ContentsAim:1Background Information:1Hypothesis:3Materials:3Variables:4Procedure:4Results:6Discussion:8Conclusion:9Bibliography:10

Calculating Acceleration Due to Gravity Experimental ReportInsert name hereAim: To determine an experimental value for acceleration due to gravity at the Earth's surface.Background Information:The symbol (g) denotes the acceleration an object will experience caused by the Earths gravitational field neglecting air resistance. On and near the surface of the Earth this value can be calculated by applying the formulas for Newtons Law of universal gravitation and 2nd Law of motion[footnoteRef:1]: [1: (Henderson, The Physics Classroom, 1996-2012)]

Where: g = acceleration due to gravity (ms-2) me = approximate mass of the Earth[footnoteRef:2] = 5.9736 1024 kg [2: (Williams, 2010)]

m1 = mass of object at the Earths surface r = approximate volumetric mean radius of the Earth[footnoteRef:3] = 6 371 000 m [3: Ibid.]

G = universal gravitational constant[footnoteRef:4] =6.673 84 x 10-11 m3 kg-1 s-2 [4: (Physical Measurement Laboratory of NIST, 2010)]

There are variations that occur to the established numerical value of acceleration due to gravity (. These can be attributed to changes in the radius of the Earth, the presence of dense ore bodies, changes in the Earths lithosphere and centrifugal effects of the spinning Earth[footnoteRef:5]. The Earth has an average equatorial radius of approximately 6378100 m and an average polar radius of approximately 6 356 800 m.[footnoteRef:6] These quantitative values demonstrate changes in the radius of the Earth and show that it is different at any individual geographical location[footnoteRef:7]. Therefore, the calculated value of ms-2 is not a complete accurate representation of acceleration due to gravity. [5: (Warren, 2008)] [6: (Williams, 2010)] [7: (Briney, 2012)]

Research was conducted with the aim of investigating a means to find the distance from the centre of the Earth to the location where the experiment was conducted (-28511.2S, 153351.1E)[footnoteRef:8]. However no means could be located and thus the approximate volumetric mean value for the radius of the Earth was used to establish a controlled value for acceleration due to gravity. As a consequence of this, the very nature of the established value () is merely an estimate in itself. [8: (ItouchMap, 2007-2012)]

Newtons 1st Law of Inertia states a body in motion will continue to move at a constant velocity unless acted upon by an external and unbalanced force[footnoteRef:9]. This can be applied to the motion of a pendulum on earth as a pendulum undergoing simple harmonic motion will eventually come to a stop due to the unbalanced forces of friction caused by both air resistance and at the pivot point. This will have a small but certain effect on the motion of the pendulum. [9: (Weisstein, 1996-2007)]

A simple pendulum is a basic harmonic oscillator that consists of a point mass suspended from a pivot by a string of negligible mass. When ignoring air resistance, the simple harmonic oscillation of a pendulums period (T) is completely independent of the mass of its bob and nearly independent of amplitude, especially if is less than about 15.[footnoteRef:10] Therefore, the period of oscillation of a simple pendulum can be calculated by applying[footnoteRef:11]: [10: (OpenStax College, 2012)] [11: (Nave)]

Where: T = the time period of a single oscillation (s) l = the length of the string (m) g = acceleration due to gravity (ms-2)

Thus, if both the period and length of a simple pendulum are established a value for acceleration due to gravity (g) can be calculated through a graphical representation with the period2 (T2) plotted on the y axis and length (l) on the x axis. The gradient of the line of best fit obtained from this graph will be able to be substituted into the known equation[footnoteRef:12]. [12: (Millennium Mathematics Project, 2008)]

Where: g = acceleration due to gravity (ms-2) m = the gradient of the line of best fit Hypothesis:Given the parameters of this experiment the measured value of acceleration due to gravity at the Earths surface will be within a deviation of approximately 5% from ms-2.Materials: 1.2 m length of string Stopwatch 1 m ruler 0.2 kg Mass Protractor Retort stand Cork Boss head Clamp Safety glasses

Variables: Independent

Length of string

Dependent

Time per period of oscillation

Control

Acceleration due to gravityApproximate 15 angle of oscillationNumber of periods per swingAll materials

Procedure:1. Ensure the work area is free from any potential hazards including water and sources of electrical energy. 2. Employ the use of safety glasses in order to avoid the swinging pendulum coming into contact with eyes.3. With a permanent marker and 1 metre ruler, accurately measure and mark lengths of 0.05 m increments from 0.1 0.7 metres from the centre of the mass up the piece of string (the independent variable).4. Assemble the following apparatus:

Note that the pendulum must be in a state of suspension and able to swing freely without colliding into any foreign objects. 5. Loosen the clamped cork and run the string through to the marked length of 0.1 m. Restore the cork to its tightened state when completed.6. Draw the pendulum aside to an angle of approximately 15 relative to the vertical, measured with the protractor, keeping the string taut. 7. Simultaneously start the timer and drop the pendulum. Allow the pendulum to swing for 7 full periods, pressing stop on the stopwatch at the completion of the 7th period (the dependent variable).8. Record Results.9. Complete steps 4 7 to collect results for a total of 3 trials for each marked length. 10. Take the average of the trial results for each length (the independent variable) and divide this number by 7 to calculate a value for the average amount of time per period of oscillation, T (the dependent variable). Exponentially increase this number by a factor of 2 to find a value for the period squared, T2.11. Using the formula where g is the acceleration due to gravity (ms-2), l is the length of string (m) and T2 is the period of oscillation squared (s), calculate a value for the acceleration due to gravity for each marked length of the string.12. Compose a graph with the period2 (T2) plotted on the y axis and length (l) on the x axis. 13. Substitute the gradient of the line of best fit into the equation where g is the acceleration due to gravity and m is the gradient of the line of best fit to determine an experimental value for the acceleration due to gravity.

Results:Length (m)Trial 1 (s)Trial 2 (s)Trial 3 (s)Average (s)Period T (s)Period T2 (s)Calculated g (ms-2)

0.14.975.174.975.0370.71950.51777.626

0.155.925.975.745.8770.83950.70488.402

0.26.576.526.586.5570.93670.87739.000

0.257.217.257.37.2531.0361.0749.192

0.37.87.957.977.9071.1301.2769.283

0.358.478.528.598.5271.2181.4849.312

0.49.128.949.139.0631.2951.6769.420

0.459.339.699.729.5801.3691.8739.485

0.510.0510.1110.0510.071.4392.0699.538

0.5510.4410.5210.5610.511.5012.2539.638

0.610.9810.8110.8110.871.5522.4109.829

0.6511.4711.4911.4511.471.6392.6859.557

0.711.7811.8111.8911.831.6902.8549.681

Using the gradient of our line of best fit and the formentioned equation, acceleration due to gravity can be calculated by applying:

= 10.10117381069965...10.101 ms-2Degree of percentage error can be calculated by applying:

= 2.84233160965...% 2.84%Metre ruler attributed percentage error: Stopwatch attributed percentage error:Measured length (m)Error of reading (%)Average measured time (s)Error of reading (%)

0.10.50005.0370.0199

0.150.33335.8770.0170

0.20.25006.5570.0153

0.250.20007.2530.0138

0.30.16667.9070.0126

0.350.14298.5270.0117

0.40.12509.0630.0110

0.450.11119.5800.0104

0.50.100010.070.0099

0.550.090910.510.0095

0.60.083310.870.0092

0.650.076911.470.0087

0.70.071411.830.0085

Calculated by applying:

Discussion:The results of this experiment demonstrate that the value of acceleration due to gravity at the Earths surface is approximately equal to 10.101 ms-2. This differs from the established value of ms-2 by approximately 2.84% thus suggesting a degree of error in the experimental design of this investigation. As mentioned in the background information, ms-2 is an averaged value and hence does not give a complete and accurate depiction of acceleration due to gravity at any separate location on or around the Earths crust. This is because the Earth's circumference and diameter differ due to its shape being an oblate spheroid or ellipsoid, instead of a true sphere. Consequently instead of the Earth being of equal circumference in all areas, there is a bulge at the equator and thus a larger circumference and diameter there[footnoteRef:13]. The very nature of the established value of acceleration due to gravity at the Earths surface () is merely an approximation in itself and not an entirely accurate representation of acceleration due to gravity at the location of the investigation (-28511.2S, 153351.1E)[footnoteRef:14]. This estimation used in the calculation of the established value may have contributed to this percentage error of 2.84%. [13: (Briney, 2012)] [14: (ItouchMap, 2007-2012)]

Faults in the materials used could have also propagated the error discrepancy. For example, if the clamps grip on the cork had not been tight enough. This could have resulted in the string slipping through the cork amidst the duration of the 7 periods of oscillation. Thus, altering the length of string and unintentionally increasing the independent variable. This increase could have potentially jeopardised the recorded measurements of the duration of the 7 full periods of oscillation and thus the average time for the duration of 1 complete period (dependent variable). This experimental limitation could have been prevented if the length of string had been measured before and after the 7 periods of oscillation were recorded. Faults and/or frailties in components of the boss head, clamp, string, cork and/or retort stand could also have resulted in experimental inaccuracies through the materials not sufficiently serving their purpose. Another potential source of the error of 2.84% can be attributed to such uncontrolled systematic variables as the error of reading on each set of equipment ( 0.0005 m on the metre ruler, 0.001 s on the stopwatch). Throughout all resultant calculations and graphical representations the metre rulers error of reading may have impacted the reliability of the investigations results by a range of approximately 0.0714% 0.5% and the stopwatch by approximately 0.0085% 0.0199%. These errors could have been reduced with the use of more precise measuring equipment. This would allow for more accurate measurements of the independent and dependent variables.The validity of this experiment could have been impacted as both random and systematic errors could have occurred through the involvement of humans in the data collection process. An example of this could have occurred each time the pendulum was brought from its state of equilibrium to the position of 15. Upon the pendulums release, if any amount of kinetic energy had been transferred from the scientists hand into the pendulum or if any tension had been lost from the string, this would have impacted the amount of force acting on the pendulum, disrupting the pendulums rhythm of simple harmonic motion and potentially leading to both inaccurate results and an unreliable conclusion. Each calculation of acceleration due to gravity at individual lengths shows that 0.1 m is furthest away from ms-2 with a value of approximately 7.6255 ms-2. This reflects a trend that from lengths 0.4m 0.1m the measurement of acceleration due to gravity deviates further and further away from ms-2. I believe this error in experimental design can be attributed to the fact that a decrease in the independent variable (length of string) subsequently leads to shorter periods of oscillation (dependent variable). These shorter periods would have hindered the ability of the timekeeper to make accurate decisions of when to start and stop the stopwatch, thus becoming more susceptible to the impact of human reaction time. Because the formula is heavily reliant on time, as it is exponentially increased (T2), it was necessary to time 7 full periods of oscillation to minimise the impact of these errors of judgment and allow the timekeeper to accurately anticipate when the pendulum would complete its 7th period.To reduce the impact of experimental limitations such as inherent equipment failure, human reaction time, the coefficient of friction, inaccuracies relating to parallax and other such errors 7 complete periods of oscillation were timed, a line of best fit was used, repeat trials were conducted with averages, and the apparatus was kept in a fixed position. This was all to increase the accuracy of the investigations results thereby enhancing the reliability of this experiment. The software program Microsoft Excel was used to conduct data analysis and minimise the impact of human involvement.The experimental design and results of this experiment are valid because they allow the hypothesis to be tested with accuracy. If this investigation were to be repeated, I would modify the experimental procedure to reduce the impact of human reaction time by filming the pendulums 7 periods of oscillation at each length in order to more accurately determine the period by video analysis. I would also factor in attributed percentage errors in my analysis; conduct a larger number of trials, measuring more periods of oscillation. This would increase the accuracy of the line of best fit and thus enhance the reliability of the investigations results.Conclusion:The results of this experiment show a determined value of 10.101 ms-2 and deviation of approximately 2.84% from the known value of acceleration due to gravity at the Earth's surface. This supports my hypothesis that the measured value of acceleration due to gravity at the Earth's surface will be within a deviation of approximately 5% from ms-2. Other findings suggest that a reduction in the independent variable, length of string, will result in a diminution of the dependent variable, time per period of oscillation and that experimental procedure could have been improved to minimise the impact of random and systematic errors that can result from human involvement. I suggest further study be conducted to more accurately and reliably determine a value for acceleration due to gravity.

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