Alternatively, Milgrom discovered that Here we report the results of a laboratory experiment to Newton,s second law can be modified with a single addi- test Newton's second law using a torsion pendulum. Our tional parameter ao to describe the measured galactic system is significantly different from that used in an earlier rotation curves extremely well without invoking dark mat- experiment by Abramovici and Vager [6], who interfero- ter [2, 3]. While Milgrom's full formalism MOND (modi- metrically measured the acceleration of a pendulum mass fied Newtonian dynamics) is untestable in the laboratory, in response to an applied electric field. Abramovici and since it requires the absence of accelerations in all direc- Vager found agreement with Newton's second law at ac tions, a modification of Newtonian dynamics provides a celerations as small as 3 X 10- m/s In our experiment simple explanation of the galactic rotation curves. Milgrom we utilize the fact that as a torsion pendulum passes suggested that Newtons second law would smoothly tran- through equilibrium its acceleration (relative to the labo- sition from F o a to F o a/ao at a=ao. Hence, for a< ratory)is zero. If the torsional amplitude is small, the time ao a force would yield a larger acceleration as compared to spent experiencing small accelerations and small forces 0031-9007/07/98(15)150801(3) 150801-1 c 2007 The American Physical Society

Laboratory Test of Newton’s Second Law for Small Accelerations J. H. Gundlach, S. Schlamminger, C. D. Spitzer, and K.-Y. Choi Center for Experimental Nuclear Physics and Astrophysics, University of Washington, Seattle, Washington 98195, USA B. A. Woodahl Physics Department, Indiana University-Purdue University, Indianapolis, Indiana 46202, USA J. J. Coy Earth and Space Science Department, Saint Joseph’s College, Rensselaer, Indiana 47978, USA E. Fischbach Physics Department, Purdue University, West Lafayette, Indiana 47907, USA (Received 12 February 2007; published 13 April 2007) We have tested the proportionality of force and acceleration in Newton’s second law, F
ma, in the limit of small forces and accelerations. Our tests reach well below the acceleration scales relevant to understanding several current astrophysical puzzles such as the flatness of galactic rotation curves, the Pioneer anomaly, and the Hubble acceleration. We find good agreement with Newton’s second law at accelerations as small as 5 1014 m=s2. DOI: 10.1103/PhysRevLett.98.150801 PACS numbers: 06.30.Gv, 04.80.Cc Newton’s second law is the equation of motion defining the field of dynamics. In its nonrelativistic form, F~
ma~ is perhaps the most famous and most often used equation of physics. Together with its relativistic and quantum me￾chanical variants, this law is implicitly tested in many applications and experiments, and its validity is simply assumed at all acceleration scales. Any deviation from F~
ma~ would have profound consequences as it would imply a violation of crucial conservation laws such as energy and momentum in their conventional definition. At very small accelerations a deviation from Newton’s second law could remain hidden in most laboratory scale experiments, but might appear in astrophysical and cos￾mological observations. One observed fact is the flatness of galactic rotation curves. The tangential velocity of stars measured as a function of distance from the galactic center rises first and flattens for larger distances. Newton’s second law together with the gravitational effect of known matter predicts a decrease in the velocities for larger distances, and dark matter has been introduced to resolve this dis￾crepancy [1]. Alternatively, Milgrom discovered that Newton’s second law can be modified with a single addi￾tional parameter a0 to describe the measured galactic rotation curves extremely well without invoking dark mat￾ter [2,3]. While Milgrom’s full formalism MOND (modi- fied Newtonian dynamics) is untestable in the laboratory, since it requires the absence of accelerations in all direc￾tions, a modification of Newtonian dynamics provides a simple explanation of the galactic rotation curves. Milgrom suggested that Newton’s second law would smoothly tran￾sition from F / a to F / a2=a0 at a  a0. Hence, for a  a0 a force would yield a larger acceleration as compared to standard Newtonian dynamics. The functional form of the transition between the two regimes is not specified. A smooth transition can be obtained by multiplying the right side of F~
ma~ by
a=a0
a=a01a2=a2 0 1=2, so that for a a0 the function
’ 1 and standard Newtonian mechanics is recovered. The characteristic ac￾celeration a0 was determined from fits [4] to galactic rotation curves to be a0  1:2 1010 m=s2. Further testing of Newton’s second law is motivated by the Pioneer anomaly. Doppler-tracking data of the Pioneer 10 and 11 spacecraft shows an unmodeled acceleration at distances >15 AU of a0  9 1010 m=s2 roughly point￾ing towards the Sun [5]. Exhaustive efforts have been undertaken to find a conventional explanation for this effect, so far without success. It is also interesting to note that the Hubble acceleration aH
cH  7 1010 m=s2 provides a natural acceleration scale, with H being the Hubble constant. Furthermore, it is also conceivable that a violation of Newton’s second law could play a role in explaining the acceleration of the Universe at large dis￾tances, which is usually attributed to dark energy. Here we report the results of a laboratory experiment to test Newton’s second law using a torsion pendulum. Our system is significantly different from that used in an earlier experiment by Abramovici and Vager [6], who interfero￾metrically measured the acceleration of a pendulum mass in response to an applied electric field. Abramovici and Vager found agreement with Newton’s second law at ac￾celerations as small as 3 1011 m=s2. In our experiment we utilize the fact that as a torsion pendulum passes through equilibrium its acceleration (relative to the labo￾ratory) is zero. If the torsional amplitude is small, the time spent experiencing small accelerations and small forces PRL 98, 150801 (2007) PHYSICAL REVIEW LETTERS week ending 13 APRIL 2007 0031-9007=07=98(15)=150801(3) 150801-1 © 2007 The American Physical Society

PRL 150 PHYSICAL REVIEW LETTERS 13 aPRIL 2007 can be made sufficiently long, so that the free torsional table. The pendulum is suspended from a 1.07 m long, period would be altered. Therefore measuring the pendu- 20 um thick tungsten wire, with K=2. 36X lums period as a function of amplitude is a direct test of 10-9Nm/rad. The natural period of the pendulum Newton's second law 795 s and the amplitude decay time is 15 days, correspond In the simplified case of a pendulum where all the mass ing to a quality factor, 2, of approximately 5000. The is concentrated at one radius, as for a hollow cylinder with pendulum is suspended inside a vacuum chamber at a radius r, a MOND-like free equation of motion relates the pressure of 10-S Pa, maintained by an ion pump acceleration a to the inertial torque on the pendulum T: Before each measurement the torsion pendulum was rl, a,ao=a actively damped to very small amplitudes by rotating the 火(a/a0, (1) turntable on which the torsion balance is installed. The turntable was then kept stationary and the torsional period where I is the moment of inertia of the torsion pendulum. was measured as a function of amplitude. The damping For a pendulum with an arbitrary mass distribution given was necessary since the pendulum gained on average a by the radial mass density p(r)the equation of motion torsional amplitude of about 60 nrad in approximately one implied by (I) is then period, mostly due to Brownian excitation in the fiber. Once the pendulum gained a substantial amplitude the u(lrel/aop(r)rav)e+ke=0.(2) data taking was stopped. The acquired torsional amplitude was removed by instantly rotating the turntable so that no Here 8 is the angular displacement of the torsion pendu- energy remained stored in the twist of the fiber, and after lum, r0=a is the acceleration, k is the fibers torsional that the data taking was resumed. Over a period of 20 days constant, and the overdots denote time derivatives. Figure I we operated the pendulum at various amplitudes ranging shows a numerical solution of Eq. (2)in comparison with a from approximately 13 nrad to 19 urad. A typical data measured trace. The simulation demonstrates that in the trace is shown as the solid line in Fig. 1. The linear drift in MOND-like regime the period of oscillations is shorter the data is caused by a slow and constant relaxation of the than predicted by Newton,s second law, just as is the torsion fiber. The noise at low frequencies (=10 mHz)is predominantly caused by Brownian excitation, while at case for stars far out in the galaxy which experience higher frequencies it is dominated by the optical readout. similarly small forces The instrument used fo test is the University of The angular position of the pendulum was measured Washington's equivalen ith an autocollimator and was recorded every 2.74S sists of a torsion balance ed on an air bearing turn- The data analysis began by averaging every 20 data points The data were then prefiltered with an algorithm that dentified large instantaneous changes in angular momen- um( spikes). These impulses were often related to pressure bursts of the ion pump, and sections containing such spikes were eliminated from the data, leading to 326 spike free data sections of various lengths. We used the zero crossings of the second derivatives of the data to find the pendulums equilibrium position and hence detect the linear drift which was then subtracted from the raw(nonaveraged) data. To each section of this drift corrected data we fit a variable frequency sinusoid using a standard Levenberg-Marquardt technique [7]. For each section we obtain the measured frequency and amplitude and their corresponding uncertainties To convert the measured angular variables to linear vari- time ables we used the mean effective radius r.=0.023 m, calculated from the pendulum's mass distribution. Com FIG. I(color online). The solid line shows a typical measure- bining this radius with the measured amplitudes A and nent of the angular position of the torsion pendulum. The frequencies o/(2), we calculate the maximum force upward drift in the data is due to a slow unwinding of the torsion using Hooke's law, F=AK/re, and the maximum accel- fiber. The noise in the measured trace is due to brownian excitation and readout noise. For purposes of comparison the eration for the torsion pendulum, which occurs at the turn- simulated motion of the pendulum in a MOND-like regime with ing points and is given by a= Ar, a2. Figure 2 shows the o=1.0 X 10 m/s is plotted with a dashed line. Note that inferred accelerations versus force for all fitted data sec- the MOND-like dynamics would result in a much shorter period tions. Also shown is the deviation in acceleration and force of oscillation from F= ma with m as the only fit parameter. Our data 150801-2

can be made sufficiently long, so that the free torsional period would be altered. Therefore measuring the pendu￾lum’s period as a function of amplitude is a direct test of Newton’s second law. In the simplified case of a pendulum where all the mass is concentrated at one radius, as for a hollow cylinder with radius r, a MOND-like free equation of motion relates the acceleration a to the inertial torque on the pendulum : I; a; a0
Ia r
a=a0; (1) where I is the moment of inertia of the torsion pendulum. For a pendulum with an arbitrary mass distribution given by the radial mass density r the equation of motion implied by (1) is then
Z
jr
j=a0rr2dV 
 
0: (2) Here  is the angular displacement of the torsion pendu￾lum, r

a is the acceleration,  is the fiber’s torsional constant, and the overdots denote time derivatives. Figure 1 shows a numerical solution of Eq. (2) in comparison with a measured trace. The simulation demonstrates that in the MOND-like regime the period of oscillations is shorter than predicted by Newton’s second law, just as is the case for stars far out in the galaxy which experience similarly small forces. The instrument used for this test is the University of Washington’s equivalence principle apparatus, which con￾sists of a torsion balance installed on an air bearing turn￾table. The pendulum is suspended from a 1.07 m long, 20
m thick tungsten wire, with 
2:36 109 N m=rad. The natural period of the pendulum is 795 s and the amplitude decay time is 15 days, correspond￾ing to a quality factor, Q, of approximately 5000. The pendulum is suspended inside a vacuum chamber at a pressure of 105 Pa, maintained by an ion pump. Before each measurement the torsion pendulum was actively damped to very small amplitudes by rotating the turntable on which the torsion balance is installed. The turntable was then kept stationary and the torsional period was measured as a function of amplitude. The damping was necessary since the pendulum gained on average a torsional amplitude of about 60 nrad in approximately one period, mostly due to Brownian excitation in the fiber. Once the pendulum gained a substantial amplitude the data taking was stopped. The acquired torsional amplitude was removed by instantly rotating the turntable so that no energy remained stored in the twist of the fiber, and after that the data taking was resumed. Over a period of 20 days we operated the pendulum at various amplitudes ranging from approximately 13 nrad to 19
rad. A typical data trace is shown as the solid line in Fig. 1. The linear drift in the data is caused by a slow and constant relaxation of the torsion fiber. The noise at low frequencies ( 10 mHz) is predominantly caused by Brownian excitation, while at higher frequencies it is dominated by the optical readout. The angular position of the pendulum was measured with an autocollimator and was recorded every 2.74 s. The data analysis began by averaging every 20 data points. The data were then prefiltered with an algorithm that identified large instantaneous changes in angular momen￾tum (spikes). These impulses were often related to pressure bursts of the ion pump, and sections containing such spikes were eliminated from the data, leading to 326 spike free data sections of various lengths. We used the zero crossings of the second derivatives of the data to find the pendulum’s equilibrium position and hence detect the linear drift which was then subtracted from the raw (nonaveraged) data. To each section of this drift corrected data we fit a variable frequency sinusoid using a standard Levenberg-Marquardt technique [7]. For each section we obtain the measured frequency and amplitude and their corresponding uncertainties. To convert the measured angular variables to linear vari￾ables we used the mean effective radius re
0:023 m, calculated from the pendulum’s mass distribution. Com￾bining this radius with the measured amplitudes A and frequencies !=2, we calculate the maximum force using Hooke’s law, F
A=re , and the maximum accel￾eration for the torsion pendulum, which occurs at the turn￾ing points and is given by a
Are!2. Figure 2 shows the inferred accelerations versus force for all fitted data sec￾tions. Also shown is the deviation in acceleration and force from F
ma with m as the only fit parameter. Our data -400 -200 0 200 400 600 800 0 500 1000 1500 angle (nrad) time (s) FIG. 1 (color online). The solid line shows a typical measure￾ment of the angular position of the torsion pendulum. The upward drift in the data is due to a slow unwinding of the torsion fiber. The noise in the measured trace is due to Brownian excitation and readout noise. For purposes of comparison the simulated motion of the pendulum in a MOND-like regime with a0
1:0 1012 m=s2 is plotted with a dashed line. Note that the MOND-like dynamics would result in a much shorter period of oscillation. PRL 98, 150801 (2007) PHYSICAL REVIEW LETTERS week ending 13 APRIL 2007 150801-2

week ending PRL98,150801(2007 PHYSICAL REVIEW LETTERS 13 APRIL 2007 10 FIG. 2(color online). The measured force versus the measured acceleration The solid line is the best fit for accelera- 81012 tion, a being exactly proportional to force F. Our data agree very well with the curve. The insets on the right and top f the m raph give the residuals of 13 force(N) res(10N) indicate that force and acceleration remain proportional expected to adhere ideally to Hooke's law because it is down to accelerations as small as 5 x 10-4 m/s2. operated far from its inelastic limit and also because the /e conducted numerous checks for bias and systematic oscillator has a high Q. Our test does not invalidate MOND uncertainties in our data analysis. We performed an inde- directly, since MOND requires that the measurement must pendent alternative data analysis in which the data were be carried out in the absence of any other larger acceler- divided into 1200 and 2000 s long sections irrespective of ations, such as those due to the earth and our solar system the spikes, which were not removed. In this analysis drift However, our results constrain any theoretical formalism was accounted for by introducing a linear and quadratic seeking to derive MOND from fundamental principles [8] large x were eliminated. This data analysis also produced laboratory conditions similar to those in our experime term in the fit. Data sections that exhibited exceedingly by requiring that formalism to reproduce F= ma unde good agreement with Newtons law, but had slightly larger For a future measurement we plan to test Fg=ma at uncertainties small accelerations, where Fg is a gravitational force as Our main data analysis was tested with two sets the galactic dynamics simulated data. The first set contained a realistic spectral The work at the University of Washington is supported oise distribution of a harmonic oscillator and white read- by the NSF under Grant No. NSF PHY-0355012 and E.F. out noise. The frequencies were recovered properly for is supported in part by the U.S. Department of Energy amplitudes comparable to the measured traces, as well as under Contract No. DE-ACo2-76ERO71428 for smaller amplitudes. The uncertainties and the x2 be- haved similarly to the corresponding quantities in the mea- sured data. The second set contained simulated monD- like data with a range of ao from 10-16-10-m/s2.All the [1] Y Sofue and V. Rubin, Annu. Rev. Astron. Astrophys.39 accelerations were recovered within uncertainties 137(2001) ummary,we have found no deviation from the [2] M. Milgrom, Astrophys. J. 270, 365(1983) proportionality in Newtons second law down to acceler 3] M. Milgrom, Astrophys. J. 270, 371(1983) ations of 5 x 10-14 m/s2, which is approximately [4] R H. Sanders and sS. MGGaugh, Annu. Rev. Astron 1000 times smaller than the previous 1986 test. In fact, Astrophys. 40, 263(2002) all our data points were measured at accelerations that [5] J.D. Anderson et al, Phys. Rev. D 65, 082004(2002) were smaller than the smallest acceleration in Ref. [6] [6] A. Abramovici and Z. Vager, Phys. Rev. D 34, 3240 Our ability to measure at even smaller accelerations is [7] W H. Press, S. A. Teukolsky, w.T. Vetterling, and B.P. complicated by the thermal excitation of the torsion pen- Flannery, Numerical Recipes in C: The Art of scientific dulum. Since we find good agreement with F= ma it is Computing( Cambridge University Press, Cambridge, unlikely that a possible violation of Hooke's law would England, 2002), 2nd ed exactly hide a violation of F=ma. Our torsion fiber is [8] J D Bekenstein, Phys. Rev. D 70, 083509(2004) 150801-3

indicate that force and acceleration remain proportional down to accelerations as small as 5 1014 m=s2. We conducted numerous checks for bias and systematic uncertainties in our data analysis. We performed an inde￾pendent alternative data analysis in which the data were divided into 1200 and 2000 s long sections irrespective of the spikes, which were not removed. In this analysis drift was accounted for by introducing a linear and quadratic term in the fit. Data sections that exhibited exceedingly large 2 were eliminated. This data analysis also produced good agreement with Newton’s law, but had slightly larger uncertainties. Our main data analysis was tested with two sets of simulated data. The first set contained a realistic spectral noise distribution of a harmonic oscillator and white read￾out noise. The frequencies were recovered properly for amplitudes comparable to the measured traces, as well as for smaller amplitudes. The uncertainties and the 2 be￾haved similarly to the corresponding quantities in the mea￾sured data. The second set contained simulated MOND￾like data with a range of a0 from 1016–109 m=s2. All the accelerations were recovered within uncertainties. In summary, we have found no deviation from the proportionality in Newton’s second law down to acceler￾ations of 5 1014 m=s2, which is approximately 1000 times smaller than the previous 1986 test. In fact, all our data points were measured at accelerations that were smaller than the smallest acceleration in Ref. [6]. Our ability to measure at even smaller accelerations is complicated by the thermal excitation of the torsion pen￾dulum. Since we find good agreement with F
ma it is unlikely that a possible violation of Hooke’s law would exactly hide a violation of F
ma. Our torsion fiber is expected to adhere ideally to Hooke’s law because it is operated far from its inelastic limit and also because the oscillator has a high Q. Our test does not invalidate MOND directly, since MOND requires that the measurement must be carried out in the absence of any other larger acceler￾ations, such as those due to the earth and our solar system. However, our results constrain any theoretical formalism seeking to derive MOND from fundamental principles [8] by requiring that formalism to reproduce F
ma under laboratory conditions similar to those in our experiment. For a future measurement we plan to test Fg
ma at small accelerations, where Fg is a gravitational force as in the galactic dynamics. The work at the University of Washington is supported by the NSF under Grant No. NSF PHY-0355012 and E. F. is supported in part by the U. S. Department of Energy under Contract No. DE-AC02-76ER071428. [1] Y. Sofue and V. Rubin, Annu. Rev. Astron. Astrophys. 39, 137 (2001). [2] M. Milgrom, Astrophys. J. 270, 365 (1983). [3] M. Milgrom, Astrophys. J. 270, 371 (1983). [4] R. H. Sanders and S. S. McGaugh, Annu. Rev. Astron. Astrophys. 40, 263 (2002). [5] J. D. Anderson et al., Phys. Rev. D 65, 082004 (2002). [6] A. Abramovici and Z. Vager, Phys. Rev. D 34, 3240 (1986). [7] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University Press, Cambridge, England, 2002), 2nd ed. [8] J. D. Bekenstein, Phys. Rev. D 70, 083509 (2004). 10-13 10-12 10-11 10-15 10-14 10-13 10-12 acceleration (m/s2 ) force (N) -5 0 5 res. (10-14 m/s2 ) -3 0 3 res. (10-15 N) FIG. 2 (color online). The measured force versus the measured acceleration. The solid line is the best fit for accelera￾tion, a being exactly proportional to force F. Our data agree very well with the curve. The insets on the right and top of the main graph give the residuals of the data to the fitted line. PRL 98, 150801 (2007) PHYSICAL REVIEW LETTERS week ending 13 APRIL 2007 150801-3