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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 is calculated using only known or given values. If the pulley’s

mass 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:

Therefore, the theoretical acceleration of the two masses can be expressed as follows:

[1]

FIGURE 1: Schematic of Atwood’s Machine

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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 will

serve as the study’s measure of accuracy. After and 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 must be

calculated in terms of and – 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

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 . Upon its release, the time

(A)

(C)

(B)

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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

.ListContourPlot

ListPlot: vs.

ListPlot: vs.

The four graphs in FIGURE 3 all indicate that reaches its maximum when is between 100

and 300 grams and when 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 and when

is in the interval .

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: vs.

FIGURE 5 suggests that precision is greatest when 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 is in the interval

and when 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 and when is

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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 greater

than 350 grams and trials with 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.