Abstract
In this lab, we set out to test if energy is conserved in the motion of circular objects rolling down
an incline. I rolled a cut section of PVC pipe down an incline, and measured its velocity and position.
From this, I calculated potential energy, as well as both rotational and translational kinetic energy. We
found that energy is conserved for rolling objects.
Objective
To test the principle of conservation of energy in the case of rolling motion for objects.
Methods & Set up
Using a lab jack, the track was raised to give a slope in order to roll the pipe down the incline.
The motion sensor was placed at the top of the slope and pointed along the track. For the PVC pipe, we
simply rolled it down the track (it was let to roll from rest at t=0). Set at 20Hz, the motion sensor
gathered data on distance from the top of the track and velocity, from which all calculations were made.
Data, Analysis, and Results
Mass (g)
Measurement
Error, +/-
Outer Radius (cm)
Inner radius (cm)
Moment of inertia
215.3
5.739
5.1
0.0006346
.05
.05
.05
1.31 x 10-6
The following data is just a small sample of the whole data – putting it all here would take up
too much space.
PVC Pipe
Time
Distance
Rotational
Total
Velocity
Potential Energy
Translational Energy
(s)
from top
Energy
Energy
0.05
0.19
0.173
0.365
0.004
0.003
0.373
0.50
0.55
0.291
0.314
0.033
0.029
0.375
0.80
0.77
0.439
0.249
0.064
0.057
0.370
1.10
1.01
0.645
0.158
0.110
0.098
0.366
1.45
1.29
0.955
0.022
0.179
0.160
0.362
The mean energy observed was 0.37008 J with a std. deviation of 0.00647 – a variation of ± 1.75%. The
standard deviation of the mean was found to be .0118.
The following is a visual representation of the data, with Translational and Rotational energy having
been combined into “Energy of motion” because they had similar values and the purpose of this
experiment is not to compare them, but to confirm conservation of energy.
0.40
0.35
Energy (Joules)
0.30
0.25
0.20
Potential Energy
0.15
Total Energy
0.10
Energy of motion
0.05
0.00
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
Time (seconds)
The error in the calculation for total energy was found by defining total energy as U, and
1
1
𝑈 = 2 𝑚𝑣 2 + 𝑚𝑔ℎ + 2 𝐼𝜔2
(1)
Which gives:
1 𝑣 2
1
𝐼𝑣 2
𝑑𝑈 = 2 𝑑𝑚[𝑣 2 + 𝑔𝑥𝑠𝑖𝑛(𝜃)] + 𝑑𝑥[𝑚𝑔 ∗ sin(𝜃)] + 2 (𝑅) 𝑑𝐼 + [− 𝑅3 𝑑𝑅]
(2)
Where h has been replaced with x*sin(θ), x being the distance along the inclined track, and θ being the
angle of incline. ω has been replaced with v/R. For reference, in SI units, dm=0.00005, dx=0.0005,
dI=0.00000131, and dR=0.00005.
In order to properly calculate root mean squared error, each term in the above statement will be
squared, then added together before taking the square root again. In other words,
1
2
2
1 𝑣 2
2 𝑅
2
𝑑𝑈𝑟𝑚𝑠 = √( 𝑑𝑚[𝑣 2 + 𝑔𝑥𝑠𝑖𝑛(𝜃)]) + (𝑑𝑥[𝑚𝑔 ∗ sin(𝜃)])2 + ( ( ) 𝑑𝐼) + [
𝐼𝑣 2
𝑑𝑅]^2
𝑅3
(3)
This gives an rms error of 3.03 x 10-4, or less than .1%. This seems odd and too small, but it also is made
under the assumption that the data gathered is perfect (that is, no instrument error). Including that
variation brings the rms error up to .141, which is 38% of the mean. This seems excessively large. The
data itself is actually confined to a narrow distribution around where one would expect the value of U to
lie. Therefore, while the calculations show that the data could be off by a large amount, the fact that the
data is quite precise leads to the conclusion that the data is good and confirms what was predicted
analytically (within a very few percent error).
Conclusion
A large source of error in this experiment came from the motion sensor not being very accurate
at gathering data on a rapidly-moving target (regardless of frequency). Because there were so many
possible different sources of error and the error calculations were long and tedious, there may be an
error in there I’m not aware of. If I wanted to reduce error as much as possible, I would have gathered
data multiple times, and then averaged each value (x and v, using the same t for both). In theory, this
would smooth out the line to perfectly flat if done n -> ∞times. However, given the circumstances, and
the condition of the data, I deemed it unnecessarily wasteful of time resources.
What this experiment shows is that energy is conserved in the rolling of a round object down an
incline. Proportionally, the amount of energy that goes into rotational rather than translational motion
depends solely on the shape of the object (ring, sphere, shell, etc.), not its mass or radius. Overall, I
would consider this lab a success – it was shown, within a small margin of error that energy is conserved
– friction does not play an appreciable role in the motion of rolling objects.