Brad Der
Confirming the Law of Conservation of Energy
The goal of this experiment was to test the validity of the Law of Conservation
of Energy, which states that the sum of gravitational potential energy, translational
kinetic energy, and rotational kinetic energy shall remain constant. To test this, various
constant torques were applied to a capstan-like system.
System and Procedure
The system involved an apparatus of a platter, with a radius given to be 12.7
centimeters, which was free to rotate, and a spindle, with radius given to be 1.5 cm,
which was attached to the platter. A string was anchored to the spindle, wound about
said spindle, and thread through a pulley over the edge of a table. A hanging mass was
attached to the end of the string over the edge. A photogate was set up such that the
edge of the photogateʼs wheel touched the edge of the platter, rotating in unison. As the
mass was released from height h1 = 75.8 cm ± 0.1 cm, the stringʼs tension applied a
torque to the spindle-platter system, causing it to rotate. When the mass had fallen to
the point where the string was completely unwound, the platterʼs momentum continued
through, the string was rewound around the spindle, and the mass was pulled back up
again. When the string was completely unwound and the mass was at its lowest point,
h2 = 7.4 cm ± 0.1 cm, the velocity of the platter would have been at its highest, and the
velocity of the mass would have been the lowest. The tension of the string would have
pulled the platter thus far, but after the mass returned from its lowest point, the platter
would have been pulling against gravity, causing a deceleration. By considering the
points at the beginning of motion and where tangential velocity of the platter was
highest as the points of interest, two points of energy were obtained and compared.
Trial Set 1
Data:
For the first set of trials, where the mass of the hanging mass was 50 g, three
trials yielded the following results:
Trial
Time (s)
Height of Mass (cm)
Tangential Velocity (m/s2)
1a
0.000
75.8 ± 0.1
0.00000
1b
9.262
7.4 ± 0.1
1.00171
2a
0.000
75.8 ± 0.1
0.00000
2b
8.191
7.4 ± 0.1
1.03421
3a
0.000
75.8 ± 0.1
0.00000
3b
9.334
7.4 ± 0.1
1.01261
As a generalization, the data gave the following graphs:
Calculations:
The gravitational potential energy of the hanging mass is represented by the
equation U=mgh. The translational kinetic energy of the hanging mass is represented by
the equation KT=½mv2. The rotational kinetic energy of the platter is represented by the
equation KR=½Iω2. The Law of Conservation of Energy states that the magnitude of
potential energy lost will equal the magnitude of kinetic energy gained, such that
-∆U = ∆K
or rather, with the above equations substituted:
-mg∆h = ½m∆v2 + ½I∆ω2
The substitutions to the above equation involved the velocity of the mass and the
angular velocity of the platter. The velocity of the hanging mass at its lowest point, as it
was changing direction at that point, was zero.
The angular velocity used to calculate rotational kinetic energy was equal to
tangental velocity divided by radius of the platter. Thus, the conservation of energy
equation can be interpreted as:
-mg∆h = ½I∆(v/rp)2
where the moment of inertia of the platter I is given to be 7.59 • 10-3 kg•m2.
To calculate the change in gravitational potential energy in the first trial, mass
was multiplied by the gravity constant and the difference between final height and initial
height, such that:
∆U = mg∆h
U = 0.050 kg • 9.8 m/s2 • (0.074 m - 0.758 m)
U = -0.33516 J
To calculate the change in rotational kinetic energy, the moment of inertia of the
platter was multiplied by 0.5 and the difference of the squares of angular velocity, such
that:
∆KR = ½I∆(v/rp)2
KR = ½ • 7.59•10-3 kg•m2 • (1.00171 m/s2/0.127 m)2
KR = 0.23623 J
Thus, substituting into the equation -∆U = ∆K, the data shows the magnitude of the
Change in Potential Energy was 0.33516 J and the magnitude of the Change in Kinetic
Energy was 0.23623 J.
Error Analysis:
The imprecise measurements, however, yielded a range of possible energies.
The error in the calculations of potential energy involved the actual mass of the
hanging mass, the validity of the gravity constant of acceleration, and the error in height
measurements. A sample of six 50 g masses was weighed and the masses of 56.32 g,
56.48 g, 56.54 g, 56.51 g, 56.54 g, and 56.52 g were recorded. While the scale used
was not calibrated and noted to be unreliable and inaccurate, the sample data yielded a
variance among the masses of 0.06 g, excluding the outlier at 56.32 g. This outlier was
excluded due to a noticeable difference in mass from the average weight, indicating that
some sort of damage or defect was present in the mass to distance it from the labelled
50 g. Thus, the error involved in the mass of the weight was 0.06 g, or 0.12%.
For the error associated with gravity, the equation
gh = g0 • [re/(re +h)]2
where gh is gravity at a height, g0 is the gravitational constant of acceleration, re is the
average radius of the earth, and h is the height above sea level, was used to
approximate the gravitational acceleration. The location of the experiment, Schaumburg
IL, is located at 242 m above sea level. With g0 given to be 9.8 m/s2 and re given to be
6,371 kilometers, the gravity in Schaumburg was approximated as:
g0 = 9.8 m/s2 • [6371000 m/(6371000 m + 242 m)]2
g0 = 9.79936 m/s2.
This would be an error in gravitational acceleration of 7.44456 • 10-4 m/s2, or
7.59649•10-5%.
The error in height was 1 millimeter, the smallest increment of measurement on
the meter stick and the estimated error due to the parallax effect. This error would yield
an aggregate error of 2 millimeters, or an 0.29240% error for the change in height.
To calculate the total error in potential energy, the relative errors were added
together, such that:
eU = em + eg + e∆h
eU = 0.12% + 7.59649•10-5% + 0.29240%
eU = 0.41248%
Thus, the error of 0.41248% gave a range of 0.33378 J to 0.33654 J of Change in
Potential Energy.
The error in rotational kinetic energy involved the error in moment of inertia of the
platter, velocity, and radius of the platter.
For velocity, the photogate took data in split-second points, as opposed to a
continuous reading. The differences between the velocity at 9.262 seconds at the two
closest points are 0.00473 m/s2 and 0.00179 m/s2. The potential error therefore was
0.00473 m/s2, or 0.47201%.
The radii in the system were listed in a series of measurement that came with the
equipment, but the error between the listed measurements and actual radii was 1 mm
for the platter, or 0.78740%.
The error in moment of inertia was listed in the same series of measurements as
7.59 • 10-3 kg•m2. However, if the machine that formed the platter was accurate yet
imprecise to the nearest gram and the error of 1 mm in radius remained, the error in the
moment of inertia became em+2er, which equaled 3.24806%.
To calculate the error in rotational kinetic energy, the relative errors were added
together, such that:
eK = eI + 2(ev-erP)
eK = 3.24806% + 2(0.47201% - 0.78740%)
eK = 4.44697%
Thus, the error of 4.44697% gave a range for possible Changes in Rotational Kinetic
Energie of 0.22572 J to 0.24674 J.
To compare the change in potential energy and the change in total kinetic energy,
the ranges of Change in Potential Energy, from 0.33378 J to 0.33654 J, and of Change
in Kinetic Energy, from 0.22603 J to 0.24713 J, were used.
In the second trial, using the same computing methods demonstrated above,
ranges of Change in Potential Energy, from 0.33378 J to 0.33654 J, and of Change in
Kinetic Energy, from 0.24491 J to 0.25843 J, were obtained.
In the third trial, using the same computing methods demonstrated above, ranges
of ranges of Change in Potential Energy, from 0.33378 J to 0.33654 J, and of Change in
Kinetic Energy, from 0.23571 J to 0.24681 J, were obtained.
Thus, the first set of trials with the 50 gram mass yielded the following ranges:
0.4
0.3
0.2
0.1
0
U1
K1
U2
K2
U3
K3
Trial Set 2
Data:
For the second set of trial, where the mass of the hanging mass was 100 grams,
the two trial yielded the following results:
Trial
Time (s)
Height of Mass (cm)
Tangential Velocity (m/s2)
1a
0.000
75.8 ± 0.1
0.00000
1b
6.133
7.4 ± 0.1
1.50205
2a
0.000
75.8 ± 0.1
0.00000
2b
6.342
7.4 ± 0.1
1.50235
Calculations:
The change in gravitational potential energy in the first trial was
∆U = mg∆h
U = 0.100 kg • 9.8 m/s2 • (0.074 m - 0.758 m)
U = -0.67032 J
The change in rotational kinetic energy in the first trial was
∆KR = ½I∆(v/rp)2
KR = ½ • 7.59•10-3 kg•m2 • (1.50205 m/s2 / 0.127 m)2
KR = 0.53085 J
Thus, substituting into the equation -∆U = ∆K, the data showed the magnitude of
the Change in Potential Energy was 0.67032 J and the magnitude of the Change in
Kinetic Energy was 0.53242 J.
Error Analysis:
In the second trial, using the same computing methods demonstrated above, ranges of
Change in Potential Energy, from 0.33378 J to 0.33654 J, and of Change in Kinetic
Energy, from 0.24491 J to 0.25843 J, were obtained.
The error, which was calculated in the same manner and method as those in the
first series of trials, gave ranges for the first trial in the second series for Change in
Potential Energy, from 0.66675 J to 0.67389 J, and Change in Kinetic Energy, from
0.51858 J to 0.54312 J.
In the second trial, using the same computing methods demonstrated above,
ranges of Change in Potential Energy, from 0.66675 J to 0.67389 J, and of Change in
Kinetic Energy, from 0.51426 J to 0.54786 J, were obtained.
Thus, the second set of trials with the 100 gram mass yielded the following
ranges:
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
U1
K1
U2
K2
Interpretation and Conclusion
In both sets of trials, the Law of Conservation of Energy was not supported. The
total kinetic energy in all trials was biased to be lower than the potential energy,
indicating a loss of energy and/or a systematic error.
By comparing the ratio of kinetic energy to potential energy, it was found that the
trials of the first series lost 19.5%, 24.9%, and 18.0% of the initial potential energy. It
was found that the trials of the second series lost 20.8% of the initial potential energy in
both trials. These lost energies were not unreasonable considering the effects of friction
and air drag. Friction along the string, which could be found around the spindle and
other sections of string, and air drag against the massʼs movement would both have
impeded acceleration, resulting in a lower system velocity when the mass reached its
lowest point and an overall lower kinetic energy. In much the same way that a bouncing
ball does not return to its original height due to lost energy, so too did the mass not
return to its original height or energy.
Possible errors to also lead to this bias included the parallax effect when
measuring the height of the hanging mass. If the actual height was smaller than
recorded, the potential energy would have been calculated to be lower and the Law of
Conservation of Energy may have been supported.
Also a possible systematic error was the mass of the hanging weight. The mass
was not determined to be exactly 50 g, and as such, the calculations of potential energy
and translational kinetic energy were inherently inaccurate.
Another possible error was the measurement of velocity. The photogate
measured in split-second data points, which, while precise, are not as accurate as a
continuous reading. If the real maximum velocity was higher than the observed
maximum, both translational and rotational kinetic energy would have been calculated
to be higher and the Law of Conservation of Energy may have been confirmed.
If this experiment was to be repeated, necessary changes to the procedure
would have to include a method of determining perpendicularity of the meter stick to the
ground, so measurements may be done as close to the mass as possible without visual
discrepancies due to angled measuring tools. Also necessary would be a closedcondition scale to accurately measure the mass of the hanging weight. More advanced
equipment for the reading of velocity would also increase the accuracy of the
measurements. With these changes, the results may indicate a confirmation of the Law
of Conservation of Energy, something this data was unable to produce.