Measuring Energy Using a
Pendulum
Asif Nazerally,
Syafiqah R., Andrew W., Savannah W.
B. C. Boivin
SPH 3UE, Grade 11 University Physics, Rockland District High School
Thursday, November 15th, 2012
2
Introduction
In this experiment, a pendulum will be used to measure potential energy and kinetic
energy. Gravitational energy is the energy that an object falling in a gravitational field has, but it
is also called potential energy due to its potential to be changed into
another form of energy (Anonymous, What is gravitational energy
definition?, 2012). The more mass in an object and the higher it is
Figure 1. The
equation for
relative to a certain point, the more gravitational potential energy there
potential energy
(Anonymous,
is, which is measured in ‘Joules’ (J) (Anonymous, Work, Energy, and
Physics , 2011).
Power, 2012). Potential energy is typically measured relative to the ground, but for the purpose
of this experiment, it will be measured to the lowest swing point of the
pendulum (see Figure 1).
The other form of energy being measured in this experiment is
kinetic energy, the energy of motion (Nave, 2000). Kinetic energy is also
Figure 2. The
equation for kinetic
energy (Sportsman,
2010).
measured in ‘Joules’ (J) and is dependent on the mass of an object and the
speed at which it is travelling at (see Figure 2) (Anonymous, Kinetic Energy, 1999). A simple
example of kinetic energy is the energy of a car. Taking the mass (in kilograms) of the car and
halving it, then multiplying it by the squared speed that the car is travelling at (in metres per
second) will determine the amount of kinetic energy that the car has, changing in accordance
with the acceleration of the car.
The piece of equipment being used in this experiment to measure potential and kinetic
energy is a pendulum. There will be a weight that will swing from side to side, attached to a
fixed point by a string. In Figure 3, the weight will stop moving for a fraction of a second at
3
points 1 and 5. This is where the weight will have no kinetic energy (since there is no
movement), but it is also where the highest amount of potential energy is, because it is at its
highest point from the relativity reference point (point 3). Point
3 is the relativity reference point (where the potential energy is
being measured from) because it is as low as the weight will go.
This is also the point where there is no potential energy and has
the highest amount of kinetic energy since it is at its top speed.
Figure 3. A diagram of a
pendulum (Anonymous,
Pendulum Experiment,
2001).
To better understand a pendulum, it is important to
understand The Law of Conservation of Energy. It states that
energy cannot be created, nor can it be destroyed, but it is capable of changing forms. In this
experiment, potential energy will be changing into kinetic energy and vice versa. A mechanical
system will maintain a constant amount of energy over time, but this energy could be impacted
by gravitational fields, air resistance, and other sources where the energy isn’t destroyed, but is
outputted and transformed into other energy (Anonymous, Conservation of energy, 2012). In
the instance of a pendulum, it is impossible for it to rise above its initial height when given no
initial acceleration (Anonymous, Law of Conservation of Energy, 2012). If this experiment were
to be done in a vacuum (no air resistance, a quixotic, hypothetical environment) with a
frictionless pendulum, the pendulum would stay in perpetual motion according to the Law of
Conservation of Energy.
The purpose of this experiment is to use a pendulum to experimentally prove that the
amount of energy in a closed system obeys the equations 𝐸𝑇 = 𝐸𝑔 + 𝐸𝑘 and 𝐸𝑔 = 𝐸𝑘 .
4
Theory
𝐸𝑔 = 𝑚𝑔ℎ
1
𝐸𝑘 = 𝑚𝑣 2
2
𝐸𝑇 = 𝐸𝑔 + 𝐸𝑘
Variables
𝐸𝑔 – Potential/Gravitational Energy 𝐸𝑘 – Kinetic Energy
𝐸𝑇 – Total Energy
𝑚 – Mass
𝑔 – Gravitational Constant
ℎ – Height
𝑣 – Speed
Materials and Methods
The materials and methods used were that of the Measuring Potential and Kinetic Energy using
a Pendulum handout on November 7nd, 2012.
Figure 4. Lab setup
(Rudeen, 2012).
5
Results and Observations
Table 1. Determining potential energy of the pendulum.
Potential Energy, Eg (J)
Mass (kg)
Drop Height (m)
[𝐸𝑔 = 𝑚𝑔ℎ]
0.05
0.41
0.2
0.200
0.46
0.9
0.500
0.51
2.5
Table 2. Determining kinetic energy of the pendulum.
Distance
Time for 20
Period
Mass (kg)
Travelled for
Complete
(secs.)
One Swing (m) Swings (secs.)
[time/20]
0.05
3.2
36.8
1.84
0.200
3.6
36.2
1.81
0.500
4.0
39.9
2.00
Calculations
% of Difference for 50g weight
% of Difference =
𝐸𝑔 − 𝐸𝑘
x 100%
𝐸𝑔
% of Difference =
0.2 − 0.0756
x 100%
0.2
% of Difference = 62%
% of Difference for 200g weight
% of Difference =
𝐸𝑔 − 𝐸𝑘
x 100%
𝐸𝑔
% of Difference =
0.9025 − 0.4044
x 100%
0.9025
% of Difference = 45%
Average
Speed, v
(m/s)
1.739
2.011
2.004
Kinetic Energy,
Ek (J)
1
[𝐸𝑘 = 2 𝑚𝑣 2 ]
0.0756
0.4044
1.0045
6
% of Difference for 500g weight
% of Difference =
𝐸𝑔 − 𝐸𝑘
x 100%
𝐸𝑔
% of Difference =
2.5 − 1.0045
x 100%
2.5
% of Difference = 60%
Discussion
It was determined mathematically that the potential energy was 0.2J for the 50g weight,
0.9J for the 200g weight and 2.5J for the 500g weight. It was then determined experimentally
that the kinetic energy was 0.0756J for the 50g weight, 0.4044J for the 200g weight and 1.0045J
for the 500g weight, after twenty complete swings of the weights. These results appear to be
inaccurate due to the large margin of error as seen in a percentage loss of potential energy
ranging from 45% to a maximum of 62%. These are primarily due to the pendulum losing a
great deal of momentum and energy due to the instability of the pendulum. Some small
amount could also have been lost to gravitational fields and air resistance, although they would
be very small.
There were many possible sources of errors throughout this experiment. One of them
would be that a lot of the potential energy is being lost through the pendulum. The reason is
because the pendulum being used were quite old, therefore unstable and wobbly, and making
the results of the experiment inaccurate. A possible solution to this would be to perhaps invest
in more modern or reliable equipment or to clamp down the pendulum (but that would still
leave the upright pole unstable).
7
Another possible source of error would be the energy lost in the string. The string that
the weight was hung from was tied around a metal ring, which was attached to the pendulum.
When one of the heavier weights were used, the string was susceptible to sliding, which would
render the results inaccurate since there is energy being lost. A solution to this issue would be
to tape down or glue down the string to the metal ring.
A third possible source of error could be that the desks that the pendulum stood on
were uneven or crooked, thus making the swinging difficult to go one way and easier to go the
other. To fix this issue, the desk’s legs could have been measured until they were made even.
All of these possible sources of error could have negatively affected the results of this
experiment.
Conclusion
With three weights of different masses, it was determined experimentally that the
amount of energy lost after twenty swings (the percentage of difference between 𝐸𝑔 and 𝐸𝑘
after twenty complete swings) was 62% for the 50g weight, 45% for the 200g weight and 60%
for the 500g weight.
Questions
1. Calculate the percentage difference between Eg and Ek for the mass that gave you
the most precise results (closest) using the formula: % of Difference =
Ensure that you comment about how precise you feel this value is.
𝐸𝑔 − 𝐸𝑘
𝐸𝑔
x 100% .
8
(Please refer to the ‘Calculations’ section on page 4)
I feel that the percentage of difference is quite accurate due to the amount of energy lost to
the unstable pendulum and the slight movement of the string (please refer to the ‘Discussions’
section on page 4 for more details on sources of error).
2. Calculate the percentage difference between Eg and Ek for the mass that gave you
the worst results (furthest) using the formula from the previous question. Ensure you
comment about how precise you feel this value is.
(Please refer to the ‘Calculations’ section on page 4)
I am not sure why the smaller weight would attract more inaccurate results than the
larger weights, because the larger weights should cause the pendulum to wobble and shake
more, taking energy away from the swinging itself. I do feel that the percentage of difference is
accurate because there were many sources of errors to take into account like the instability of
the pendulum (please refer to the ‘Discussions’ section on page 4 for more details on sources of
error).
3. List a few sources of error that you feel could have affected your results.
(Please refer to the ‘Discussions’ section on page 5)
9
Selected References
Anonymous. (2012, November 15). Conservation of energy. Retrieved November 14, 2012, from
Wikipedia: http://en.wikipedia.org/wiki/Conservation_of_energy
Anonymous. (1999, September 29). Kinetic Energy. Retrieved November 14, 2012, from Theory
UWinnipeg: http://theory.uwinnipeg.ca/mod_tech/node30.html
Anonymous. (2012). Law of Conservation of Energy. Retrieved November 14, 2012, from City
Collegiate: http://www.citycollegiate.com/workpowerenergy_Xd.htm
Anonymous. (2001). Pendulum Experiment. Retrieved November 14, 2012, from Material
Worlds: http://www.materialworlds.com/sims/Pendulum/pendulums.gif
Anonymous. (2011 йил 11-February). Physics . Retrieved 2012 йил 13-November from
Wikibooks:
http://upload.wikimedia.org/math/4/7/0/470ec22866c4b54b959c8559cd27c25d.png
Anonymous. (2012). What is gravitational energy definition? Retrieved 2012 йил 13-November
from Answers: http://wiki.answers.com/Q/What_is_gravitational_energy_definition
Anonymous. (2012). Work, Energy, and Power. Retrieved 2012 йил 13-November from The
Physics Classroom: http://www.physicsclassroom.com/class/energy/u5l1b.cfm
Nave, C. R. (2000, August). Kinetic Energy. Retrieved November 14, 2012, from HyperPhysics:
http://hyperphysics.phy-astr.gsu.edu/hbase/ke.html
Rudeen, S. (2012 йил 7-November). Physics 11/12 2012. Retrieved 2012 йил 12-November
from Facebook: http://sphotos-a.ak.fbcdn.net/hphotos-akash3/533604_10151129833702810_1480149778_n.jpg
Sportsman. (2010, August 10). Kinetic Energy Recovery Systems. Retrieved November 15, 2012,
from Formula 1 Journal: http://i.ehow.com/images/a04/9c/4m/calculate-kinetic-energy-1.5800X800.jpg