Experiment 4: Propagation of Uncertainties
PHY 1110
Section: 151
Jospeh LoDuca
Mitchell Colton
Created: 2/8/24
Date the report was submitted: 2/13/24
Purpose:
This experiment is being conducted to get a better understanding of how uncertainties in a calculated
quantity get influenced by uncertainties in measured quantities. The relevant equations for this
experiment will be discussed with the work below.
Setting up:
To set up this experiment it would require string, meter sticks, a pendulum bob, and timer.
Procedure:
In this experiment, the length L, using a two-meter stick to measure the distance from the support to the
center of the pendulum by recording the start and end position of the two-meter stick (π₯1 πππ π₯2) at
different start positions. Then we will calculate the average time per cycle it takes by pulling the bob to
the side at a designated angle and timing it with a stopwatch. Then the acceleration due to gravity and
its uncertainty will be determined from those recorded measurements.
First the length L from the support to the center of the pendulum was recorded from the center 5 different
times, starting with different start positions each time:
π₯2 (End Position)
107.8 cm
111.3 cm
116.6 cm
121.6 cm
156.7 cm
π₯1 (Start Position)
1 cm
5 cm
10 cm
15 cm
50 cm
Length (π₯2 − π₯1 )
106.8 cm
106.3 cm
106.6 cm
106.6 cm
106.7 cm
After finding the data for the length we can calculate the πΏππ£π and π’πΏ using these equations:
∑ πΏ 533
=
= 106.6 ππ = πΏππ£π
π
5
π’πΏ =
ππ‘
√π
=
0.187083
√5
= 0.0837 ππ
πΏ = 106.6 ππ ± 0.0837 ππ
There could also be other sources of systematic errors when measuring the length. One of those factors
being that the measurements were only taken by two people meaning that there could have been
different perceptions of how they were measuring it with the two-meter stick.
Now to find the period and its uncertainty, we have to time how long it takes for the ball to perform 20
oscillations. The pendulum bob will be pulled to side to produce an angle of 2°. The following times were
recorded as such:
Time (Seconds per 20 oscillations)
39.62s
41.36s
41.27s
39.20s
39.48s
Now we can find the πππ£π and the π’ π using these equations since we know that we recorded how long it
took to perform 20 cycles:
πππ£π =
π’π =
∑π
(π)
ππ‘
√π
πΆ
=
200.93
5
=
= 2.0095π
20
0.052238
√5
= 0.02345π
π = 2.0095π ± 0.02345π
After getting the recorded times and doing the calculations, our uncertainty of the period does look
reasonable to the average time per cycle.
Some of the significant sources of systematic errors could be that the times were only recorded by two
people. This could have a potential flaw in the time recording as human reaction timing differs in every
person and it could have impacted the time values. Another systematic error that came up was that we
used only two stopwatches, so the stopwatches could have potentially been somewhat flawed with its
accuracy of starting and stopping the time.
Now that we have found the average length and time per cycle, we can calculate the value of πππ₯π by
substituting the average values of L and T into this equation below:
But first the average length must be converted to meters first for future equations:
106.6 ππ = 1.066 π
π = 4π2
πΏ
1.066
π
= 4π2
= 10.422 2
2
2
π
2.0095
π
Now that the value of πππ₯π has been found, we can find the uncertainty for g by substituting g, L, π’πΏ , T,
and π’ π in this equation:
π’π
π
π’
π’
0.0837
0.02345
= √( πΏπΏ )2 + (2 ππ )2 = √( 1.066 )2 + (2 2.0095 )2 ∗ 10.422 = 0.256
π = 10.422 ± 0.256
Comparing our g value to the accepted value of 9.803, it would make our values be somewhat in the
range of the accepted value.
However, to get to that equation above, we would have need to derive it first. The first step would be to
start with a mathematical equation that would relate to the final quantity to all the other measured
quantities. For this case, the equation would be:
π = 4π2
πΏ
π2
From there the function must be differentiated using the partial derivatives. The derivatives will look like:
ππ
1
= 4π2 2 ππΏ
ππΏ
π
ππ
2πΏ
= 4π2 3 ππ
ππ
π
Then each partial derivative will be squared to ignore the negatives of the uncertainties, and plugged
ππ΄
ππ
ππ΄
ππ
into the equation that substitutes ππ₯ ππ₯ for ππΏ and ππ ππ¦ for ππ .
ππ = √(4π2
1
2πΏ
2 + (4π2
ππΏ)
ππ)2
π2
π3
Now the differentials dL and DT get replaced by the uncertainty quantities, d = u. The equation would
then look like:
π’π = √(4π2
1
2πΏ
π’πΏ )2 + (4π2 3 π’ π )2
2
π
π
Then by dividing both sides by gravity and ignoring the 4π2 because it’s a constant, we get the
uncertainty of gravity which would make the equation be:
π’π
π’πΏ
π’π
= √( )2 + (2 )2
π
πΏ
π
Now doing the correction to g with this equation:
π = 4π2
π = 4π2
πΏ
1
θ 2
2
[1
+
π ππ
(
)]
π2
4
2
1.066
1
2
[1 + π ππ2 ( )]2
2
2.0095
4
2
π
π = 10.423 2
π
Comparing our g value to the accepted value of 9.803, it would make our values to be again somewhat in
the range of the accepted value.
Experimental Insight:
Swinging the pendulum ball from side to side was quite satisfying to watch. Deriving the gravity equation
and getting the length L was a bit tedious however, but the process of it made sense as we wanted to get
a sense of uncertainty by switching the people who measure the lengths of the radius of the ball to the
end of the string. This experiment did help further our understanding of uncertainties get influenced
with how we approach to record the data in the experiment.
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