Lab 4 PHY105M (1)
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University of Texas *
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105M
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Chemistry
Date
Apr 29, 2024
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Introduction:
This week our objective was to examine the elastic properties of a chosen extensible material to
determine whether Hooke’s law applied. For this experiment, we opted to investigate the
behavior of a rubber band that was provided in the lab. To investigate the mechanical properties
of a band under varying loads, we utilized a meter stick to measure the amount of stretch in
meters. The experiment involved applying masses of weights 1 kg, 0.7 kg, 0.5 kg, 1.7 kg, and
1.2 kg to the band. We aimed to determine the spring constant, k, in the equation F=-kx, where
F represents force in newtons and x denotes the displacement in meters.
The uncertainty that was noted for force due to the multiplication of mass and gravity was +/-
.0098N and the uncertainty for the displacement based on the use of the meter stick was +/-
.0005m. To determine k, we first calculated the force exerted by each mass using the formula
F=mg, where m is the mass and g is the acceleration due to gravity that is 9.8 m/s ^2. Then, we
divided the force by the change in length of the band when it stretched, yielding the spring
constant k in newtons/meter. We also calculated the uncertainty propagation as there are
uncertainties in both force and displacement which contribute to the uncertainty of k.
Subsequently, we plotted the data obtained from the experiment into two different graphs of k
vs.x and f vs. x.
The resultant graphs allowed us to observe the relationship between force and displacement
and verify if it followed a linear trend as described by Hooke's Law. Despite the diverse range of
masses utilized, our hypothesis was that the relationship between force and displacement would
maintain a linear trend, consistent with Hooke's law. However, it's important to note that the
uncertainty in our experiment primarily stems from measurement errors in determining both the
masses and the corresponding displacements of the rubber band..
Conducting Your Experiment:
ΔF (N)
Δx (m)
k
𝑖
K
δ
𝑖
9.8 +/- 0.0098N
0.035 +/- 0.0005m
280 +/- 4.01N/m
4.01
6.86 +/- 0.0098N
0.19 +/- 0.0005m
361.05 +/- 9.52 N/m
9.52
4.9+/- 0.0098N
0.01+/- 0.0005m
490 +/- 24.52N/m
24.52
11.76+/- 0.0098N
0.048 +/- 0.0005m
245 +/- 2.56 N/m
2.56
16.66+/- 0.0098N
0.0705 +/- 0.0005m
236.3 +/- 1.68 N/m
1.68
Note:
We used 5 different masses to determine the linearity of the rubber band with force in N,
displacement in m, and the spring constant in N/m.
Conclusion:
Upon analysis of the force versus displacement plot, it was evident that Hooke's law
applied consistently across the entire range of forces tested. Additionally, the plot exhibited a
linear relationship between force and displacement, further confirming the validity of Hooke's
law within the tested regime. This outcome aligns with our hypothesis that the behavior of the
rubber band would adhere to a Hookean regime, thereby failing to reject our initial prediction.
However, the experiment could be enhanced in future iterations by implementing more precise
measurements of force and displacement, as well as considering environmental factors that
may influence the material's behavior. Additionally, upon examination of the k versus
displacement graph, a downward slope was observed. This suggests that as the rubber band
stretches further, it may be experiencing a decrease in stiffness or an increase in elasticity,
possibly due to material fatigue. Though we viewed a linear regime as plotted in the scatterplot
of f vs. x, the uncertainties in the spring constant ki varied, ranging from approximately
+/-4.01N/m to +/-24.52 N/m. This variation of uncertainties for the spring constant can be
attributed to the uncertainties of both force and displacement, which is why the uncertainty
propagation formula was used as they affect the determination of k. Reducing these
uncertainties through improved measurement techniques or instrumentation could lead to even
clearer validation of Hooke's law within the tested masses. Alternative ranges that could be
explored could be higher ranges in the 2,000 kg range, and a possibility of improvement could
be using a different measuring device instead of a meter stick as well as using a different
material that could possibly stretch more as ours was very sturdy.
Part 2:
Introduction:
In the initial phase of the experiment, we successfully validated Hooke’s law
across the entire range of forces applied to the rubber band. Considering this, we sought to
explore a different range of forces in search of potential non-linear behavior, indicating a
deviation from Hooke's law. For this purpose, we utilized masses of 0.4 kg, 1.9 kg, 2.5 kg, 3.0
kg, and 3.4 kg, extending our investigation beyond the previous range. Following the
established methodology from the first part, we employed a meter stick to measure the stretch in
meters as we applied the varied masses to the rubber band. Our objective remained to
determine the elastic properties of our material and we first had to find the spring constant, k, in
the equation F=−kx, where F represents force in newtons and x signifies displacement in
meters.
The uncertainties for Force and displacement were +/-.0098N and +/- .0005m as we used the
meterstick to determine the length the rubber band stretched. To derive k, we utilized the
formula F=mg to calculate the force exerted by each mass, where m represents the mass in kg
and g denotes the acceleration due to gravity which is approximately 9.8 m/s^2. We made use
of the uncertainty propagation equation as there is an effect on the value of k based on the
uncertainties of force and displacement. Subsequently, we determined k by dividing the force by
the change in length of the rubber band during stretching, or displacement. Following the data
collection, we plotted the force that we obtained from multiplying the mass and gravity, against
the corresponding displacement of the rubber band.
This plotting enabled us to observe the relationship between force and displacement, allowing
for the identification of any potential non-linear trends in the elastic response of the rubber band.
We anticipate that as the applied forces increase within this extended range, the relationship
between force and displacement will maintain linearity, following the principles outlined by
Hooke's law. It's essential to note that the primary source of uncertainty in our experiment stems
from measurement errors in determining both the masses and the corresponding displacements
of the rubber band.
Conducting Your Experiment:
ΔF (N)
Δx (m)
k
𝑖
k
δ
𝑖
3.92 +/- 0.0098 N
0.08 +/- 0.0005m
490 +/-30.65 N/m
30.65
18.62 +/- 0.0098 N
0.079 +/- 0.0005m
235.7 +/-1.5 N/m
1.5
24.5 +/- 0.0098 N
0.087 +/- 0.0005m
281.01 +/- 1.62 N/m
1.62
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