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Wednesday, March 21, 2012

The simple pendulum and its basic workings

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1.1 Relationship between initial angle of release and period, T


Using the results it was evident that for angles of release greater than 0° the data did not follow the SHM theory and that the period was not independent of the angle of release. Instead, the data followed the extended formula, where the period became longer when the angle of release became larger. Therefore, we can conclude that the period is proportional to the angle of release and therefore, the amplitude of oscillation.


1. Relationship between length of pendulum cord and period, T


Using the results, we could see that there was an extremely strong power relationship between the period and the length of the pendulum cord. Using logarithms, we were able to confirm the relationship to within % accuracy. Therefore, we can conclude that the length of the string does affect the period of the pendulum swing and this relationship can be represented by the formula


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1. Relationship between mass of bob and period, T


Using the results, we could see that there was no significant change in the period as the bob’s mass grew. Therefore, we can conclude that there is no relationship between the mass of the bob and the period, T. The period of the pendulum swing is unaffected by the bob’s mass.


1.4 Confirming the SHM formula


Using the results we were able to confirm that the relationship





existed to within % of the theoretical results. This was a very accurate finding, suggesting that the relationship between the two variables is extremely likely.


Furthermore, we went on to confirm that the constant for the above relationship was


where j is a constant


within % accuracy once more, again suggesting that the constant was extremely accurate.


By proving the relationship, , existed and by proving that its constant was we were able to confirm the SHM formula





1.5 Definition of a small angle


From the theory at the start, it was noted that the difference between sin è and è only becomes significant when the angle of release reaches 0°. Also, from the results obtained, it was clear that when the angle of release reached 0° that the period suddenly started to move away from the SHM model and started to follow the extended formula. As the results only followed SHM for the angles of release and SHM only works for small angles of release it can be said that a small angle of release occurs when the angle of release is between (0°, 0°] to the vertical.


1.6 Verifying the extended formula for large angles of release


The theoretical results proved that for larger angles the extended formula must be used to get a more accurate period. The results also proved that for large angles of release that the period follows the extended formula to within 1.5% of the extended formula’s theoretical values for the period. Therefore, we can conclude that the extended formula is accurate in determining the period for large angles of release ie angles of release between (0°, 0°] to the vertical. (It is then implied that the extended formula does work for large angles of release and that because it works we have verified it).


1.7 Verifying the value of gravity to be .8N/kg


Using all the above formulas, we came to the conclusion that the strength of gravity was equal to . This not only confirmed the accepted average value of gravity of .8N/kg to within .7% but also confirmed the theory that the strength of gravity to changes around the world.





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