Hypothesis
The length of the pendulum is in direct correlation with the average period for one swing
Data collection
The data needed to provide the evidence needed to support the hypothesis would be the length
of the pendulum, which was chosen randomly, to have a wide range of values that were larger
than 0 and smaller or 1m. As for the length of the pendulum, we set the uncertainty to be at
around 0.05m, due to us not being able to measure the length more precisely (+-(0.05)
To evaluate the effect that the length of the pendulum has on the period, the time taken to
complete 10 swings was measured. This was set as 10 swings since the effect of human
reaction time on the data can then be minimized by increasing the number of swings, due to the
fact that the uncertainty added by delays in measuring were spread out over 10 swings, Which
was repeated 5 times individually, to evaluate the most precise value, since human error in
measurement can be further minimized.The average time evaluated over the 5 trials was then
calculated by adding all of the trials and dividing by the # of trials. This will give us the time
taken for 10 swings.
avg. T(for ten swings)= ((Trial) T1 + T2+T3+T4+T5)/5
This value can then be further clarified by finding out the
uncertainty, by finding out the uncertainties, which are
calculated by
(Avg. Val)-(Value with the largest difference, meaning
either highest or lowest)
. As we can see here already, the higher the average time, the
higher the length, but more importantly, the period can be
evaluated, to further compare the values and prove our
hypothesis either wrong or correct.
Period= (Avg. T)/(10)
To further visualize the value, by turning it into the period, the percentage uncertainty needs to
be found out, meaning that the uncertainty is divided by the average time and the resulting
value is then multiplied by 100. The resulting value is the percentage uncertainty of the average
time for 10 swings. The uncertainty is th same for the period, since
(Avg. Time for 10 swings/10)= Period
(Uncertainty of period=percentage uncertainty of avg. time + %age uncertainty of 10),
meaning that the percentage uncertainty of the period is the same, due to the percentage
uncertainty of 10 being 0, since we measured exactly 10 swings. This uncertainty is important in
determining whether the length actually has something to do with the length of the pendulum,
since it shows for example that the smallest value for length 1m-(1m*0.04209) and the largest
value for 37cm (0.37+(0.37*0.03411) still adhere to the hypothesis, which says that the period is
in direct correlation with the length of the pendulum, meaning the longer the pendulum, the
longer the period. As we can see, even with the
uncertainty factored in, this is true for all of the lengths
of the pendulum, meaning that the hypothesis is true.
To show this relationship in a more detailed way, the
results can be graphed, which will show how the values
lie on one axis, and thus their direct correlation. To
obtain a linear graph, I will plot the length (m) on the y
axis, and the Period^2 on the x-axis. The line of best fit
then needs to be graphed, through which we can see
that there are close to no random errors, with the points
lying on a straight graph. With the uncertainties of
0.05m and the percentage uncertainties*2, due to the period being squared, added to the
points, we can see that the line of best fit lies within all of the uncertainties, proving the direct
correlation stated in the hypothesis.
In order to now prove that the values measured are accurate, we can use the formula
Length=(Gravity/4pi)*period^2 ,having the same form as y=mx (add values from graph)
This is possible due to the fact that we know that gravity on earth is approximately 9.81ms^-2.
Knowing this, we can add the other values that were evaluated from measurements into the
equation, proving the accuracy of the experiment. The uncertainties will also need to be factored
in, in order to evaluate the minimum and maximum value for the gradient.
Gradient = change in y/change in x
Min gradient
Max. gradient
Avg. gradient
0.219ms^-2
0.301ms^-2
0.260ms^-2
=(gravity/4pi^2)
=(gravity/4pi^2)
=(gravity/4pi^2)
gravity=8.645ms^-2
11.8803ms^-2
=10.26ms^-2
Calculation:
Gravity=(4pi^2)*(gradients ms^-2)
Percentage uncertainty gradient = (10.26ms^-2-11.8803ms^-2)/(10.26ms^-2)*100=15.79%
Finally, we can find the discrepancy between the actual and the experimental value, being
discrepancy= ((9.81ms^-2)-(10.26ms^-2)+-15.76%)/(9.81ms^-2)=(8.76%+-15.76% of 8.76%)
Conclusion
The question: does the length of the pendulum affect the period? Was answered and the
hypothesis, that the length of the pendulum is in direct correlation with the period, was also
confirmed. This is both supported by the data visualized in the graph, where we can see that the
line of best fit passes through 3 of 5 points and is not outside of the uncertainty of any of the
points. The findings link back to the originally set equation which is that gravity while doing the
experiment is 4pi^2 multiplied by the gradient, for which we got the values for gravity, which
were very close to the set value of gravity on earth, pointing to the correctness of the line of best
fit, and thus to the general trend of the data being true, confirming the conformation of the
hypothesis.
Evaluation
During the experiment, we hung a piece of metal onto a stand using a piece of string. This
would be the pendulum. The length of the pendulum was then determined by the use of a wood
ruler, which could have been a major source of error in the final results, since the time for 10
swings was taken multiple times, minimizing the error, but the measurement with the ruler was
only taken once, not minimizing the chance of random error that could have occurred. For this
reason, the uncertainty of 5cm was chosen, showing that the length has a larger uncertainty
value.
When calculating the average value for 10 swings, the random errors were very surprising, with
for example 1 trial taking almost 1 second more than the average including that trial. But since
this is also data that was collected, it needs to be included into the data. When this was then
turned into the average period, a certain trend already became clear, which was that the longer
the pendulum, the longer the period would be. But to confirm that it actually had a direct
correlation, the uncertainties would need to be factored in, since it became apparent to me that
the uncertainty could lead to the data being misrepresentative, meaning that if the length of 1m,
if it has a percentage uncertainty of 100%, could be smaller than 37 cm with a uncertainty of
40%. To combat this heart to evaluate data, the use of a graph would be beneficial. When
graphing the data in a linear graph, including the uncertainties, using a line of best fit, it
becomes apparent that even though there were some random errors, the hypothesis was true,
since all of the points are on a linear graph. With this proving the correlation, the magnitude of
the line could be used to determine the gravitational force in the experiment, which was
displayed in uncertainties, with the set gravity of earth being within the uncertainty, proving that
the measurements collected and the resulting figures that proved the hypothesis correct, were
valid.