Determining the gravitational constant G using a gravitational torsion balance

Transcription

1 Determining the gravitational constant G using a gravitational torsion balance Alexander Bredikin Department of Physics and Astronomy, Ithaca College, Ithaca, New York 1485 (Dated: May 8, 217) A gravitational torsion balance was used to determine the value for the gravitational constant G. A large mass was brought close to the small masses at either end of the balance s pendulum arm, which is suspended by a torsion wire. This caused the small masses to oscillate due to the gravitational attraction between the large and small masses. Attached to the torsion wire is a small mirror that reflected the laser on a distant screen, and the motion of the pendulum arm was tracked using the reflected laser light and tracking software. This position and time data was plotted, and two methods were used for analysis. The first method used the plot to estimate the period of oscillation of the small masses, as well as the displacement of the equilibrium point of the reflected laser. For the second method, a position versus time function was superimposed over the data. The parameters used to generate this function were adjusted until it fit the data. These parameters were then used to improve the initial calculations of G. The final reported value of G is (7.2±.7) 1 11 m 3 s 2 kg 1, which falls over the accepted value of G, which is m 3 s 2 kg 1. The results of this experiment indicate the capability of the advanced lab torsion balance to determine important physical constants, like G. I. INTRODUCTION Gravity is a remarkable property of matter that causes objects of any mass to be attracted to each other. Isaac Newton first described this in his famous publication, the Principia. It was in this text that he declared the law of universal gravitation as a means to explain the motion of celestial bodies, that the whole force with which one of these spheres attracts the other will be reciprocally proportional to the square of the distances of the centres. 1 This is written mathematically as F G = G Mm r 2, (1) where the masses of the two objects are M and m, the square of the distance between their centers is r 2, and there exists a constant of proportionality G, called the gravitational constant. In 1797, Henry Cavendish devised an experiment to determine the density of Earth, and the initial work completed in this experiment was able to, for the first time since Newton, provide an accurate measurement of G. 2 However, even with modern experimental methods and equipment, G is the least precise physical constant. The goal of this experiment is to measure G to understand how precisely it can be measured using advanced lab equipment. In this experiment, a gravitational torsion balance balance was used to measure the force between two masses. The balance and a laser were mounted on an optical table. The laser was directed at a mirror that is attached to a pendulum arm. A small mass rested on either end of the pendulum arm. When one of the large masses was brought close to small masses, the arm began to oscillate due to the force of attraction between the largeand small masses. This caused the mirror to move, so the reflection of the laser moved, and it was possible to track the motion of the pendulum arm using video recording and tracking software. Using this information, a value for G was derived. Two methods were used. First, by shifting the large masses to alternate positions, the small masses began to oscillate, causing the laser point to oscillate on the screen and eventually settled at a new position. This motion was recorded using a high speed camera. By analyzing the recording using motion tracking software and creating a data plot with that information, we estimated the period of these oscillations and the change in the equilibrium point of the reflected laser point. These estimates were used to determine G using a derived calculation. The second method utilized the tracking software, but instead of using estimates for the period T and correction factor, the data were fit using the equation for a damped harmonic oscillator. The values used to generate this fit were used to repeat the calculation from the first method, but with improved values and not just estimates. II. THEORY A. Initial calculations for G The gravitational torsion balance employs two sets of masses one large set and one small set to measure the gravitational attraction between them. We begin by considering Newton s law of universal gravitation in the context of this experiment, given by F G = G Mm b 2, (2) where G is the gravitational constant, M is a large mass, m is a small mass, and b is the separation distance between the large and small masses. The attraction due to gravity between the large and small mass causes the

2 2 pendulum arms to rotate, and so a torque is produced, τ gravity = r F G. (3) The magnitude of the vector r is the length of the pendulum bob s lever arm between the small mass and the center of the arm s rotation. Because there is a mass on each end of the pendulum arm, we multiply by two and take the cross product of r and F G. The vectors r and F G are perpendicular, and so the magnitude of the torque is τ gravity = 2F G d, (4) wheref G istheforceduetogravity,anddisthelengthof the pendulum bob lever arm. Following Newton s Third Law of Motion, the torsion band, with a torsion constant κ and an angle θ through which it is originally twisted, provides an equal and opposite torque, given in the PASCO manual 3 by τ band = κθ. (5) Though there are multiple torques acting in the system, the system will find itself at equilibrium, and an equation of motion reveals itself: This reveals that Solving for F G yields It follows, then, that = τ gravity +τ band. (6) τ gravity = τ band 2F G d = κθ. (7) F G = κθ 2d. (8) κθ 2d = GMm b 2. (9) Although the values for θ and κ are unknown, we can apply knowledge of periodic motion to solve for both. We begin with θ. As shown in Figure 1, the gravitational attraction between the large and small mass causes the pendulum arm to rotate a small distance. Assume two equilibrium positions at S 1 and S 2, the result of the large masses being in different orientations with respect to the pendulum arm. When the large mass swivel support moves to the opposite side, there is a change from S 1 to S 2 so that S = S 2 S 1. Along with a change in the equilibrium points, the angle of rotation of the torsion band also changes. When the pendulum arm and by extension, the torsion band is at rest, the torsion band is twisted by an angle of θ. When the large masses are shifted, the pendulum arm rotates the other way, and so the torsion band twists the other way as well. The torsion band firsts untwists to where the angle of torsion is. The torsion band continues to twist as FIG. 1. When the large masses are shifted to the alternate position, the small masses begin to oscillate due to the gravitational attraction between the large and small masses. The oscillation is damped, and eventually the laser resettles to a new equilibrium point on the screen. The change in the distance from the original equilibrium point to the final is S. the pendulum arm continues to rotate (while the small masses are coming to equilibrium in the opposite direction). Therefore, the torsion band twists 2θ once when it untwists to and once when it twists to the new equilibrium point. We approximate this change using a small angle approximation to create a trigonometric statement that tan(2θ) 2θ = S/2L. Solving for θ now yields θ = S 4L, (1) where L is the distance from the apparatus to the screen. Solvingforthetorsionconstantκcanbedoneusingthe same principles of oscillatory motion. As the pendulum rotates, it has a certain period T such that T 2 = 4π2 I κ. (11) Eq. 11 is the rotational analog of the period of harmonic motion. We will derive the period of a linear harmonic oscillator, and then relate it to the relationship shown in Eq. 11. For simple harmonic motion, we begin with an oscillator that moves with an amplitude A like x(t) = Acos(ωt) (12) Taking the first and second derivatives provide the motion for velocity and acceleration x (t) = v(t) = ωasin(ωt) (13) x (t) = a(t) = ω 2 Acos(ωt). (14) Note that the Acos(ωt) term in Eq. 14 is also the position function shown in Eq. 12. Eq. 12 is substituted into Eq. 14 so a(t) = ω 2 x(t). (15) For simple harmonic motion, there is a force acting on the system to cause the motion, like m a(t), and a linear

3 3 restoring force, like k x(t), where k is an intrinsic property of the oscillator (the spring constant). For a system undergoing simple harmonic motion, we find ma(t) = kx(t). (16) If we substitute Eq. 12 and Eq. 15 into Eq. 16, divide both sides by x(t), and then take the square root of both sides, we see k mω 2 x(t) = kx(t) ω = m. (17) The definition of angular frequency is ω = 2πf, where f is the ordinary frequency of the system. Frequency is also the inverse of period T, so with these substitutions, 2π T = k m m T = 2π k. (18) Squaring Eq. 18 yields T 2 = 4π2 m. (19) k Note the similarity of Eq. 19 to Eq. 11. Because we are working with a rotational system, we used the moment of inertia I in place of the ordinary inertia m, and we replace the spring constant k with the torsion constant κ. This yields the equation originally seen in Eq. 11, shown again here: T 2 = 4π2 I κ. (2) In Eq. 2, I represents the moment of inertia of the level arm-small mass system, I = 2m (d ) r2, (21) where d is the distance from the level arm center to the small mass m and r is the radius of the small mass. A brief derivation of this moment of inertia a modification of the parallel-axis theorem follows because of its importance to this experiment. By breaking up a larger body, it is possible to find the moment of inertia for each of the smaller pieces of the mass and then sum up the contribution by each piece using a Riemann sum to find the total moment of inertia. This can be written as I = r 2 m n = r 2 dm. (22) n The r 2 term in Eq. 22 can be rewritten as the sum of the components of the distance between the center of mass and the piece of the mass x 2 n + yn 2 and the parallel axis and the piece of mass d 2 x + d2 y. This method is seen in Eric Mazur s Principles of Physics. 4 After substituting FIG. 2. The spherical coordinate system, as shown in the Cartesian coordinate system. ρ is the distance from the origin toapoint p, φis theangle betweenzenithand thesegment ρ, and θ is the angle formed between the x-axis and the projection of the segment ρ in the xy plane. these values and simplifying, Eq. 23 shows the result to be I = md 2 +I cm. (23) Continuingthe derivationofeq.21, wemustfindthe moment of inertia of the center of mass I cm of a solid sphere with uniform density. To derive this value, a spherical coordinate system will be used, shown in Figure 2. The density δ of the sphere is defined as m/v. By changing both m and V, we maintain that δ = dm/dv. Solve for dm = δdv. Referring back to Eq. 22, we can substitute the value for density and eliminate the dm term, and we integrate over some volume V: I = r 2 δdv. (24) Now, wefind anequationforr thatwecanintegrateover. By using trigonometry and the geometry of the spherical coordinate system, we find that r = ρsin(φ) and so I = ρ 2 sin 2 (φ)δdv. (25) We insert the limits of integration and the variables over which the integral will be calculated. Because this is a triple integral in spherical coordinates, the dv term can be expanded to ρ 2 sin(φ)dρdφdθ. In Eq. 26, the ρ 2 sin(φ) terms have been distributed already: I = 2π π r ρ 4 sin 3 (φ)δdρdφdθ. (26)

4 4 The final result shows that the moment of inertia for the center of mass for a solid sphere of uniform density is I = 2 5 mr2, (27) wheremisthe massofthesphereandr isitsradius. This is substituted into Eq. 23 and multiplied by 2 (because there are two solid spheres) to show the total moment of inertia for the pendulum arm and small mass system is I = 2md mr2 = 2m (d ) r2. (28) We return to the original reason that we derived this moment of inertia. I was a component of the period T from Eq. 11. The moment of inertia is substituted in for I, and we solve for κ to yield κ = 8π2 m(d r2 ) T 2. (29) We now substitute Eq. 1 and Eq. 29 into Eq. 8 and solve for G to yield G = π2 Sb 2 (d r2 ) T 2. (3) MLd B. Improving values for S and T The initial calculation of G utilizes estimated values of S and T. By fitting the initial data to a curve in data analysis software, we can obtain improve values of S and T, thereby improving our calculation of G. This plot will created by using [ x(t) = A 1 e t cos( t) e t ] sin( t). (31) John Taylor lays the foundation of the formulation in Eq. 31 in his Classical Mechanics text. 5 Some key components of Eq. 31 are reviewed here for clarity. The term can be expanded to = ω 2 o 2. (32) where is the damped frequency, ω o is the undamped frequency, and is the damping ratio. The sin( t) and cos( t) terms are periodic functions that reflect the oscillatory nature of the pendulum arm s movement. The e t term contributes to the damped harmonic motion of the pendulum arms. When the large masses are moved into a new position, the small masses oscillate on the end of the pendulum arm. Over time, their movement is damped and they come to an equilibrium point. Eq. 31 was plotted over the data collected from the tracking software, and the values of A,, and were adjusted until the fit matched the experimental data. Eq. 32 can be rearranged so that ω o = ω (33) FIG. 3. The large mass and small mass system. The large and small masses are separated by a distance b, and the length of the pendulum arm between the center and the small mass is given by d. The force between the adjacent large and small masses is F G, and the force between the alternate large mass and small mass is F. The angle φ is formed between the entire pendulum arm 2d and the force F. The force f, which acts in the opposite direction of F is the y-component of F. Using the value for ω o, the ordinary frequency f o can be found from ω = 2πf. The value for the ordinary frequencycanbe usedtodeterminetheperiodofthemotion because frequency and period are inversely related. This entire relationship is brought together to show the period is given by T o = 2π (34) ω The value for T o can be used to calculate G, G = π2 Sb 2 (d r2 ) T 2 o MLd. (35) C. Improving calculations for G This relationship was originally seen in Eq. 3. This value for G only accounts for the force between the small mass m and the adjacent large mass M. However, the other large mass in the system is close enough to the small mass that there is a measurable difference in the gravitational constant G. Figure 3 shows the force F of the alternate large mass on the small mass. We are interested in the component of F that opposes the gravitational force between the adjacent small and large masses, which shall be called f. The angle between F and the pendulum arm is φ. The use of trigonometry offers up the way to determine f using F and the distances b and d in the system. The forces F and f and distances b and d are related such that sin(φ) = f F (36)

5 5 and sin(φ) = b b2 +(2d) 2 (37) where we have solved for the length of the hypotenuse using the Pythagorean Theorem. Putting Eq. 36 and Eq. 37 together yields and therefore f b = (38) F b2 +(2d) 2, f = The F term is equal to bf (39) b2 +(2d) 2. GMm F = ( (4) b 2 +4d 2 ) 2, and so by substituting Eq. 4 into Eq. 39, the result is bgmm f = (b 2 +4d 2 ) (41) b 2 +4d2. By simplifying Eq. 41, the result is f = bgmm (b 2 +4d 2 ) 3 2 = F G, (42) where is a correction factor for the force due to the gravitational attraction between the masses. We can solve for the constant : ( )( ) bgmm b 2 b 3 = =. (43) (b 2 +4d 2 ) 3 2 GMm (b 2 +4d 2 ) 3 2 Now that we know the force F G between the adjacent large and small masses and the y component f of the force F between the alternate large mass and the small mass, we can calculate the net force F net acting on the small mass as F net = F G F G = F G (1 ) (44) This net force is beneficial, because we can now find a correctedgravitationalconstant, G, which takesinto account the force on the small mass as a result of both the adjacent and alternate large mass, given by G = G (1 ) G = G 1 (45) The value G is a more accurate determination of G, because it considers other forces in the torsion balance system. FIG. 4. The torsion balance as seen from a bird s eye view. There are two positions in which the large lead masses can be oriented, as indicted in the figure. When they are shifted to the alternate position (as shown by the dashed arrow, the small masses begin to oscillate because of their gravitational attraction to the large masses, the pendulum arm swings, causing the mirror (not shown) to move. On a distant screen (L = (1.5±.5) m), the laser reflected by the mirror will show the motion of the pendulum arm as it oscillates. III. EXPERIMENT A PASCO Scientific gravitational torsion balance model AP and X-Y diode laser model OS-8526A were employed to conduct this experiment. A top view of the torsion balance is shown in Figure 4 to emphasize the two positions in which the large masses can be oriented, as well as to show the motion of the pendulum arm which holds the small masses. A front view of the system is shown in Figure 5 to demonstrate the mirror and pendulum arm as they are connected to the torsion wire. The experiment was conducted on an optical table to reduce vibrations due to motion in the surroundings of the apparatus. A laser was mounted about 6 centimeters in front of the torsion balance. Two large lead balls, with a mass M of (1,5±1)g and radius R of (31.9±.1) mm each, were placed on the balance s swivel arm. The small masses have a mass m of (38.3±.2) g, and a radius r of (9.53 ±.1) mm. They are placed a distance d from the center of the pendulum arm that is (5±1)mm. The large and small masses are separated by a distance b of (46.5±.1) mm. After placing the large masses into either position I or II, the system was given about 6 hours to ensure it was settled. It is paramount to the success of this experiment that the system is not disturbed during the data-taking process. Precautions should be made to prevent individuals from approaching, touching, or otherwise interfering with the torsion balance. Motion near the optical table

6 6 4 3 Amplitude (mm) Time (s) FIG. 6. The position of the laser point on the screen was plotted against the time when it was there. This is a damped oscillation. FIG. 5. The torsion balance as seen from the front. A torsion wire holds the pendulum arm, which holds the two small masses. A mirror is mounted on the torsion wire, so that the laser canbe reflectedonadistantscreen andthemotion of the pendulum arm can be observed. The large masses can change position by rotating the swivel arm. In the figure, the large mass on the right is in front of the small mass. The torsion wire, pendulum arm, mirror, and small masses, are contained in a metal frame to prevent air currents from disrupting the system. could throw the system out of equilibrium, tainting the data. The large masses were then switched into the alternate position, and the small masses began to oscillate. The mirror, attached to the torsion wire, also oscillated, and the laser pointed at the mirrorwas reflected onto the screen. A Casio Exilim high speed camera was used to record the motion of the laser point as it travelled on the screen. The system was given 2 hours to reach an equilibrium point. The recorded video was removed from the camera, and imported into the Tracker software. This enabled motion tracking of the laser point on the screen over time. IV. RESULTS A. Calculating G using estimates from the damped harmonic oscillator plot The position and time data from the software Tracker were imported into MATLAB for analysis and are shown in Figure 6. The period of the oscillation was determined by inspection of the plot in Figure 6 and its corresponding data. First, direct inspection of the plot pro- vided a general idea of the beginning and end of a cycle (for example, inspection of the graph in Figure 6 suggests the first period begins at around 25s, and ends at around 75s). Then, the position and time data were examined to find the values for position more accurately. These values were assigned an estimated error of ± 4s, because only one frame was taken every other second, and only every other frame was analyzed. This process was repeated for the first hour of cycles (until t = 3,6s). The values of the periods in the first hour were averaged, so T = (497±2)s. The change in the point of equilibrium was also calculated by inspection, and found to be ((15±5)mm) - (( 144±5)mm) = (249±7)mm. Using these values, as well as the values for b, d, r, L, and M, I solved for G, yielding a value of m 3 s 2 kg 1. The constant G is notorious for being the least certain physical constant, so understanding error propagation in this calculation is critical. Error propagation methods presented here are adapted from John Taylor s Introduction to Error Analysis. 6 Due tothelength ofthe equation used to calculate G, we will consider the error in parts, beginning with the numerator. For the (d r2 ) term in the numerator, we must apply the rules for uncertainty in a power, uncertainty times an exact number, and uncertainty for sums. This yields δq = ( 2 [ ] 2 δd d) + d ( 2 5 { 2 [ ] 2 δr r}), (46) r where ( q is the best calculated value in the term ( d r2). Plugging values into Eq. 46 shows q = d r2) = (2,536±2)mm 2. Now, the remaining terms of the numerator are all multiplied together, so we may add the fractional uncertainties in quadrature, so that δn n = (δ S S ) 2 +( 2 [ ]) 2 δb + b ( ) 2 δq. (47) q Using Eq. 47, the fractional uncertainty in this part of the numerator is The best value for these

7 7 terms in the numerator is mm 5, and so the absolute uncertainty of this part of the numerator is mm 5. Finally, we remember that all of these terms are multiplied by the constant π 2 (and therefore so is the error), so the best value for the numerator is (1.348±.1) 1 1 mm 5 = (1.348±.1) 1 5 m 5. We now consider the uncertainty in the denominator. This analysis is more straightforward, requiring the application of the rule for uncertainty of products. For the denominator (represented by E), ( [ ]) 2 δt 2 + T ( ) 2 ( ) 2 ( ) 2 δe δm δl δd E = + +. M L d (48) Using Eq. 48, the fractional uncertainty is found to be.527. The best value for the denominator is 194,519m 2 s 2 kg, and so the absolute uncertainty is 1,m 2 s 2 kg. Therefore, the denominator is (19,±1,)m 2 s 2 kg. Adding the fractional errors in quadrature for the numerator (.7) and the denominator (.527) yields δg G = (.7) 2 +(.527) 2 =.527. (49) The final reported value for this section of our analysis is G = (6.9± ) m 3 s 2 kg 1. This method agrees with the value of G ( m 3 s 2 kg 1 ), within the margin of uncertainty. This value has a fractional uncertainty of about 5%, and there is a discrepancy of only.2± 1 11 m 3 s 2 kg 1, so we are confident in the precision of this value of G. B. Improving the calculation of G using a damped harmonic oscillator fit The previous calculation of G was completed using estimates for S and T. Using the data from the initial analysis, a new plot can be created by superimposing the fit [ x(t) = A 1 e t cos( t) e t ] sin( t) (5) over the data, where = ω 2 o 2. (51) The plot is shown in Figure 7. The curve was adjusted to fit the data, using values of A = S = 262mm, =.49, =.1262 Hz. We take into account the uncertainty in recalculated G. Uncertainties in A,, and were found by adjusting the fit over the data until the fit did not resemble the plotted data. Using this method, δa = δ S = 4mm, δ =.1 Hz, and δ =.4. By rearranging Eq. 51, we find ω = ω (52) Amplitude (mm) Time (s) FIG. 7. The position of the laser point on the screen was plotted against the time when it was there. This is a damped oscillation. This oscillation has been plotted with Eq. 5 superimposed to show the correlation between the fit and the data. The superimposed curve, in red, was designed by changing values of the amplitude A, the decay constant, and the damped angular frequency. These values were adjusted until the curve fit the data. The best valueforω o, usingeq.52,is.12,63hz. The uncertainty in ω1 2 is given by [ ] δω1 2 δω1 = 2 ω1, 2 (53) and the uncertainty in is given by [ ] δ δ 2 = 2 2. (54) Therefore, by adding in quadrature, the uncertainty for ω is ( δ ( [ ω1 2 +2) δω1 = 2 ] 2 ( ω1) [ ] ) 2 δ 2. (55) Applying Eq. 55, the uncertainty in ( ω ) ( 1 2) can be found with δ ( ω ) ( 1 2) = 1 2 ( [ ) 2 ( 2 δω1 ]ω (ω ) [ δ ) 2 ] 2 ( ω ) ( 1 2). (56) Using Eq. 56, we find δ ( ω ) ( 1 2) = δωo =.1Hz. Therefore, ω o = (.126±.1) Hz. We revisit Eq. 34 to solve for the desired quantity T o, so T o = s. Using the fractional uncertainty of ω o, we can find the absolute uncertainty for T o. We discover T o = (499±4)s. Repeating the calculation for G and δg with these improved values, we find G = (7.2±.4) 1 11 m 3 s 2 kg 1. Unlike before, the experimental and accepted value do not agree

8 8 within the margin of uncertainty for this calculation, because this value for G is higher than previously calculated. There is a discrepancy of m 3 s 2 kg 1. We go further now, returning to Eq. 45, where G = G 1. (57) We can evaluate the error in G by applying (δg ) 2 δg = + G ( ) 2 δ G. (58) Solving for G and its error provide that G = (7.2 ±.7) 1 11 m 3 s 2 kg 1. This result provides the same value of G as previously calculated (with the same discrepancy of m 3 s 2 kg 1 ), but with a greater error. Within the margins of this error, the experimental and accepted values do agree. This larger error is expected for G, because even though holds a very small value, its uncertainty must be considered in addition to the uncertainty in G. V. CONCLUSIONS The gravitational constant, G, was determined using a gravitational torsion balance. A laser was pointed onto a mirror on the apparatus. By shifting the position of the large masses, the small masses oscillated, causing the mirror to oscillate as well. On a distant screen, the laser (reflected by the mirror) showed the motion of the pendulum arm holding the small masses. The movement of the reflected laser was recorded, and motion tracking software was used to create a position vs. time plot. Using information extrapolated from this plot, G was initially calculated to be (6.9 ±.7) 1 11 m 3 s 2 kg 1. Within the margin of error, this value agrees with the accepted value of G ( m 3 s 2 kg 1 ), with a discrepancy of.2m 3 s 2 kg 1, and a fractional uncertainty of about 5%. After the data were compared to a plot fit over the curve formed by the data, a new value for G was calculated. This process found that G = (7.2±.7) 1 11 m 3 s 2 kg 1. This value has a similar fractional uncertainty, but it also has a greater discrepancy. This is unexpected, because this second method used values calculated using a data fit. These values should have been more in line with the true values, more so than their estimations. Future work with this experiment and apparatus should focus on improving measurementsfor L and T specifically. Because T is squared during calculations, it contributes significantly to the value of G and its uncertainty. Furthermore, the implementation of a photo-gate could prove to be beneficial in the data-collecting process, where instead of recording the movement of the reflected laser point and importing the video into tracking software for manual analysis, the period could be calculated from the photo-gate triggers. abredikin@ithaca.edu 1 I. Newton, A. Motte, W. Davis, W. Emerson, and J. Machin, The Mathematical Principles of Natural Philosophy, The Mathematical Principles of Natural Philosophy No. v. 1 (H.D. Symonds, 183) pp H. Cavendish, Philosophical Transactions of the Royal Society of London 88, 469 (1798). 3 Instruction Manual and Experiment Guide for the PASCO scientific Model AP-8215 Gravitational Torsion Balance, PASCO scientific, 111 Foothills Blvd, Roseville, CA (1998). 4 E. Mazur, Principles & Practice of Physics, 3rd ed. (Pearson Education Limited, 215). 5 J. Taylor, Classical Mechanics (University Science Books, 25). 6 J. Taylor, Introduction To Error Analysis: The Study of Uncertainties in Physical Measurements, A series of books in physics (University Science Books, 1997).