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Optimizing Accuracy and Precision in
Experimentation: Atwood’s Machine
Daniel Kim
AP Physics C
Period 5
ABSTRACT
Whatever the nature of the study, reasonable assumptions allow scientists to quickly arrive at
answers that approximate reality. Despite the convenience and expedience of this approach,
however, there are – quite predictably – some limitations: Under particular conditions,
assumptions start yielding answers that are neither accurate nor precise. Since quantifying
these constraints has proven to be quite challenging, an Atwood’s Machine was optimized for
experimental error. Low values of
and high values
were found to maximize accuracy
and precision. The present study proposes a generalized method to identify the optimal ranges
for any scientific instrument.
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INTRODUCTION
To mitigate the rigors of mathematics, assumptions are often made in experimentation.
Whatever the nature of the study, reasonable assumptions allow scientists to quickly arrive at
answers that approximate reality. Despite the convenience and expedience of this approach,
however, there are – quite predictably – some limitations: Under particular conditions,
assumptions start yielding answers that are neither accurate nor precise.
Since quantifying these constraints has proven to be quite challenging, the primary goals of the
present study are to (1) optimize experimental error in a simple apparatus and (2) generalize
this optimization process to other, more complicated instruments.
THEORY
A simple Atwood’s Machine was constructed to approximate the acceleration of the attached
masses. The following section specifies the concepts behind theoretical acceleration,
experimental acceleration, and the statistical methods required for analysis.
Theoretical Acceleration
Theoretical acceleration
mass
is calculated using only known or given values. If the pulley’s
is assumed to be negligible, the tension force
will be the same on both sides of the
string (FIGURE 1). Keeping this in mind, the following equation is derived:
FIGURE 1: Schematic of Atwood’s Machine
Therefore, the theoretical acceleration of the two masses can be expressed as follows:
[1]
2
Experimental Acceleration
Unlike theoretical acceleration, experimental acceleration
is calculated using only
measured values. In this study, experimental acceleration is found by considering the time
taken to travel a vertical displacement
. Starting from a simple kinematics formula, the
following equation is derived:
[2]
Accuracy
Accuracy refers to the proximity of measurements to the “true” value. Percent error
serve as the study’s measure of accuracy. After
and
will
are found using EQUATION 1 and
EQUATION 2, percent error can be found by
[3]
Precision
Precision denotes the reproducibility of a measurement. In probabilistic terms, precision
represents the general distribution of experimental data – the likelihood that repeated
measurements will produce similar results. As in the past, the a priori error estimate will
serve as the study’s relative measure of precision. Since experimental acceleration is wholly
dependent on displacement and time (EQUATION 2), the precision of acceleration
calculated in terms of
and
must be
– the a priori error estimates of displacement and time,
respectively. Error propagation is performed as follows:
[4]
After two partial derivatives of EQUATION 2 are taken, EQUATION 4 simplifies into
[5]
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Multiple Regression
After taking some measurements, scientists are often interested in how independent variables
affect a dependent variable. For this reason, scientists perform multiple regression to obtain
models with high predictive validity (FIGURE 2).
FIGURE 2: Sample Multiple Regression Table
(A)
(C)
(B)
In order to maximize this predictive power, some factors must be taken into consideration:

The p-value of the T statistic represents the significance of an individual term: The
smaller the p-value (or higher the T statistic), the more significant the independent
variable is to the dependent variable. (FIGURE 2A)

The p-value of the F statistic evaluates the statistical utility of the entire model: A small
p-value (or a high F statistic) implies that the model is a good fit. (FIGURE 2B)

The multiple coefficient of determination (R2) quantifies how well the model
represents the given data: An R2 value of 0.97 suggests that the model can adequately
account for about 97% of the data. (FIGURE 2C)
Hence, an ideal model would minimize all its p-values while maximizing the R2 coefficient. Due
to constraints in space, the present study will omit regression tables and provide only the
equation for the best model.
PROCEDURE
A simple Atwood’s Machine was assembled as shown in FIGURE 1. Various combinations of
masses
and
were attached to the ends of the string. For each trial, the height of the
heavier mass was recorded as the vertical displacement
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. Upon its release, the time
necessary to strike the floor was measured by a stopwatch. All measurements and
accompanying error estimates were organized into a table. Data from other researchers at the
Spenner Lab was incorporated into the present study’s analysis and discussion. Supplies were
generously provided by the Physics Department of The Harker Upper School.
ANALYSIS
Optimization of Accuracy
Once data was imported into Mathematica,
and
were plotted as a function of
(percent error). Some of the resulting graphics are included below:
FIGURE 3: Various Plots Generated by Mathematica
.ListPlot3D
ListPlot:
.ListContourPlot
vs.
ListPlot:
The four graphs in FIGURE 3 all indicate that
and 300 grams and when
reaches its maximum when
vs.
is between 100
is between 600 and 900 grams. To confirm this range, multiple
regression was performed on the sample data. The model with the greatest statistical utility
was found to be the following
:
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[6]
After a rigorous stepwise regression, the ideal model (EQUATION 6) was superimposed onto the
original data:
FIGURE 4: Multiple Regression Model
.Plot3D
Plot3D
The regression plots in FIGURE 4 further corroborate the evidence found in previous graphs:
Accuracy is at an optimal level when
is in the interval
is in the interval
and when
.
Optimization of Precision
Similar to the analysis above, the optimal range of precision is determined through graphical
exploration and regression analysis. Some plots of precision and mass are reproduced below:
FIGURE 5: Various Plots Generated by Mathematica
.ListPlot3D
.ListContourPlot
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ListPlot:
vs.
ListPlot:
FIGURE 5 suggests that precision is greatest when
vs.
is between 0 and 200 grams and when
is between 600 and 1000 grams. Again, multiple regression was performed to test the
validity of this range:
[7]
FIGURE 6: Multiple Regression Model
.Plot3D
Plot3D
EQUATION 7 and FIGURE 6 predict a larger region for optimal precision. This discrepancy may
be caused by the lack of data in those areas. But despite this incongruity, the regression model
still succeeds in confirming previous notions: Precision is greatest when
and when
is in the interval
is in the interval
.
DISCUSSION AND CONCLUSION
Considering the overlap of optimal ranges for accuracy and precision, the present study
concludes that minimal error is achieved when
is in the interval
7
and when
is
in the interval
. In retrospect, these optimal ranges should have been quite
predictable. A low
and a high
both work to minimize acceleration (EQUATION 1).
Since slower accelerations lend to sharper responses from human scientists, minimal error
should be expected in those conditions.
Before embracing these intervals as the answer, however, the domain of the experimental data
should be scrutinized for completeness.
FIGURE 7: Histograms for Experimental Data
. Frequency vs.
Frequency vs.
While the intervals of interest are well supported by trials, there still exist many gaps in
experimental data. As FIGURE 7 clearly shows, the experiment lacks trials with
than 350 grams and trials with
greater
less than 200 grams. Such scarcity of trials may explain
why the regression model in FIGURE 6 overestimates the optimal region for precision. To obtain
models with a wide functional domain, future studies should collect a more comprehensive set
of data.
Adjusting the mass to meet the prescribed ranges is certainly one way of maximizing accuracy
and precision. A more proactive method of optimization may be to alter experimental protocols
so that measurements are taken with the least amount of error. The use of a force plate and a
computer, for example, would have eliminated some inaccuracies in timing.
Rectifying tenuous assumptions would also help in minimizing experimental error. Previously,
the pulley’s mass was assumed to be negligible. Working off this assumption, tension in both
strings were thought to be equal. A more critical look at rotational motion, however, shows that
such the pulley’s mass
significantly affects the acceleration under certain conditions.
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FIGURE 8: Close-Up Schematic of Atwood’s Machine
If
is no longer considered negligible, the pulley gains rotational inertia, thus necessitating a
net torque. And since net torque is no longer zero,
and
cannot be equal (FIGURE 8).
Symbolically:
With substitution and some algebraic manipulation, the following is obtained:
Therefore, the theoretical acceleration of the two masses should be expressed as follows:
[8]
Although the pulley’s mass would be negligible at high values of
, low values of
would
significantly alter the acceleration (EQUATION 8). But putting too much mass on the strings
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would also have its own share of problems: As
reaches higher values, the normal force on
the pulley (by the “massless” strings) would gradually increase – ultimately converting frictional
force into a prominent source of torque. If future studies can account for factors like these,
experimental accuracy can be greatly improved.
Having successfully characterized the Atwood’s Machine, the present study now proposes a
generalized optimization process for other, more complicated instruments:
1. Explicitly acknowledge assumptions and derive necessary equations accordingly.
2. Collect experimental data under various conditions. Replicate each trial multiple times.
3. Calculate the accuracy and precision of each trial. With the use of contour plots and
scatter plots, identify approximate regions where error is minimal. If enough
representative samples have been collected, perform multiple regression to confirm
these regions.
4. Correct protocol and assumptions to decrease experimental error as much as possible.
5. Repeat the optimization process with these new adjustments.
Though extremely painstaking and time-consuming, the steps described above should lead to
dependable, robust systems with optimal accuracy and precision. Since scientific findings gain
legitimacy only when experimental error is small or negligible, sufficient time should be invested
into the optimization process.
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