Objective
In this experiment, we examined the motion of a simple pendulum constructed from a bob hanging from a long, thin string attached to the ceiling. We wish to confirm that the period of the pendulum depends not on the mass of the bob, but only on the length of the string and the acceleration due to gravity. We also wish to obtain an experimental value for earth's gravitational constant, g, which is known to be 980.35 cm/s2.
Discussion
Two approaches were used to make an experimental determination of the value of g. The first approach involved measuring a single period 100 times, using a stopwatch. The average of these periods was taken and used in the calculation of g from the formula
g1 = (2π/Tave)2Lave
The second approach involved measuring 50 continuous periods 2 times, getting the time for 50 total periods to elapse twice, and then dividing the time by 50 to get the period T50. The average of the two measurements of T50 was then used in the calculation of g from the formula
g50 = (2π/T50, ave)2Lave
Sources of Error
One of the sources of error for this experiment is the assumption that the swing of the pendulum is a simple harmonic. In reality, the motion of the pendulum only approximates simple harmonic motion. The pendulum experiences friction from both air resistance and from the point of contact with the surface that it swings from. In addition to these systematic sources of error, human error may have been involved in the reaction time of operating the stopwatch. Relying on human judgment to record the period of the pendulum is not the most accurate way to make time measurements. In addition, there may have been some error introduced by inaccuracies in the measurements of the length of the pendulum, although this source of error should have been corrected for by averaging over a series of measurements.
Questions and Analysis
Question 1
It is possible that using just one measurement of the length of the pendulum would not have resulted in a more reliable value of g, but only if that measurement was very accurate. However, using only one length measurement could not have resulted in a precise value, because precision is a measure of how close a group of measurements are to each other. By averaging the length of the pendulum over ten measurements, we obtained a precise value of g.
Question 2
The reliability of g is more sensitive to the imprecision in time rather than length, as the fractional error for both results confirm. This means that the human judgment involved in measuring the duration of a period is less precise than the ruler.
Question 3
(a) If one measurement is off by 0.2 seconds per period, and 10 periods are measured, a single period is off by 0.02 seconds. If 50 periods are measured, a single period will be off by 0.004 seconds.
(b) Two experimental methods that can be used to increase the accuracy of g50 include timing the period of the pendulum by marking when the bob goes past a certain reference point, or by measuring the duration of 50 periods more than only 2 times.
Conclusion
Both approaches to the calculation of g yielded results that were remarkably close to the expected value of 980.35 cm/s2. The first approach yielded an experimental value for g1 of 998.0 cm/s2., only a 1.8% difference from the accepted value. The second approach yielded an even more accurate value for g50 = of 988.2 cm/s2, only 0.79% away from the accepted value. The experiment does a good job of demonstrating that the period of a pendulum is independent of the mass of the bob, and the results of the experiment confirmed theoretical values within a reasonable margin of accuracy.
