Determining Gravitational Acceleration
Determination of Gravitational Acceleration g in Burlington, Ontario, Canada
Using a Simple Pendulum
By: Julia C. Muller
Table of Contents
3 Experimental Design and Observation
4.2 Determining the length of pendulum L
4.2.1 Measuring the length of brass bob l_{ b}
4.2.2 Measuring the distance between the top of the suspension rod and the steel plate l_{o}
4.2.3 Measuring the gap between brass bob and steel plate l_{ g}
4.4 Calculating the value for the gravitational acceleration g
5 Evaluation of Results Obtained
5.1 Experimental result vs. accepted value
5.2 Search for systematic error
5.2.2 "Mathematical pendulum" vs. experimental setup
5.2.2.1 Determining the pendulum's corrected length L_{ cor}
5.3 Recalculating the gravitational acceleration g using L_{ cor}
6.1 Prospecting for minerals and deposits
7 Historical Notes on Pendulum and Pendulum Clocks
Determination of Gravitational Acceleration g
in Burlington, Ontario, Canada Using a Simple Pendulum
Summary
The gravitational acceleration in Burlington, Ontario was determined by observing the motion of a simple pendulum. The value of g = 9.821 ± 0.023 m/s^{2} determined with this simple setup agrees with the accepted value 9.805 m/s^{ 2} for this area to within the experimental uncertainty; however, the mass of the "string" had to be taken into consideration.
1 Introduction and Question
Despite the fact that the magnitude of the gravitational acceleration g permeates all physics' topics such as kinematics, dynamics, circular motion, work and energy, and  "naturally"  gravitation, throughout the courses its value as 9.80 m/s^{ 2} has been accepted as "given" and has not been challenged. In this experiment, an attempt is made to verify and/or determine the magnitude of the acceleration due to gravity g in Burlington, Ontario, Canada.
A simple harmonic motion is defined as a periodic vibratory motion in which the force and the acceleration are directly proportional to the displacement (Hirsch, 2003, p. 33). A simple pendulum undergoes simple harmonic motion if the oscillations have a small amplitude. A simple pendulum consists of a mass at the end of a cord (presumed to have no mass) suspended from a support so that it can oscillate back and forth. The mass is called the bob. For small amplitudes, the equation for the period is
;where L is the length of the pendulum (distance between its suspension point and the pendulum's centre of gravity) and g is the magnitude of the acceleration due to gravity (Hirsch, 2003, p. 214). The period T is the time it takes the bob to oscillate from the peak of a swing on one side to the other side and back. For small amplitudes, the period T depends only on the length of the pendulum L and on the magnitude of gravitational acceleration g where the pendulum is located; it does neither depend on the mass of the bob nor the initial angular displacement as long as the amplitudes are kept small. The error for amplitudes of 5º is less than 0.05%, for amplitudes of 2º it is about 0.01% (Bergmann and Schaefer, 1965, p. 138).
Conversely, the gravitational acceleration can be found by solving the above equation for g:
In other words: the gravitational acceleration g at a specific location can be determined by this formula if the length of the pendulum L and the corresponding period T are known.
The law of universal gravitation states:
where F_{G} is the magnitude of the force of gravity, G is the universal gravitation constant, m is the mass of a body influenced by the gravitational field of the central body of mass M, and r is the distance between the centres of gravity of the two bodies. From Newton's second law follows F_{g} = m*g , where F_{g} is the magnitude of force of gravity acting on a body of mass m and g is the magnitude of the gravitational field strength. At any specific location, F_{g} and F_{G} are equal, thus
→ →
This means in general that the magnitude of the acceleration due to gravity varies slightly depending on location; the greater the distance from Earth's centre, the lower the acceleration due to gravity or in other words: the gravitational acceleration g decreases with increasing altitude. Not only altitude but also latitude affects the magnitude of g (Sonneborn, 2005, p.12). At the same elevation, the value for g is slightly lower at the equator than at the poles because Earth bulges outward slightly at the equator (Hirsch, 2003, p. 274ff). In addition, as Earth rotates daily on its axis, the effects of the centrifugal acceleration on objects at the surface are very small; nonetheless, they do exist. The magnitude of the centrifugal force is a maximum at the equator, and decreases to zero at the poles. The acceleration at the equator is about 0.34% less than the acceleration due to gravity alone (Hirsch, 2003, p. 136).
The textbook states the acceleration due to gravity for Toronto (latitude 45° N, altitude 162 m above NN) as 9.805 m/s 2 [down] (Hirsch, 2003, p. 33).
The website of the United States National Geodetic Survey predicts the acceleration due to gravity for the location of this experiment, Burlington, Ontario, as g = 9.80400 m/s^{ 2} ± 0.0002 m/s^{ 2} (Roman, 2005).
3 Experimental Design and Observation
3.1 Experimental Design
To determine the gravitational acceleration g with this experiment, a simple pendulum was installed on a rigid support, the length of the pendulum L was determined, the pendulum bob was displaced by less than 3º from its resting position and released. The pendulum swung in a vertical plane and the time required for 100 oscillations was measured. From the values obtained for L and T, the gravitational acceleration g was determined from the equation
Fig. 1 depicts the setup of the experiment.
Fig. 2 depicts the actual setup of the experiment including materials used and the experimenter.
Fig. 3 shows some tools in detail.
Fig. 2 Setup of experiment 


Fig. 3 Detail of materials used 
3.3 Experimental Procedure:
3.3.01 The length of the brass bob l_{ b} was measured with a digital vernier caliper over six surface lines (about 60° apart) of the brass cylinder to verify also its end planes being parallel.
3.3.02 The diameter of the brass bob was measured with a digital vernier caliper at four locations to verify its cylindrical shape.
3.3.03 The mass of brass bob was determined on a postal scale.
3.3.04 The music wire was attached to the brass bob with setscrews and a screw driver.
3.3.05 The tripod was set up.
3.3.06 The mounting platform of the tripod was leveled with the aid of a spirit level in two perpendicular directions by adjusting the legs of the tripod.
3.3.07 The steel rod was secured by clamps onto the mounting platform of the tripod.
3.3.08 The steel plate was placed on the floor vertically below the mounting platform of the tripod.
3.3.09 The leveling of the steel plate was check with the spirit level to assure it being parallel to the mounting platform for the suspension rod.
3.3.10 The brass bob was placed onto the steel plate.
3.3.11 The music wire was looped around the steel rod and the length of the wire shortened so that the brass bob "floated" above the steel plate.
3.3.12 The music wire was wrapped around itself below the suspension rod.
3.3.13 The distance l_{o} from the top of the steel rod to the top of steel plate was measured with the meter stick.
3.3.14 The gap l_{g} between brass bob and steel plate was measured with feeler gages.
3.3.15 The pendulum's length L was calculated.
3.3.16 The brass bob was displaced from its resting position about 50 mm from the vertical in the direction of the steel rod.
3.3.17 The brass bob was released and let swing.
3.3.18 Ambient conditions were recorded (location, date and time, room temperature, humidity).
3.3.19 The time for 100 periods was measured with an electronic stopwatch.
3.3.20 At the end of the experiment, the gap between the brass bob and the steel plate was checked to ensure that no changes have occurred in the setup during the experiment.
3.3.21 The time T for one period was calculated.
3.3.22 The gravitational acceleration g was calculated.
3.3.23 The total uncertainty for the calculated value was determined and compared to the accepted value.
3.3.24 The accepted value for g was found not to fall within the uncertainty of the value determined in the experiment.
3.3.25 The systematic error was found to be caused mainly by the mass of the "string".
3.3.25 The pendulum's corrected length L_{cor} was determined.
3.3.26 The gravitational acceleration g was recalculated.
3.3.27 The accepted value for g was found to fall within the uncertainty of the value determined by using the pendulum's corrected length L_{cor}.
Date: 2007 March 10
Time: 16h30  17h30 EST
Location: latitude N 43° 17' 55"; longitude W 79° 51' 20"; (Fisheries and Oceans Canada, 1979)
83 m ± 2 m above Mean Sea Level, North American Datum 1927 (Energy, Mines and Resources Canada, 1984)
Room temperature: 21° C
Relative humidity: 45% (room)
Mass of brass bob m_{b}: 1361 g ± 2 g
Diameter of bob: Ø 52.82 mm ± 0.05 mm (very uniform within the precision of the digital vernier caliper)
Diameter of music wire: 0.054 mm ± 0.05 mm (very uniform within the precision of the digital vernier caliper)
4.2 Determining the Length of the Pendulum L.
4.2.1 Measuring the length of brass bob l_{b}
The brass bob has a uniform cylindrical shape with a diameter of Ø 52.82 mm ± 0.05 mm and a mass of 1361 g ± 2 g; however, the circular planes on either end of the cylinder are not truly parallel as can be seen from the measurements above. A uniform tapped bore extends through the axial centre line of the bob with setscrews on either end. The setscrews are required to secure the music wire to the bob. The two setscrews are mounted flush with the circular end planes of the cylinder, that the centre of gravity can be assumed to be located on the axial centre line halfway between the circular end planes, i. e. at 0.5*l_{ b} from either end plane.
4.2.2 Measuring the distance from the top of the suspension rod to the top of the steel plate l_{o}
The distance l_{o} was measured on two spots each on either side of the steel rod surrounding the looped end of the music wire.
The precision of a wooden meter stick looks at first sight not commensurate for an experiment of this nature. However, the distance on the unfolded meter stick from its steeltipped origin to the highest point on the segment shown on Fig. 4 (indicated by the red line) was measured on a coordinate measuring machine having a precision of ± 0.001 mm: it was found to be 1619.338 ± 0.001 mm at 20º C.
Figure 4: Detail of meter stick 
4.2.3 Measuring the gap between brass bob and steel plate l_{g}
The value for the gap was measured by sliding feeler gauges inbetween the steel plate and the bottom of the brass bob.
L represents the length of the mathematical or "simple" pendulum; i. e. the distance from the top of the steel rod (suspension point) to the assumed centre of gravity of the brass bob.
By using the average of the measurements:
L = 1565.5 mm  0.775 mm  0.5*83.18 mm = 1523.135 mm;
L = 1523.1 mm
Total maximum uncertainty of L is the absolute value of the sum of the individual
uncertainties: ±1.0 mm + 0.025 mm + 0.05 mm = ± 1.1 mm;
Thus, L = 1523.1 mm ± 1.1 mm;
From its resting position, the bob was displaced about 50 mm in the direction of the steel rod to which the music wire is attached and then released, that it swung in a plane vertically below the steel rod. The displacement of 50 mm corresponded to an angle of sin ^{1} (50/1523) ≈ 2º resulting in an initial amplitude of ≈ 4º. From this initial displacement the pendulum was swinging throughout the entire experiment on its own without any further "push" by the experimenter; evidently with decreasing amplitude.
The reaction time of the observer to actuate the stopwatch when the pendulum was at maximum amplitude caused an uncertainty of ± 0.2 s for 100 periods, correspondingly ± 0.002 s for 1 period. The precision of the electronic stopwatch was negligible for one period in comparison to the uncertainty introduced due to the reaction time of the observer.
Average for 1 period: T = 2.473 s ± 0.002 s
4.4 Calculating the Gravitational Acceleration g
When the average values for L and T are introduced into the equation for
For the total percentage uncertainty one obtains
(Δg/g)*100% = (ΔL/L)*100% + 2*(ΔT/T)*100%;
Δg/g = (1.1/1523.1)*100% + 2*(0.002/2.473)*100% = 0.07% + 0.16% = 0.23%;
9832 mm/s² * 0.23% = 23 mm/s²
Thus, the gravitational acceleration was determined by the experiment as
g = 9.831 m/s²± 0.023 m/s²;
corresponding to a minimum of g = 9.807 m/s² and a maximum of 9.854 m/s².
5 Evaluation of Results Obtained
5.1 Experimental Result vs. Accepted Value
Since the accepted value for g = 9.804 m/s² for Burlington, Ontario falls below the minimum value obtained in the experiment and below its range of uncertainty, the experiment must be plagued by a systematic error. This systematic error might cause either the length L being determined too long or the period T being measured too short  all other elements of the equation are well defined constants and do not affect the result.
5.2 Search for Systematic Error
When looking at the times measured for 100 periods, a slight decrease for decreasing amplitude was observed, which was to expect due to the smaller amplitude. This gives reason to assume that the observer's reaction time did not necessarily introduce an uncertainty as big as stated under paragraph 4.3, because the observer's reaction time varied less than anticipated. However, decreasing the uncertainty for the period T narrows the window of uncertainty and increases the minimum value for g. Therefore, the systematic error must be sought in the determination of L. The systematic error in the determination of L can be influenced by the location of the suspension point, the mass of the "string", and the location of the bob's assumed centre of gravity.
5.2.1 Suspension "point"
The music wire does not contact the steel rod in a point; the loop is in contact with the steel rod over a portion of the latter's circumference as pointed out in Fig. 5. It is difficult, however, to determine the true "point of suspension".
Figure 5: Detail of suspension "point" 
5.2.2 "Mathematical pendulum" vs. experimental setup
The socalled "mathematical pendulum" is defined as a masspoint suspended on a string without a mass; it does not exist in reality. In this experiment, a music wire made of steel as string and a bob made of cast brass were used. Besides the remote probability that the brass bob might have cavities originating from the casting process, which would make it difficult to determine its center of gravity, there exists the certainty that the wire, which so far assumed to be without mass, has a mass m w:
where d represents the wire diameter, l_{w} the length of the wire between the suspension point and the top of the brass bob, and ρ the density of the wire material, namely steel. By introducing
d = 0.054 cm;
l _{w} = L  0.5*l _{b} = 152.31 cm  0.5*8.32 cm = 148.15 cm;
ρ = 8.75 g/cm 3;
into the above equation, it renders:
5.2.2.1 Determining the pendulum's corrected length L_{ cor}
Therefore, the pendulum used in the experiment can be considered to consist of two masses oscillating on a massfree string around the same suspension point: the mass of the music wire m_{w} at the distance l_{ w} and the mass of the brass bob m_{ b} at the distance L. The common centre of gravity of these two masses lies somewhere between l_{ w} and L, obviously very close to L. Since the two masses lie on the same axis, the common centre of gravity lies also on this axis and its distance L_{cor} from the suspension point can be calculated as follows
L_{cor} represents the distance of the combined centres of gravity for the music wire and the brass bob.
5.3 Recalculating the Gravitational Acceleration g Using the Corrected Pendulum Length L _{cor}
When the corrected value for L and the original average value for T are introduced into the equation for
Thus, the gravitational acceleration determined by the experiment and taking the mass of the string into account results in
g = 9.821 m/s²± 0.023 m/s²;
corresponding to a minimum of g = 9.798 m/s² and a maximum of 9.844 m/s².
5.4 Conclusion
The accepted value for the gravitational acceleration g for the location Burlington, Ontario, Canada is 9.804 m/s²; a fairly close value was obtained by using a simple pendulum. However, the mass of the string in relation to the mass of the bob needed to be considered to contain the accepted value within the limits of the uncertainty of the experimental value g = 9.821 m/s² ± 0.023 m/s².
Fig. 6 shows a reversible pendulum. Arrows indicate the support points. (Image taken from www.daedalon.com/pendulum.html.) 
Figure 6 



6.1 Prospecting for Minerals and Deposits
In more recent times, electronic instruments have enabled investigators to measure with high precision the time of free fall of a body from rest through one meter. Consequently, direct measurements of free fall have replaced the pendulum for absolute measurements of gravity. Today, such portable gravimeters can detect variations of one part in 10^{8} in the gravitational force and are in wide use for mineral and oil prospecting. Unusual underground deposits reveal their presence by producing local gravitational variations.
In order to efficiently find mineral deposits, once a mineral exploration team has identified a promising site, geophysicists must measure the gravity  and also magnetic and electrical properties  of the rocks. Any measurements that differ from those of the surrounding rocks are called anomalies, and these anomalies usually indicate the presence of a mineral deposit. An example of this would be that some deposits might have a higher specific gravity or density than the surrounding rocks, which results with an anomalous gravity reading. Thus, the reading of gravity aids in the search for mineral deposits ( Australian Academy of Science, 2006).
The most obvious use of the pendulum are pendulum clocks in which the pendulum regulates the intervals for the progression of the gears driving the clock hands. Since the mid 17th century pendulum clocks are manufactured until today to varying degrees of sophistication.
7 Historical Notes on Pendulum and Pendulum Clocks
The first truly scientific approach to the question of how things fall was made by the Italian scientist Galileo Galilei at the time when science and art began to stir from their dark sleep of the Middle Ages. According to the story, which is colorful but probably not true, it all started one day when young Galileo was attending a mass in the cathedral of Pisa, and absentmindedly watched a candelabrum swinging to and fro after an attendant had pulled it to the side to light the candles. Galileo noticed that although the successive swings became smaller and smaller as the candelabrum came to rest, the time of each swing (oscillation period) remained the same. He decided to check this casual observation by using a stone suspended on a string and measuring the swing period by counting his pulse or using a "water clock"  the pendulum clock still waiting for being invented. He observed that the period remained almost the same while the swings became shorter and shorter. Galileo started a series of experiments, using stones of different weights and strings of different lengths. These studies led him to an astonishing discovery. Although the swing period depended on the string’s length (being longer for longer strings), it was independent of the mass of the suspended stone. This observation was definitely contradictory to the up to then accepted dogma that heavy bodies fall faster than light ones. Indeed, the motion of a pendulum is nothing but the free fall of a mass deflected from a vertical direction by a restriction imposed by a string, which makes the bob move along an arc of a circle with the center in the suspension point.
If light and heavy objects suspended on strings of equal length and deflected by the same angle take equal time to come down, then they should also take equal time to come down if dropped simultaneously from the same height. To prove this fact to the adherents of the Aristotelian school, Galileo climbed the Leaning Tower of Pisa (perhaps another building) and dropped two weights, a light and a heavy one, which hit the ground at the same time, to the great astonishment of his opponents (Gamow, 1962, p. 2224). From his experiments with gravity, Galileo would go on to build a freely suspended pendulum.
Astronomers found that Galileo's freely suspended pendulum, merely a weight on a thread, was a more accurate timekeeper than any clock at that time. The trouble was that someone had to count the swings and to give the bob a little push from time to time.
One of the astronomers impressed by the pendulum was Johannes Hevelius, who lived in Danzig, Eastern Prussia, and owned a collection of sundials and water clocks divided into minutes like his mechanical clocks, as well as a telescope 150 feet long. In 1640, he read a book by Galileo describing the cord pendulum, which Hevelius then began to use regularly. It occurred to him that the pendulum might be applied to a clock, but he could not persuade a craftsman that this idea would work.
The same idea had struck the Dutch astronomer Christiaan Huygens, the mathematician and scientist who discovered Saturn’s rings, formulated the wave theory of light, improved the telescope and made the two most fundamental inventions in horology. Huygens developed Galileo’s mathematical theory of the pendulum and invented the practical pendulum clock. He was granted a patent for the clock in 1657, and the first version was a small wall clock constructed by Salomon Coster, a clockmaker of The Hague, in 1657.
In 1673, Huygens published a mathematical treatise on the pendulum called Horologium Oscillatorium that caused an extraordinary dispute with one Viviani, a friend and pupil of the now dead Galileo. Viviani claimed that Galileo had invented the pendulum clock, although he had previously written Galileo’s biography without mentioning such an invention. An Italian academy even published a picture of the “clock.” The drawing upset Huygens, who protested by letter.
Galileo’s “clock” might have remained a matter of purely academic dispute, had it not been for an odd coincidence, which occurred about 80 years after Viviani’s claim. An Italian professor bought some meat from his butcher and found it wrapped in some pages of an old manuscript. His curiosity aroused, he discovered that they had been written by Galileo. Another visit to the butcher enabled him to recover many more valuable manuscripts and letters which had been hidden by Viviani in a grain bin in his home in Florence and been sold by weight as wrapping paper to the butcher by Viviani’s nephew. A replica of Galileo's clock is depicted in Fig. 7 (Bruton, 1979, p. 218 220).
Fig. 7: Replica of Galileo's pendulum clock 
For guidance and direction during the preparation of this paper, I like to thank my Physics teacher, Dr. V. Zeman. Thanks also to my father, who provided materials, literature, encouragement and "food for thought".
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