Measuring Newton's Constant of Universal Gravitation using a Gravitational Torsion Balance

Journal of the Advanced Undergraduate Physics Laboratory Investigation Volume 1 Issue 1 Article 5 2013 Measuring Newton's Constant of Universal Gravitation using a Gravitational Torsion Balance Zachary C. Waldron American University, zw4784a@american.edu Follow this and additional works at: http://opus.ipfw.edu/jaupli Opus Citation Waldron, Zachary C. (2013) "Measuring Newton's Constant of Universal Gravitation using a Gravitational Torsion Balance," Journal of the Advanced Undergraduate Physics Laboratory Investigation: Vol. 1 : Iss. 1, Article 5. Available at: http://opus.ipfw.edu/jaupli/vol1/iss1/5 This Article is brought to you for free and open access by the IPFW Student Journals at Opus: Research & Creativity at IPFW. It has been accepted for inclusion in Journal of the Advanced Undergraduate Physics Laboratory Investigation by an authorized editor of Opus: Research & Creativity at IPFW. For more information, please contact admin@lib.ipfw.edu.

Measuring Newton's Constant of Universal Gravitation using a Gravitational Torsion Balance Cover Page Footnote We would like to acknowledge Dr. Gregory Harry for his instruction and guidance in this experiment. We would also like to thank the American University Physics Department for the use of their space and equipment. This article is available in Journal of the Advanced Undergraduate Physics Laboratory Investigation: http://opus.ipfw.edu/jaupli/ vol1/iss1/5

Waldron: Measuring Newton's Constant of Universal Gravitation 1 Introduction Newton s Universal Law of Gravitation states that any particle or mass will, attract every other particle in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers [1]. In mathematical form, this can be written as: F gravity = G m 1m 2 b 2 This relation was first formulated by Isaac Newton in his 1686 publication, Philosophiæ Naturalis Principia Mathematica. He predicted that there must be some proportionality constant with units of m 3 kg 1 s 2 ; this constant is now known as G, or Big G [1]. In 1798, Henry Cavendish conducted the first experiment that tested Newton s Universal Law of Gravitation in a lab, and calculated the first measured value for Big G [2]. Cavendish did his measurement using a rotating torsional pendulum with spherical masses on each end. He then used additional heavy spheres placed in specific positions relative to the small masses to cause the force of gravity between them to twist the pendulum. Knowing the mass of each sphere, the torsional qualities of the pendulum s fiber, the lengths of pendulum s, and the measured movement of the masses allowed him to carefully calculate a measured value for G. His experiment was so accurate that a more precise measurement for Big G was not made for nearly 100 years, when C.V. Boys officially proposed the Newtonian Constant of Gravitation to the Royal Institution of Great Britain in 1894 [3]. Having an accurate measurement for Big G has implications on calculations throughout Newtonian Mechanics and General Relativity. Having a more precise measurement can have a subtle impact on the various formulations it is involved in such as Einstein s Field Equations, Newton s law of gravitation, and in orbital mechanics [2]. Published by Opus: Research & Creativity at IPFW, 2013 1

Journal of the Advanced Undergraduate Physics Laboratory Investigation, Vol. 1 [2013], Iss. 1, Art. 5 2 Methodology This experiment is performed using the PASCO scientific AP-8215A Gravitational Torsion Balance, which allows the users to replicate Henry Cavendish s famous 1798 experiment using a similar, yet more modern, method. Figure 1 depicts the apparatus and some of its high level components. Figure 1: A front-forward view of the apparatus and its high level components. Taken from PASCO Manual 012-11032A. Used with permission. Figure 2: A view inside the box that surrounds the small mass, torsional pendulum. Taken from PASCO Manual 012-11032A. Used with permission. The apparatus consists of two 38.3 ± 0.2 gram masses that are suspended from a beryllium copper ribbon that is 260 mm long and has a cross section of 0.017x 0.15mm. Together they make up a torsional pendulum whose oscillations are independent of Earth s gravity. The torsional ribbon and rotational pendulum with small masses are housed inside of a box with a protruding rod to limit their interaction with the environment; this can be seen in Figure 2. Next, we used two 1500 ± 10-gram lead masses that can be positioned in two different configurations relative to the smaller masses in the pendulum. Figure 3 shows these two reference positions for the large masses. As the larger masses are brought close to the smaller masses, their gravitational attraction can be observed via the movement of the pendulum. Since the pendulum s rotation is very small, a mirror is affixed to the pendulum and a laser is reflected from the mirror, forming an optical lever. This http://opus.ipfw.edu/jaupli/vol1/iss1/5 2

Waldron: Measuring Newton's Constant of Universal Gravitation amplifies the deflection of the pendulum, which becomes easily measurable on a distant wall. This effect can also be seen in Figure 3. We used a laser rangefinder to measure the distance to the wall to be L=11.52 ± 0.01m. Figure 2: A top down view of the effects of the optical lever. The two positions of the large masses are labeled. Taken from PASCO Manual 012-11032A. Used with permission. A crucial step for the success of this experiment is the initial set-up, alignment, and calibration of the apparatus with the optical lever. This step involves making sure that the laser s reflected point from the mirror affixed to the pendulum aligns with the point reflecting off the glass panel on the base of the apparatus. The latter reflected point serves as the equilibrium position for the whole apparatus and all measurements were made relative to this equilibrium. We used three different methods to measure Big G that were suggested in the PASCO manual: final deflection (Method 1), equilibrium positions (Method 2), and acceleration of gravity (Method 3). Published by Opus: Research & Creativity at IPFW, 2013 3

Journal of the Advanced Undergraduate Physics Laboratory Investigation, Vol. 1 [2013], Iss. 1, Art. 5 2.1 Theory The gravitational force between each small mass (m1) and its respective large mass (m2) is given by: Equation 1.1 F = Gm 1m 2 b 2 Where b is the distance between the centers of the two masses. The force of gravity between the small masses and the large masses produces a torque τ on the pendulum: Equation 1.2 τ grav = 2Fd Where d is the length of the lever holding the pendulum s cross piece. By Newton s 3 rd Law, there must be an equal and opposite force being supplied by the band which is quantified as: Equation 1.3 τ band = kθ Where k is the torsion constant of the band and θ is the angle by which it is twisted. Therefore, we can combine these three equations and rearrange to give the relation: Equation 1.4 G = k θ b2 2 d m 1 m 2 This equation tells us that the only unknowns we must solve for are dependent upon the movement of the pendulum. Fortunately, this movement can be tracked quite easily using our optical lever set up. In order to determine the values of k and θ, we can cause the system to oscillate by moving our large masses from Position 1 to Position 2. Now our system will oscillate from equilibrium position 1 (S1) until it settles in a new equilibrium position (S2) determined by position 2 of the masses. θ is now twice its original magnitude, and its angle is doubled by the optical lever. Some rearranging of this gives: Equation 1.5 θ = (S 2 S 1 ) 4 L The torsion constant k can be determined by observing the period of oscillation and using the relation between period and moment of inertia (I): Equation 1.6 and 1.7 T 2 = 4 π2 I k I = 2 m 2 (d 2 + 2 5 r2 ) Combining these two equations and solving for k gives: Equation 1.8 k = 8 π 2 m 2 (d2 + 2 5 r2 ) T 2 http://opus.ipfw.edu/jaupli/vol1/iss1/5 4

Waldron: Measuring Newton's Constant of Universal Gravitation Plugging the derived equations for k and θ into our equation for G gives our final equation for Big G with all variables being either known or measurable: Equation 1.9 G = π 2 (S 2 S 1 ) b 2 (d2 + 2 5 r2 ) T 2 m 1 L d 2.2 Method 1: Measurement by Final Deflection For this method, we placed the lead masses in Position 1 as shown in Figure 3 and allowed the system to come to equilibrium. After measuring Position 1 s equilibrium as S1, we moved the masses to be in Position 2, also shown in Figure 3, and used a stopwatch to measure three separate periods of oscillation. We then took the average of these three periods as a more accurate measurement. After the oscillations settled, we recorded the final position of the laser as Position 2 s equilibrium point S2. 2.3 Method 2: Measurement by Equilibrium For this method, we once again started in Position 1 equilibrium and rotated the masses to be in Position 2. As soon as the rotation was made, we measured the position of the moving laser point on the distant wall every 15 seconds for 30 minutes. After 30 minutes, we rotated the masses back to Position 1 and measured the position of the light point again every 15 seconds for another 30 minutes. 2.4 Method 3: Measurement by Acceleration The theory behind this measurement is slightly different than the one for the other two methods. We know that the force between large masses and the small masses will be given by the universal law of gravitation: Equation 3.1 F = Gm 1m 2 b 2 The system remains in equilibrium with the counteractive forces of the torsion wire exerting a torque against the direction of the gravitational attraction. When the masses are first put into position and the masses begin moving, the force of gravity before the torsion wire exerts its force is given by: Equation 3.2 F total = 2 F = 2 Gm 1m 2 b 2 By Newton s 2 nd law, this force can be written as: Equation 3.3 m 2 a 0 = 2 G m 1m 2 b 2 Published by Opus: Research & Creativity at IPFW, 2013 5

Journal of the Advanced Undergraduate Physics Laboratory Investigation, Vol. 1 [2013], Iss. 1, Art. 5 If the position of the movement of the laser point is measured for some time before the torsion wire begins acting on the system causing it to damp, then the acceleration of the point can be reasoned to be only caused by the force of gravity and the following rearrangement of Equation 3.3 can be used: Equation 3.4 G = b2 a 0 2 m 1 The acceleration (a 0 ) of the masses can be calculated from the kinematic equations and S can be found in the same way it was in Method 1, taking into account the double angle: Equation 3.5 Equation 3.6 a 0 = 2 s t 2 S d = t 2 L S = s ( 2L d ) Starting in Position 1 equilibrium, we once again rotated the masses to be in Position 2. We then recorded the position of the mass every 15 seconds for two minutes. According to the above theory, this gives us a measurement for acceleration that is only caused by the force of gravity. We can then use Equation 3.5 and 3.4 to calculate G. 2.4 Correction for Extra Forces Our value calculated using Equation 1.9 is subject to an extra source of error: the small sphere is attracted to the close large sphere and also the more distant large sphere, though with a much smaller force. Figure 3 A diagram showing the force acting on each small mass from the slightly more distant large mass. Taken from PASCO Manual 012-11032A. Used with permission. Based on Figure 4, we can see that: Equation 4.1 f = F 0 sin φ http://opus.ipfw.edu/jaupli/vol1/iss1/5 6

Waldron: Measuring Newton's Constant of Universal Gravitation Equation 4.2 F 0 = G m 1m 2 (b 2 +4d 2 ) Where F0 is the force felt from the distant large sphere. f is equal and opposite to the force felt by the close sphere, so: Equation 4.3 f = G m 1 m 2 (b 2 +4d 2 )((b 2 +4d 2 ) 1 2 = βf Where β is a parameter that is the ratio of f to F. Therefore, β has the form: Equation 4.4 β = b 3 (b 2 +4d 2 ) 3 2 Therefore, G is experimentally determined to be: Equation 4.5 G = G 0 (1 β) Therefore, rearranging gives: Equation 4.6 G 0 = G (1 β) This value for G0 compensates for the force acting on each small sphere from both large masses. 3 Results 3.1 Method 1: Measurement by Final Deflection Our recorded value for the equilibrium positions S1 and S2 are: Our recorded time for the period T is: S1= -30.2 ± 0.1cm S2= -5.3 ±0.1 cm T1= 513 ± 1s Using Equation 1.9, our calculated value for G was 5.93 10-11 m 3 kg 1 s 2. We had to account for the systematic error from the extra forces acting on the small masses as detailed in Section 2.4. Considering the uncertainties surrounding our measurements of the mass, position, and time in Method 1, and taking into account the force correction, our final value for G1 became: Published by Opus: Research & Creativity at IPFW, 2013 7

Journal of the Advanced Undergraduate Physics Laboratory Investigation, Vol. 1 [2013], Iss. 1, Art. 5 G1= 6.41 10-11 ±.05 10-11 m 3 kg 1 s 2 Given the NIST CODATA value of 6.67408 10-11 m 3 kg 1 s 2, this value for G1 is 5.1 times our uncertainty. 3.2 Method 2: Measurement by Equilibrium Mathematica was used to graphically determine the values for S1, S2, and T2 from the data we collected in Method 2. The Mathematica code can be found in the Appendix. 20 Position 1and 2Displacement v.time 0 Position (cm) - 20-40 - 60 0 500 1000 1500 2000 2500 3000 3500 Time (s) Graph 1: Graphical analysis in Mathematica was used to determine the above equilibrium positions. The blue curve shows Position 1 displacement and the orange curve shows Position 2 displacement. From Mathematica: S1 = -5.5 cm S2 = -29.9 cm T2 = 510.5 seconds Using Equation 1.9, our calculated value for G was 5.88 10-11 m 3 kg 1 s 2. We again accounted for the force correction as detailed in Section 2.4. Considering the uncertainty surrounding our http://opus.ipfw.edu/jaupli/vol1/iss1/5 8

Waldron: Measuring Newton's Constant of Universal Gravitation measurements of the masses, position, and time in Method 2, in addition to the force correction, our final value for G2 was: G2 = 6.36 10-11 ±.06 10-11 m 3 kg 1 s 2 Given the NIST CODATA value of 6.67408 10-11 m 3 kg 1 s 2, this value for G2 is 5.3 times our uncertainty. 3.3 Method 3: Measurement by Acceleration Below is a graph of ( s vs. t 2 ) based on the data collected using the procedures in Section 2.4. s was determined by taking our measured values for S and subtracting the equilibrium position S1 from each data point. Graph 2: A trend line for the first minute of data in Method 3 plotted as displacement versus time squared. Graph 2 above shows that the acceleration of the small mass is constant in the first minute, represented by the red data points, before it is impacted by the forces of the torsional wire. This constant acceleration means that we can use kinematics to solve for the acceleration as given by equation 3.5, then use equation 3.4 to solve for Big G. The acceleration of small mass (a0) was 1.1 10-7 m s -2 and our calculated value for G was 7.81 10-11 m 3 kg 1 s 2. Published by Opus: Research & Creativity at IPFW, 2013 9

Journal of the Advanced Undergraduate Physics Laboratory Investigation, Vol. 1 [2013], Iss. 1, Art. 5 We again had to account for the force correction as detailed in Section 2.4. Considering the uncertainty surrounding our measurements of the masses, position, and time in Method 3, in addition to the force correction, our final value for G3 was: G3 = 8.44 10-11 ±.13 10-11 m 3 kg 1 s 2 Given the NIST CODATA value of 6.67408 10-11 m 3 kg 1 s 2, this value for G3 is 13.5 times our uncertainty. 3.4 Propagation of Error We performed a propagation of error analysis by taking the partial derivatives of each of the measured values with respect to variables that had some inherent uncertainty attributed to them. These variables were the mass, the position, and the time for each measurement. The result for each calculated propagation of uncertainty for each method was: δg1 = 5.16 10-13 m 3 kg 1 s 2 δg2 = 5.88 10-13 m 3 kg 1 s 2 δg3 = 1.31 10-12 m 3 kg 1 s 2 Our measured values for G1, G2, and G3 were 5.1, 5.3 and 13.5 times our uncertainties, respectively. The reason for this is possibly that we underestimated how much error there was in the measurement of time and displacement for each of the methods. Specifically, for Method 3, there was a potential for error if the movement of the large mass, the measurements of the laser point and the timing of the two-minute period did not all start at the exact same moment. This is because the time period was so much shorter than the measurements for Method 1 and 2, so getting accurate results in this smaller interval was a challenge. Conclusion This experiment is a modern reproduction of the measurement of the gravitational constant as performed by Cavendish. In this treatment, a reflected laser is used to amplify the deflection of the pendulum. Proper and careful calibration technique was applied, and three measurement methods were used to garner support for our final determined value and to demonstrate multiple ways to measure G. Our most precise measurement for the constant of universal gravitation was 6.41 10-11 ± 5.16 10-13 m 3 kg 1 s 2. This measurement, given the NIST CODATA value of 6.67408 10-11 m 3 kg 1 s 2, was 5.1 times our uncertainty. http://opus.ipfw.edu/jaupli/vol1/iss1/5 10

Waldron: Measuring Newton's Constant of Universal Gravitation References [1] The Mathematical Principles of Natural Philosophy (1846)/BookIII-Prop1. (2011, March 16). In Wikisource. Retrieved 07:39, April 18, 2017, from https://en.wikisource.org/w/index.php?title=the_mathematical_principles_of_natu ral_philosophy_(1846)/bookiii-prop1&oldid=2506374 [2] Gottlieb, Michael. "Theory of Gravitation." The Feynman Lectures on Physics Vol. I Ch. 7: The Theory of Gravitation. California Institute of Technology, 2013. Web. 12 Apr. 2017, http://www.feynmanlectures.caltech.edu/i_07.html [3] Boys, C. V. (1895). On the Newtonian constant of gravitation. Philosophical Transactions of the Royal Society of London. A, 186, 1-72. [4] "Gravitational Torsion Balance Lab Manual." Lab Manual 012-11032A. PASCO. Web. Published by Opus: Research & Creativity at IPFW, 2013 11

Journal of the Advanced Undergraduate Physics Laboratory Investigation, Vol. 1 [2013], Iss. 1, Art. 5 Appendix A: Mathematica Code for Graphically Finding Equilibrium Position in Method 2 http://opus.ipfw.edu/jaupli/vol1/iss1/5 12

Waldron: Measuring Newton's Constant of Universal Gravitation Published by Opus: Research & Creativity at IPFW, 2013 13

Journal of the Advanced Undergraduate Physics Laboratory Investigation, Vol. 1 [2013], Iss. 1, Art. 5 http://opus.ipfw.edu/jaupli/vol1/iss1/5 14

Waldron: Measuring Newton's Constant of Universal Gravitation Published by Opus: Research & Creativity at IPFW, 2013 15