Aspect 2:
1. Calculating the period average:
e10avg = (19.400 + 0.005s) + (19.020 + 0.005s) / 2
= 38.42 + 0.01s / 2
= 19.21
Error = ((0.005/19.4) + (0.005/19.02))*19.21
2. Calculating time for one period:
T1 = (19.210 + 0.005s) / 10 = 1.9210
eT1 = (0.005 / 19.210) (1.9210) = 0.0005
3. Calculating frequency
f = 1 / T
f = 1 / (1.9210 + 0.0005s) = 0.520562Hz
ef = (0.0005 / 1.9210) (0.52056) = 0.0001
4. Calculating the period (T2)
T2 = (1.9210 + 0.0005s) 2 = 3.690241s2
eT2 = [(0.0005 / 1.9210) + (0.0005 / 1.9210)] * 3.690241 s2
eT2 = 0.002
T2 = 3.690 + 0.002 s2
Aspect 3:
Graph #1: Frequency vs. length
Graph #1
Graph #1 plot the frequency of the pendulum against the length of the pendulum. The points are connected with a curved line indicating that the length of the pendulum does affect the frequency but not in a linear fashion.
Graph #2
Graph #2 plots the period squared against the length of the pendulum. The points are connected with a straight line indicating that the period of the pendulum is effected by the square root of the length of the pendulum.
Graph #2: Period Squared (T2) vs. length of a pendulum
Slope of graph #2:
Slope = rise / run =
Slope = (4.460 – 0.847) / (1.080 – 0.200) = 4.110s2 / m
Slope max = (4.500 – 0.400) / (1.06 0– 0.110) = 4.270s2 / m
Slope min. = (4.000 – 1.000) / (1.100 – 0.100) = 3.000s2 / m
Error: max – min / 2
Error = 4. 320s2 / m – 3.000s2 / m divided by 2
Error = 0.1
Percent discrepancy and comparison of gravity:
And,
9.81 – 9.605454405 / 9.81 * 100%
= 2.01% difference
Error: max – min / 2
Error = 4. 320s2 / m – 3.000s2 / m divided by 2
Error = 0.1
Using proportionality constant to predict the frequency of length 140 cm
140 cm = 0.14m
y = mx = 4.1 * 0.14 = 0.5740 Hz
Effect of Amplitude on the frequency:
The effect of amplitude made the frequency of the pendulum longer as the pendulum took more time to complete one cycle. While the time for each trial was made shorter the frequency kept increasing.
Conclusion and Evaluation
The aim of the experiment was to determine the effects or contribution of the length of the string on the period for the simple pendulum and find out a mathematical relationship between the length and the period. According to the data taken during the experiment, the period depends on the length of the string. When the length of the string is increased, the value of the period also increases. As it is shown in Graph, the period of the pendulum square and the length of the string are directly proportional to each other. The slope of the graph was 4.1s2 / m, slope max (4.3s2 / m) and slope min (3.0s2 / m). Gravitational acceleration was calculated from the slope of the graph 10.01 m s-2 and if theoretical gravitational acceleration is taken 9.81 m s-2, relative error was found from that formula which was (2.01%)
There were some problems done during the experiment which caused the relative error to increase. Stopwatches were used that do not have uncertainty interval. Reaction time is also other error factor. Flexibility of the string is a problem, non-flexible string was chosen but a change in the length of the string might cause error.
There were not any other errors caused by the apparatus. It was stable enough for the experiment. Therefore, there is no need to change the apparatus if this experiment would be done again. However, some other materials could be used such as the computer/multimedia system and a photo gate timer, to measure the period of the pendulum. They would decrease the relative error. Making the experiment in air vacuumed place will decrease the air friction.