Year 12 Physics- HSC Course Task: 1
Vincent Ryan

First Hand Investigation: Determining a value for Acceleration due to gravity using a pendulum

I declare that this report is solely my own work and all sources have been referenced
Vincent Ryan
___________
Term 4 2014 Teacher Mr. K. Todd
Contents
Aim: 3 Hypothesis: 3 Essential Theory: 3, 2 Variables 4, 5 Method: 5 Equipment list: 5 Practical Method Steps: 5 Data analysis steps: 6 Diagram: 2 Qualitative Observations: 7 Table of Results: Raw Data 7, 2, 9 Calculated Quantities: 20 Graphs: 20 Analysis of Data: 21 Percentage Difference: 21 Conclusion 21 Discussions: 12, 2, 14 Bibliography 2

Aim: To determine a value of acceleration due to gravity at Earth’s surface by measuring, calculating and observing the motion of a simple pendulum.
Hypothesis: An experimental value of the acceleration due to gravity will be obtained however given the parameters of the procedure, experimental error and also considering that gravity within itself is inconsistent on Earth the result may vary ±5% of the averaged 9.84m-2.
Essential Theory:
Where:
* g= Acceleration due to gravity * G= Universal gravitational constant * M=Mass of the earth * d= distance from the centre of the earth or the radius

Using values for Earth: * G= (6.67 x 10-11) * Mearth= (6.0 x1024)kg * dearth = (6.378 x106)m
∴g=(6.67×10-11)(6.0 ×1024)(6378 ×103)2
∴g≈9.84ms-2
Fg=Fw
Fg=GmMd2
Fw=mg
GmMd2=mg
∴g=GMd2

The symbol ‘g’ represents acceleration due to gravity acting on an object due to the presence of a gravitational field. Sir Isaac Newton stated in his law of universal gravitation that for only two masses separated in space, there is a force of attraction between them due to the interaction of their gravitational fields (Wu and Farr 2009). On and near the surface of a planet the force of attraction or ‘g’ can be calculated by applying the formulas of universal gravity and the 2nd law of motion (Wu and Farr):

There are variations that occur to the averaged mathematical value of acceleration due to gravity of 9.84ms-2 from Wu and Farr (2009). These variations can be attributed to the deviation in factors influencing the acceleration due to gravity. Thus the common value of 9.84ms-2 around the globe is an approximation within itself as condition vary from place to place.
These factors as described by Wu and Farr (2009) are: * Altitude: Since acceleration due to gravity is inversely proportional to distance, an increase in altitude, or distance from the centre of the earth results in the value of gravity decreasing. * Local crust density: Since acceleration due to gravity is proportional to mass, then areas of high crust density i.e. where there are mineral deposits, will have a greater value of gravity. * The rotation of the Earth: The Earth is rotating once every 24-hours so therefore regions of lesser latitude i.e. the equator, will experience a higher centrifugal force acting outwards. This force some slightly counteracts the acceleration due to gravity and as the centrifugal force differs from place to place so does the value of ‘g’. * Shape of the Earth and distance from core: due to its rotation the Earth slightly bulges at the equator the resultant shape being an ellipsoid. Consequently the poles are closer to the centre in comparison to the equator. As stated previously ‘g’ is inversely proportional to distance, therefore gravity will increase as the distance from the centre decreases.
According to Louviere (2005) Newton’s First law of motion states “an object will remain at rest unless acted on by an external force. An object in motion continues in motion with the same speed and in the same direction unless acted upon by an external force.” This law can be applied to the motion of a pendulum. According to Sprott (Date: N/A) the presence of the atmosphere will result in the swinging pendulum eventually stopping at a point of equilibrium. The unbalanced forces of friction caused by both air resistance and surface friction at the pivot point can be held accountable for causing the pendulum to stop.
Where:
* T= the time period of one oscillation (s) * l= length of string (m) * g= acceleration due to gravity (ms-2)

T=2πlg
In reference to Nave R. (2012) A simple pendulum is a basic harmonic oscillator consisting of an object with an insignificant size hanging from a pivot point. If air resistance is ignored, then the period (T) of an oscillation is completely independent of the mass hung, especially if the angle of release is less than 10˚ (Wu and Farr 2009). By adopting these requirements the period of an oscillation can be given by the formula:

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

T2=2π2lg
∴g= 4π2lT2

Thus using this, a formula for ‘g’ can be derived:

Therefore, if both period and length of a simple pendulum are established acceleration due to gravity can be calculated. This can be done by plotting period2 (T2) on the y-axis against the length (l) on the x-axis. The inverse gradient of the line of best fit obtained from the graphical representation can be substituted into the above equation to find ‘g’ (Wu and Farr 2009).
Variables
Variable | | Independent | String length | Dependent | Time taken per period of Oscillation | Controlled | Angle of release of mass at5˚Number of oscillations per trial: 10Acceleration due to gravityMass of the pendulumAll materials |
Variables:
Uncontrollable variables: * Friction of the Pivot point: The pivoting point was a clamp with string tied off, this didn’t let the pendulum swing completely free, therefore the momentum of the mass was slowed down with each oscillation. This is due to the law of conservation of energy as kinetic energy is transformed into heat energy from the friction between the string and clamp. * Swing pathway: Although the angle of release was somewhat consistent the pathway the pendulum followed was not so. This potentially influenced the result as the diagonal pathway travelled would be of a greater distance compared to that of the swing perpendicular to the light gate. * Pendulum release height: The pendulum was released using our hands, this was inconsistent as the height of the release could possibly vary from the 5° in each trial due to human error. Since Fw=mg,∴F=Gm1m2d2. (Wu and Farr 2009) Therefore since weight force is inversely proportional to distance, the force will change as distance does. Consequently the resulting force will determine the period of the oscillation whether it be faster or slower. * Pendulum release method: On releasing the pendulum the hand could add force to the mass by pushing it rather than cleanly releasing it. * Centre of gravity of the carrier: The mass carrier was tied to the string at its top, thus since its centre of gravity was at a much lower height it could have potentially had an internal swinging motion inside that of the pendulum.
Method:
Equipment list:

Item | Quantity | Item | Quantity | String | 1 metre | Ruler | 1 metre | Light gate | 1 | Data logger | 1 | Slotted masses | 1x 0.05 kg | Mass carrier | 1 | Retort stand | 2 | Clamp | 2 | Boss head | 2 | | |
Practical Method Steps: 1. All equipment was gathered and the Apparatus was set up according to the diagram below 2. The data logging device program was set to measure time and was connected to the light gate. 3. A piece of paper with an angle of 5˚ was drawn up and blue tacked to the clamp, keeping the angle of release consistent across each trial assisting the validity of the result. 4. A mass of 0.1 kg initially hung from a height of 1m (length measured from pivot point to centre of the 0.1 kg mass). 5. The mass of the pendulum was brought back to the angle of release (5˚). 6. The data logger was initiated. 7. The mass was released simultaneously after the data logger was started. 8. The pendulum was allowed to swing for a total of ten oscillations. 9. The data logger was stopped after 10 oscillations to get the maximum oscillations without being affected by the friction and air resistance. 10. The results obtained from the trial were sent to an excel spread sheet 11. Steps 5-8 were repeated three times for 1m. 12. After the 3 trials were completed, the length of the string was changed from 1m to 0.9m. 13. Steps 5- 10 were repeated for all lengths (1 - 0.6m) going down in increments of 0.1m for each trial.
Data analysis steps: 1. The excel spread sheets were opened for trials one, two and three of the 1m test. 2. The results of two and three were brought to trial one and tabulated. 3. The results were rounded to three decimal places to ease the calculations but to also keep a moderate accuracy. 4. The average period for each oscillation was calculated on excel by entering the sum of the periods divided by the number of periods (10). 5. An overall period average was then calculated by entering the sum of the averages and then dividing it by the number of averages (3). 6. Steps 1-5 were repeated for each separate string length. 7. Period squared was then calculated for each string length by squaring the averaged period of an oscillation. 8. A scatter graph of period (s) versus length (m) was plotted. 9. A scatter graph of period2 (T2) versus length was plotted to find the relationship between the two and a line of best fit was drawn to show the gradient. 10. The gradient of this graphical representation was substituted in the equation to calculate acceleration due to gravity as shown below.
Finding ‘g’
Where:
* m= gradient * T= period * l= length of string * g= acceleration due to gravity g= 4π2lT2
∴g=4π2×lT2
∴g=4π2×1m
∴g=4π2m
Gradient= RiseRun
∴Gradient=period2(T2)Length
∴m=T2l
∴1m=lT2

Diagram

Qualitative Observations: * When lining up the string of the pendulum with the 5˚ release point, sometimes this would not be perfectly in line. This could alter the results as it gives a greater gravitational potential energy which is transformed into kinetic energy. However this observation was ignored given the restrictions of the procedure and the inability to control such a fine aspect. * Since the shape of the mass carrier was irregular, the centre of the object was difficult to identify. Consequently it was noticed that the length of the string of the pendulum was inconsistent giving an approximation to the centre rather than the desired length. This observation was ultimately ignored as the margin of error would be a matter of millimetres and again given the restrictions of the experiment, such a fine detail couldn’t be fully controlled
Table of Results: Raw Data
1m Pendulum data

90 cm Pendulum data

80 cm Pendulum data

70 cm pendulum data

60 cm pendulum data

Calculated Quantities: Length of String | Average period of Swing | (Average period of swing)2 | m | s | s | 1 | 2.002 | 4.008 | 0.9 | 1.855 | 3.441 | 0.8 | 1.786 | 3.19 | 0.7 | 1.66 | 2.7556 | 0.6 | 1.537 | 2.362 |
Graph 1:

Graph 2: Analysis of Data:
Using the gradient of the line of best fit and the formula to calculate acceleration due to gravity, a value for ‘g’ can be determined for this experiment as below:
Gradient of period squared versus length m=RiseRun m= 3.977
Acceleration due to gravity: substituting 1m into the equation for ‘g’ g=4π2lT2 ∴g=4π2m
∴g=4π2m
∴g=9.926ms-2
Percentage Difference:
The degree of error can be calculated using:
Percentage Difference %=(Theoretical Value-Experimental Value)Theoretical Value×100
∴Percentage difference%=(9.8-9.926)(9.8)×100
∴Percentage difference=-1.2857…%
∴Percentage Difference ≈-1.286%
Conclusion:
This experiment has produced results leading towards definitive findings that have been able to determine an experimental value of acceleration due to gravity or ‘g’ at the surface of the Earth. By using the data received from the experiment, the value for acceleration due to gravity at a point of 34˚ S 151˚ E 158m above sea level (Itouch map 2007) was found to be 9.926ms-2.This had a percentage error of 1.286% from the theoretical acceleration of gravity. Therefore the experiment has achieved above average results that contribute to both the aim and hypothesis. This system has determined a value of acceleration due to gravity and obtained a value that did vary within 5% of hypothetical value due to experimental errors, and factors that affect the value of ‘g’. However in order to approach a conclusive finding the procedure as a whole must be repeated several times to account for systemic errors combined with the use of sophisticated equipment that reduce these can reduce the significance of the errors.
Discussions:
1. In this project an experimental value of acceleration due to gravity has been determined as 9.926ms-2. This differs from the common value of acceleration due to gravity as 9.84ms-2on the surface of the Earth (Wu and Farr 2009) by approximately 1.286% thus suggesting the high level of experimental design of this investigation.
As mentioned previously in the essential theory, according to Wu and Farr (2009) the averaged mathematical value of acceleration due to gravity varies in each location due to a range of factors including, altitude, local crust density, rotation, shape of the Earth and the distance from the core. As a result of the alterations in geographical condition of spatial locations these four key factors contribute to the varying values of ‘g’ around the world. Therefore the averaged value for acceleration due to gravity is merely an approximation within itself and consequently cannot entirely represent the value of ‘g’ at the point of investigation (34˚ S 151˚ E) (ITouch map 2007). Thus this estimation used in calculations could potentially attribute to the percentage error of 1.286%.
Validity refers to whether an experiment tests what it has proposed to test (Shuttleworth, 2008); a key indicator of validity is percentage difference which has been calculated above. The percentage difference between the experimental and theoretical value is 1.286%, thus showing how close the experiment was to the common number of acceleration due to gravity. Consequently it is evident that this experiment is externally valid as it has achieved a value for acceleration due to gravity so close to the accepted numerical value. Therefore this experiment has ‘determined a value for ‘g’ at the surface of the Earth’.
Accuracy is a term which alludes to how close a measuring result is to the actual result using the best measuring device according to David (1996). Contributing to the validity of this result was the use of accurate results, obtained from the data harvest light gate. This measuring device was capable of obtaining results with up to seven or more decimal places, illustrating the high degree of accuracy of the findings. This accuracy carried through as values were only rounded to three places to get a balance between accuracy and ease of calculation. Thus the value of ‘g’ being 9.926ms-2 was very accurate as minimal value would have been lost in calculation errors.
However the minute percentage difference of 1.286% can be attributed to various sources in the form of systemic errors. Firstly Faults in the equipment and the setup of the procedure could have potentially contributed to the error. As mentioned in the background theory sources of error could arise from the uncontrollable variables in the experiment. This includes friction between the string and the pivot point, where the law of conservation of energy states; energy cannot be created nor destroyed (Nave 2012). This means that the kinetic energy of the pendulum is transformed into heat energy from the friction, decreasing the velocity of the pendulum. Furthermore the swing pathway was very inconsistent, as the pathway deviated from perpendicular to the light gate. Consequently this furthered the distance required to travel and consequently increased the period of an oscillation in relation to other trials. Moreover the pendulum release height was inconsistent with small variations due to parallax error. In reference to Wu and Farr (2009) F=mg ∴F=Gm1m2d2. From this equation it is evident that acceleration due to gravity is inversely proportional to drop height and thus the change in height results in an alteration in ‘g’. In addition to this the pendulum release method could add further force to that of the gravitational potential energy of the mass bob. This could affect the results as the addition in force could consequently amplify the acceleration, influencing the findings. Finally the irregular shape of the mass created a centre of gravity that was difficult to distinguish. Due to this the string was joined to the mass at a high point of the mass which could result in an internal swing to that of the motion of the pendulum due to the law of conservation of momentum (Nave 2012). These uncontrollable variables are all sources of validity error and can account for the 1.286% difference between theoretical and experimental values.
2. This experiment has yielded consistent result across each trial in each length category, consequently outlining the reliability of the procedure for this experiment. In each length category three trials where undertaken to ensure any systemic errors where accounted for if any discrepancies where present amongst the findings. Furthermore the scientific setup of the procedure was of an above average grade, in which the majority of all aspects where kept constant. In particular the light gate was essential as it removed human error when recording the period of each oscillation. Although, in observing the graphical representation of both period versus length and period squared versus length there is a distinct outlier at the length of 0.9m. By examining the three trials for 0.9m, all results were approximately 1.855s with a deviation of 0.001s. Thus it appears that an error must have occurred in the assembly of the system that has remained constant across each trial which justifies the consistent results. This error was most likely the string length that wasn’t measured precisely and therefore has altered the outcome. Otherwise all other results have been repeated three times, each with consistent results that have been averaged. A final mean was found across the three trials and used in the graphical representation. By using the averages of each trial, outliers in recordings became somewhat irrelevant and decreased their significance in the graph. However to reinforce the reliability of this experiment it must be repeated entirely in a separate environment. This will evaluate the reliability of the method to remove any systemic errors that may have occurred in the first procedure.
3. The experimental design of the procedure was of a high grade and internally valid, this enabled both the aim and the hypothesis to be tested with above average accuracy. The percentage difference between the theoretical and experimental values of ‘g’ was 1.286%. This indicates an above average validity, however to account for this percentage error changes could be made in future to improve the findings in relation to the theoretical result. Firstly changes could be made in the equipment used including the permanence of the angle measurement, to ensure that it remains constant throughout the experiment. Also the string used to hang the mass from, and the method of shortening the string could be improved by having a system where the desired length can be adjusted easily, for example a pulley. Furthermore the use of a pulley at the pivoting point would assist in declining the degree of surface friction between the string and clamp. Additionally the technique of releasing the mass by hand could also be adopted by a mechanism to remove human error and reduce any force that could be added to the swing. Finally the mass itself should be a spherical mass where the centre of gravity is easily identifiable and also allow the string to be tied closer to this centre, minimising the severity of an internal swing from the momentum. These are just a few suggestions that could go towards improving the validity of the result.

In addition to this, due to the restrictions of the process the reliability was good with a moderate extent of repetition throughout the experiment however the experiment within itself needs to be repeated. This is required to remove systemic errors that occurred initially and carried through the whole experiment. Repetition additionally accounts for recalculating a numerical value for ‘g’ to inspect for any calculation errors and observing any difference in the two results to evaluate the reliability. Furthermore, to increase the reliability a wider range of string length would produce a larger pool of results that could be utilised in the graph to observe the trend line and any deviations evident at larger or smaller lengths of string. Moreover each string length should include further trials to allow for more definitive averages that remove significant outliers influencing the final results.
The method of obtaining the period of the pendulum was through the use of a light gate, this is a very accurate device that uses light travelling at approximately 300, 000, 000 ms-1. Because of this the results are essentially instantaneous, compared to the human error involved in using a stop watch. So for the equipment available the process provided accurate measurements compared to that of the actual results. However in the calculations the numerical value of period was rounded to three decimals to ease calculations, to further add to the accuracy keeping the values unrounded instead would assist reinforcing the accuracy of the findings. The experimental design of this process accounted for a high level of accuracy and thus there was only a small room for improvement for future reference in regards to accuracy.
Bibliography

David, D. (1996). Chemicool. Retrieved November 4, 2014, from http://www.chemicool.com/about.html#cite
ItouchMap. (2007-2012). Latitude and Longitude of a Point. Retrieved November 24, 2012, from http://itouchmap.com/latlong.html
Louviere, G. (2005, October 8). Newton's 3 Laws of Motion. Retrieved October 25, 2014, from Teacher Tech: http://teachertech.rice.edu/Participants/louviere/index.html
Nave, R. (2012). Simple Pendulum. Retrieved October 28, 2012, from HyperPhysics: http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html
Shuttleworth, M. (2008, October 20). Explorable. Retrieved Novemeber 4, 2014, from https://explorable.com/validity-and-reliability
Sprott, C. (-, - -). Sprott's Gateway. Retrieved from Physics Demonstrations- Motion: http://sprott.physics.wisc.edu/demobook/chapter1.htm

Wu, X. L., 2009, Physics in Focus HSC course/ Xiao L. Wu and Robert Farr. McGraw-Hill Australia Pty Lt, Level 2 82 Waterloo Road, North Ryde NSW 2113.