Research question
To establish a relationship between the separation distance in a coupled pendulum system
and the time taken for a coupled pendulum to transfer all energy from one pendulum bob to
the other.
Introduction
I chose this experiment because I have always been interested in how energy is transferred.
For example in tennis or in badminton, the principle of physics is the same. The shuttlecock
or the tennis ball hits the racquet and all that stored potential energy in the racquet is
transferred through contact to the shuttlecock or ball. The distance the shuttlecock or ball
travels is proportional to the force exerted onto it. This means that the further the distance,
the more energy is needed to hit the ball to make it travel that distance. That theory applies
in this experiment because the further the two pendulum bobs are from each other, the more
time it takes for them to transfer the energy from one another.
Background
Couple pendulum refers to a system of two equal-length pendulums connected by a
third-string as shown in the figure 1 below:
Figure 1: structure of a coupled pendulum
When one pendulum starts to swing in the system, it triggers a pull on the third string. As the
second pendulum is also connected to the third string, it will experience a small force from
the third string and begin to swing slightly out of phase. At the same time, the second
pendulum exerts a tension to the third string which pulls back the first pendulum. As a result,
the amplitude of oscillation in the first pendulum decreases and that of the second pendulum
increases. The initial energy of the first pendulum is completely converted to the second
pendulum when the first pendulum comes to rest. The time taken in this process is
investigated with different separations between the two pendulum strings.
Hypothesis
My hypothesis is: the further pendulums are from each other, the longer it will take to transfer
its energy:
In my experiment, when the first pendulum starts to swing, a small tug acts upon the third
string connecting the pendulum, causing a wave (energy) to propagate from one pendulum
to the other. This wave carries energy signals from the first pendulum to the second one.
(see the Figure 2 below)
Figure 2: energy propagation from one pendulum to the other
It takes time for this signal to travel back and forth between the pendulums. The time
required for the signal to travel between the pendulum depends on the distance from one
pendulum to the other. Therefore, my hypothesis is that the time of energy transfer and the
separation of the two pendulums has a positive correlation. i.e., the larger the separation of
the pendulums, the more time it takes for one pendulum to transfer its energy to the other
and completely stop.
Variables in the experiment
Variable type
Variable
Details
Independent variable
Separation distance between the
two pendulum bobs
Starts with a length of 5 cm,
increases by 5 cm interval.
Measure this with a meter
rule or a smaller ruler. To
change the distance first use
a ruler and mark out all the
different points on the rope
its self. You can do this
before you attach the rope
to the G stand or whil you
are conducting the
experiment itself
Dependent variable
Time taken for energy
The amount of time take for
the pendulum being struck
to transfer all its energy to
the other pendulum to stop
swinging and back again. To
measure this you use a
stopwatch
Length of pendulum
The length of the pendulum
has to be the same and
should not be changed
because your results
depend on the length
Height of stand
The height of the stand
holding everything up should
not change.
Initial displacement of oscillation
Can be difficult at first but try
and make sure when you hit
the pendulum, it is always
the same amount of force
hitting it
Control variables
Experimental Setup
The setup of the experiment is shown in the picture below:
Items needed:
Stands with boss heads x 2
G clamp x 2
Strings x 3 (2 of equal length, 1 longer string )
Pendulum bobs x 2
Stopwatch x 1
Meter ruler x 1
Marker pen x 1
Procedure:
1. Place two stands with boss heads a certain distance apart from each other keep the
distance between the stands the same distance at all times as the string tension
depends on this
2. Attach G clamps on the bottom to a hard surface to keep it stable; this is to make
sure the stands do not fall if the pendulum has too much force added to it.
3. Attach string from one side to the other and cut any string remaining to make sure
nothing is getting in the way
4. Set up two vertical strings with a starting distance 5 cm apart by marking the points
with a marker ( I did this so that my experiment could be done with efficiency). You
can either mark the points on the horizontal string before you start the experiment e.g
on a flat surface using a bigger ruler or you can mark down the points during the
course of the experiment itself.
5. Make a loop around the large string on the string with the bob so that you can
change the distance between the pendulums.
6. Displace a pendulum with a fixed amplitude to trigger an oscillation. Try to make sure
that the amount of power is used every time when hitting the pendulum.
7. Release the pendulum and start the timer spontaneously ( if this proves difficult, ask
someone to help you start the timer )
8. Stop the timer right after the second pendulum transfers back the energy it got from
the first pendulum and comes to a complete stop.
9. Repeat step 6 - 8 by increasing the separation between the pendulums with a 5 cm
interval, i.e. adjusting the separation of pendulums to 5 cm,10 cm, 15 cm,20 cm,25
cm,30 cm,35 cm 40 cm,45 cm and 50 cm
Safety
When conducting the experiment, make sure you wear goggles to prevent your eyes
from being hit by the pendulum
- Make sure no one is within a 1.5 meter perimeter of your pendulum experiment while
it is taking place because the pendulum can hit them in the eye or hurt them or it can
hit them elsewhere on the body and hurt them quite badly as these pendulums are
quite heavy.
- Always conduct the experiment while standing up so you are alert.
- Make sure when using the G clamps, the clamps are securely holding down the
stands as they can fall and seriously injure someone’s feet.
- Make sure the pendulum bobs are properly tied to the rope as it will hurt someone if it
falls off during the course of the experiment.
Raw Data and error analysis
The raw data of the measurements in the 5 trials of each set of the experiment are shown
as the table below:
Table B : The time taken for energy transfer and the length between the two pendulums.
The length
between two
pendulums /
cm (± 0.3 cm)
Time taken for energy to transfer from the first pendulum to the
second pendulum
/ s ( ±0.6 s)
1st trial
2nd trial
3rd trial
4th trial
5th trial
Average time
5.1
4.0
3.9
4.3
4.1
4.5
4.2
10.2
4.8
4.5
5.6
4.7
4.9
5.0
15.2
5.1
5.5
4.9
5.5
5.0
5.2
20.1
5.0
5.2
5.5
5.0
5.5
5.2
25.3
6.3
5.2
5.8
5.8
6.0
5.8
30.2
6.1
5.6
5.7
5.4
6.0
5.9
35.2
6.9
7.5
7.6
6.8
7.1
7.2
40.1
7.5
7.1
7.8
8.3
8.1
7.7
45.2
9.4
9.8
9.9
10.0
9.3
9.7
50.1
10.2
9.0
9.7
10.2
11.4
10.1
Note that the value of the trials is in 1 d.p. so as its uncertainty based on the sensitivity of the
stopwatch.
Decimal places and uncertainties
The length between the two pendulums is reported in 1 d.p. as the sensitivity of the ruler is
0.1 cm. The absolute uncertainty of the length Δs is based on the smallest scale on the ruler
Δl = ± 0.1 cm and the width of the maker mark on the string Δw = ± 0.2 cm:
Δs = Δl + Δw = ± 0.3 cm
Time taken to transfer energy from one pendulum to the other is reported in 1 d.p. as the
stopwatch can measure accrue to 0.1 s. The uncertainty of the average time Δt is based on
the maximum fluctuation of the data set, it is calculated by:
Δt =
­
tmax tmin
2
According to table B, the measured time fluctuated the most in the last trial (with a string
length of 50.1 cm), the uncertainty is calculated by:
Δt = 11.42­9.0 = ± 1.2 s
Note that the degree of fluctuation of the time is different for each set of experiments. The
largest fluctuation is taken as the uncertainty of the time of energy transfer.
Result
The correlation of between the separation of pendulum and the time of energy transfer in a
double pendulum system is shown in the graph below:
Result graph 1 - correlation between time of energy transfer and separation
The result graph 1 suggests is that there is a positive relation between the time and
separation, i.e. the time of energy transfer increases with the separation between the
pendulums. The line of best fit shown in the Result graph passes through all the time error
bars, demonstrating the uncertainty in time is reasonably estimated.
The relation between the time of energy transfer T and the separation of the pendulums d is
given by T = 0.127 d + 3.09 (3 s.f.). The R2 value is calculated by microsoft excel program.
R2 =0.9005 suggests that there is a strong linear correlation between the time and the
separation.
The experimental result is consistent with the hypothesis I stated. As it takes time for a wave
(energy) to travel in between the pendulum system, the longer the third string is, the longer
time it takes to receive an energy signal from one pendulum to the other.
Further analysis about the correlation
The result graph may also suggest a different correlation rather than a straight line. By
looking at the curvature of the graph, a square model of the relation T ime = a × distance 2 + b
(where a and b are constant) is also tested. This is done by using the method of
linearization.
To analyze the uncertainty of the square value of the distance, error propagation method of
indices is applied:
δ d2 = 2 δd
d
Based on the calculation, the uncertainty of the squared value of the distance is
insignificantly small (in the order of 0.01).
Table C below shows the processed data in order to investigate the likelihood of a square
correlation between the time and distance between the pendulums.
Table C - New table of processed data with squared value of distance
The length
between two
pendulums /
cm (± 0.3 cm)
Square value of
the length d2
/ cm2
Average time for energy to transfer
from the first pendulum to the second
pendulum
/ s ( ±0.6 s)
5.1
26.0
4.2
10.2
104
5.0
15.2
231
5.2
20.1
404
5.2
25.3
640
5.8
30.2
912
5.9
35.2
1240
7.2
40.1
1610
7.7
45.2
2040
9.7
50.1
2510
10.1
The square value is rounded to 3 s.f. ,based on the significant figure and sensitivity of the
instrument.
Result graph 2 - Correlation between time of energy transfer and separation squared
Note that the error bar of the square values of separation is too small to be shown in the
graph. Based on the line of best fit of the linearized graph is determined by microsoft excel
program, the relation I obtained is T ime = 0.0023 × distance 2 + 4.3457 . Furthermore, the
uncertainty of the gradient of the linearized graph is also estimated by two lines of best fit
with a maximum gradient and minimum gradient:
The uncertainty of the gradient a is determined by:
δa=
­
amax amin
2
=
­
0.003 0.0016
2
= ± 7 x 10-4 s cm-2
The R2 values of the line of best fit is between the linear model (in the previous part of
result) and the quadratic model are compared. The R2 value is 0.9684 (quadratic model) >
0.9005 (linear model), demonstrating the correlation is more likely to be a quadratic relation.
Conclusion
In conclusion, this experiment was conducted to establish a relationship between the
separation distance in a coupled pendulum system and the time taken for a coupled
pendulum to transfer all energy from one pendulum bob to the other. This relation is studied
by connecting a string through two pendulums with the same length and mass. A pendulum
is given a fixed amplitude to swing and the time taken for energy transfer is recorded when
the pendulum stops swinging momentary (as energy is transferred fully to the second one).
A linear and quadratic relation is investigated using the linear regression method. The
quadratic model is linearized by graphing time vs. distance2. The linear model I found
regarding the time of energy transfer T and the separation d is T = 0.127 d + 3.09 , whereas
the quadratic model is T = 0.0023 × d 2 + 4.3457 . Both the models support my hypothesis
that the time increases when the separation of the pendulum increases. By comparing the
R2 value of the quadratic (0.9684) and linear model (0.9005), I conclude that the relation is
more likely to be quadratic. The final relation I obtained is T = 0.0023 × d 2 + 4.3457 with a ±
7 x 10-4 s cm-2 uncertainty of the gradient.
Evaluation
If I were to do this experiment again, there are some changes that I would make to the actual
set up. First,I would not use a timer, I would use a light gate connected to a computer that
can help accurately determine when the pendulum bob has been hit and when the bob
finally comes to rest. Moreover, the other change I would make is I would make a
mechanism that would hit the pendulum bob at exactly the same power each time as I felt
sometimes, i hit the bob too hard thus making it faster and therefore making the results less
reliable. This method,did allow me to find a relationship between time taken for a coupled
pendulum to transfer all energy from one pendulum bob to the other.You can see in the raw
data table, there is an anomaly of 12.4, I am not quite sure why this value came up but, I
believe it may been because I turned away to look at something and the timer still kept
running after the pendulum had stopped however, this does not seem to have a huge impact
on the graph/outcome. As you can see there is a positive correlation in the raw and
processed data graph as the graph is increasing and shows a positive relation. The line of
best fit falls through the error bars the uncertainty in time which was calculated using the
equation : Δt =
­
tmax tmin
2
. I used this formula to calculate the uncertainty as there were
systematic and random errors in the relationship. There were also instrumental and random
errors/uncertainties as the stop watch as an uncertainty of 0.01. Add that to the random error
of the experiment conductor as the time where the pendulum bob is hit and the timer is
started is very small thus causing an uncertainty in the data of ±0.6 s. Moreover, the
uncertainty of the ruler is ± 0.1 add that to the uncertainty of the marker it self ± 0.2, the total
uncertainty of the measurement in length is ± 0.3 The relation between the time of energy
transfer T and the separation of the pendulums d is given by T = 0.127 d + 3.09 The R^2
value 0.9684 shows the relationship between the distance and time is strong and that it is a
quadratic relationship.
References:
https://www.exploratorium.edu/snacks/coupled-resonant-pendulums
https://www.physics.wisc.edu/ingersollmuseum/exhibits/mechanics/coupledpendulums/
https://en.wikipedia.org/wiki/Pendulum
https://en.wikipedia.org/wiki/Double_pendulum
https://courses.lumenlearning.com/physics/chapter/16-4-the-simple-pendulum/