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Finding and Modeling the Effect of a Force Applied to a System on the System’s Acceleration
Experiments performed on September 24th and October 1st, 2018 by Aaron Eiben, John Blasing, and Henry Leach
1 Experimental Design Template Research Question:
How does the acceleration of a system change when the applied force changes? [1]
Dependent Variable Acceleration of the system Independent Variable Applied force Control Variable System mass (187.69 g) Hypothesis A system’s acceleration is affected by force applied to it.
Prediction
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2 Data Table and Graph Table 2-1: Data for testing the effect of applied force.
Hanging Mass (Âą0.05g) * 5.0 10.0 14.7 19.6 24.4 29.1 34.0 38.7 Average:
Applied Force (Âą0.49mN) â€
Acceleration (m/s2)
Accel. Uncertainty (m/s2) ‥
49 98 144 192 239 285 333 379
0.257 0.510 0.737 0.992 1.230 1.470 1.720 1.970 1.111
0.0011 0.0050 0.0016 0.0017 0.0050 0.0050 0.0051 0.0057
All error bars set to 2đ?œŽđ?œŽ
Figure 2-1: Graph of applied force vs. acceleration data. §
Applied Force vs. Acceleration
Acceleration (m/s2)
2.5 2.0 1.5
Average
1.0 0.5 0.0
0
50
100
150
200
250
300
Applied Force (mN)
Mass uncertainty set to scale uncertainty of measuring device. Force uncertainty derived from mass uncertainty. ‥ Acceleration uncertainty set to larger of scale uncertainty or fit uncertainty. § Applied force and acceleration error bars are small but visible at these scales. See section 3.5 Uncertainty and Error Analysis for details. * â€
y = 0.0052x - 0.0015 R² = 0.9999
350
400
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3 Discussion and Conclusion An experiment is performed to address the research question “how does the acceleration of a system change when the applied force changes?� [1] The experimental apparatus is a sled on a level air track attached to a string running over a pulley with a mass hanging from the free end. The hanging and sliding masses as well as the string connecting them and the pulley over which the string runs defines the experimental system. The analysis assumes that the string and Figure 3-2: A diagram of the the pulley are massless so that the total mass of the system is the experimental apparatus taken from [1]. sum of the hanging and sliding masses. The experiment also assumes that any friction between the sliding mass and the air track and within the pulley axle is negligible so that the primary force acting on the system is the force of gravity exerted upon the hanging mass. Finally, one also assumes that the string does not slide over the pulley. Instead, the string and pulley maintain good contact such that any motion of the string transfers to the pully. An encoder attached to the pulley records its angular position and feeds this data to a computer which computes translational velocity values for the system and plots a velocity vs. time graph. The computer then uses a least-squares Figure 3-1: A screenshot of a velocity vs. time graph. The method to fit a linear function to the velocity vs. acceleration of the system is defined as the slope of a linear fit to time data, the slope of which represents the the highlighted portion of this data set. acceleration of the system. One adjusts the force applied to the system by transferring mass between the hanging and sliding components.
3.1 Experimental Results Figure 2-1 shows no error bar overlap across each data point since the data uncertainties are far smaller than the ranges of values explored in both force and acceleration. Consequently, it is improbable that these data represent equivalent acceleration values for different applied forces and far more likely that different system accelerations correspond to different hanging weights. This observation is consistent with the hypothesis that the force applied to a system affects the system’s acceleration.
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3.2 Comparison with Other Groups 3.2.1 Group 5 in PHYS 1051L-015 18FS Figure 3-4 shows that Group 5’s data are also consistent with their hypothesis that a correlation exists between the force applied to a system and the system’s acceleration. As is the case in Figure 2-1, small, non-overlapping error bars suggest that each acceleration value corresponding to different applied forces is distinct. Thus, one cannot rule out the possibility that a correlation between the force acting upon and the acceleration Figure 3-3: Presentation of Group 5’s data. Image credit: Claire Allen [6]. of Group 5’s system does indeed exist.
Table 3-1: Group 5’s data for testing the effect of applied force.
Trial 1 2 3 4 5 6 7 8 Average:
* †
Applied Force (±0.98mN) *
Acceleration (m/s2)
Accel. Uncertainty (m/s2) †
46.0 94.1 151.0 198.9 245.0 300.9 348.9 396.9
0.204 0.435 0.677 0.838 1.110 1.340 1.620 1.820 1.006
0.0011 0.0015 0.0026 0.0035 0.0050 0.0098 0.0077 0.0190
Force uncertainty derived from mass uncertainty of 0.1g Acceleration uncertainty set to larger of scale uncertainty or fit uncertainty.
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All error bars set to 2đ?œŽđ?œŽ Figure 3-4: Graph of Group 5’s applied force vs. acceleration data. *
Acceleration (m/s2)
Applied Force vs. Acceleration 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0
y = 0.0046x - 0.0216 R² = 0.9974
Average
0
50
100
150
200
250
300
350
400
450
Applied Force (mN)
3.2.2 Group 2 in PHYS 1051L-015 18FS Figure 3-6 shows that Group 2’s data is also consistent with their hypothesis that the acceleration of a system is directly affected by force applied to it. Again, non-overlapping error bars suggest that each acceleration value corresponding to different applied forces is distinct so that one cannot rule out the possibility that a correlation between the force acting upon and the acceleration of Figure 3-5: Presentation of Group 5’s data. Group 2’s system also exists. Image credit: Claire Allen [6].
*
Applied force and acceleration error bars are small but visible at these scales.
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Table 3-2: Group 2’s data for testing the effect of applied force.
Hanging Mass (Âą0.05g) * 4.4 8.9 13.4 17.8 22.3 Average:
Applied Force (Âą0.49mN) â€
Acceleration (m/s2)
Accel. Uncertainty (m/s2) ‥
43.12 87.22 131.32 174.44 218.54
0.247 0.505 0.755 0.897 1.090 0.699
0.0010 0.0025 0.0047 0.0010 0.0011
All error bars set to 2đ?œŽđ?œŽ
Figure 3-6: Graph of Group 2’s applied force vs. acceleration data. §
Applied Force vs. Acceleration 1.2
y = 0.0047x + 0.0775 R² = 0.9885
Acceleration (m/s2)
1 0.8 0.6
Average
0.4 0.2 0
0
50
100
150
Applied Force (mN)
Mass uncertainty set to scale uncertainty of measuring device. Force uncertainty derived from mass uncertainty. ‥ Acceleration uncertainty set to larger of scale uncertainty or fit uncertainty. § Applied force and acceleration error bars are small but visible at these scales. * â€
200
250
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3.3 Comparison with Theory 3.3.1 Basic theory Newton’s second law of motion states that the acceleration of a system đ?‘Žđ?‘Žâƒ—đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ is proportional to the net force acting upon that system đ??šđ??šâƒ—đ?‘›đ?‘›đ?‘›đ?‘›đ?‘›đ?‘› and inversely proportional to the system’s inertial mass đ?‘šđ?‘šđ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ : đ?‘Žđ?‘Žâƒ—đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ =
đ??šđ??šâƒ—net đ?‘šđ?‘šđ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘
Equation 3-1 [2].
In other words, when plotting the force acting on a system versus its acceleration, theory predicts a linear relationship, the slope of which is the reciprocal of the system’s mass. For a system mass of 187.69 g, the corresponding slope is therefore 0.0053 đ?‘”đ?‘”−1. Figure 2-1 shows that a linear function does indeed explain nearly all the variation seen within the experimental data, having an đ?‘…đ?‘… 2 value of 0.9999. The slope of the fitted line is 0.0052 đ?‘”đ?‘”−1, having a relative error with respect to the theoretical value of 2.40%. A variation on this analysis finds an experimentally determined system mass of 192.31 g, which differs from the measured mass by 2.46%. Thus, one finds little contradiction between the experimental data presented here and Newton’s second law of motion, lending support to Equation 3-1’s status as a true law of nature.
3.3.2 Enhanced theory Astute readers will point out Equation 3-1 does not appear to have a y-intercept value, unlike the experimental model presented in Figure 2-1. What is more, the model seems to suggest that when no force acts upon the system, a backward acceleration of 0.0015 m/s 2 will still be observed, running counter to Newton’s first law of motion, which states that an object’s state of motion does not change unless external forces act upon it! [3] However, one may rectify this seeming contradiction by noting that the force in Equation 3-1 is the net force acting upon the system. While the experimental apparatus is designed to minimize all external forces except those imposed by the experimenter, these various external factors are not eliminated entierly. A more precise theoretical model, therefore, takes these forces into account. By substituting đ??šđ??šâƒ—net → đ??šđ??šâƒ— + đ??šđ??šâƒ—ext into Equation 3-1, one derives the enhanced theoretical model: đ?‘Žđ?‘Žâƒ—đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ =
đ??šđ??šâƒ— đ??šđ??šâƒ—ext + đ?‘šđ?‘šđ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘šđ?‘šđ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘
Equation 3-2.
Within Equation 3-2, đ??šđ??šâƒ— is the force imposed by the experimenter and đ??šđ??šâƒ—ext represents all the unknown external forces, which might include any friction not eliminated by the air track, any gravitational force components caused by the track not being perfectly level, any aerodynamic forces, etc. Upon making further substitutions đ?‘Žđ?‘Žâƒ—đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ → đ?‘Śđ?‘Ś m/s 2 , đ??šđ??šâƒ— → đ?‘Ľđ?‘Ľ mN, đ??šđ??šâƒ—ext → đ?‘?đ?‘? đ?‘šđ?‘šđ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ m/s2 , đ?‘šđ?‘šđ?‘ đ?‘ đ?‘ đ?‘ đ?‘ đ?‘ → 187.69 g, Equation 3-2 becomes
8|Eiben, et al. đ?‘Śđ?‘Ś = 0.0053 đ?‘Ľđ?‘Ľ + đ?‘?đ?‘?
Equation 3-3.
đ?‘Śđ?‘Ś = 0.0052 đ?‘Ľđ?‘Ľ − 0.0015
Equation 3-4.
Equation 3-3 is, in turn, directly comparable to the experimental model in Figure 2-1:
It is still the case that the slopes of the two models differ by 2.40%. However, now, a comparison of the y-intercepts of Equations 3-4, 3-3, and by extension 3-2 permits the quantification of the net unknown external force. This comparison suggests that the y-intercept đ?‘?đ?‘? in Equation 3-3 has a value close to −0.0015. Multiplying this by the system mass suggests a magnitude of 0.28 mN, far smaller than any force applied as an experimental variable. The negative value of đ?‘?đ?‘? indicates that the direction of this unknown net force is opposite that of the applied force. These results remain consistent with Newton’s laws of motion.
3.4 The Case of a Heavy String An interesting scenario to consider is one in which the string connecting the hanging and sliding masses is sufficiently massive so that its mass contributes significantly to the system mass. The dynamic situation introduces the complication that as more of the string runs over the pully, its mass contributes to the hanging weight, which is the force applied to the system and thus the primary cause of the system’s acceleration. When the string is assumed to be massless, the hanging weight and resulting acceleration are both constant throughout time. The system’s velocity versus time graph is, therefore, a linear function as shown in Figure 3-1. However, if the hanging weight increases over time, then so too will the system’s acceleration. In that case, the velocity versus time graph will not be a line with a constant slope, but instead a function with a continually increasing slope, such as an exponential or hyperbolic function or a polynomial of any order higher than one. Consequently, a fit of the velocity versus time data will not produce one specific value for the acceleration of the system, but instead a function of the system’s acceleration over time. If a similar function for the hanging weight over time is obtainable, a comparison of these two expressions should remain consistent with Newton’s second law of motion.
3.5 Uncertainty and Error Analysis 3.5.1 Mass measurement All mass measurements are made using a triple beam balance with a scale uncertainty of Âą0.05g. One maintains measurement uncertainties at the scale level by measuring the individual components of a total mass together as a whole, rather than measuring each component separately and summing these values to find the total mass. A past report describes details of this method [4].
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3.5.2 Force calculation Applied force values derive from hanging masses by multiplying each mass by �� = 9.8 m/s 2 , which is assumed to be an exact value. The applied force uncertainty is therefore calculated from the mass uncertainty following the constant uncertainty propagation rule in [5] and found to be ¹0.49mN.
3.5.3 Acceleration determination As the introduction to Section 3 describes, system accelerations derive from least-squares fits of system velocity versus time graphs. The fitting software reports the slope of each graph along with an uncertainty value, which one compares to the scale uncertainty of the slope value. The acceleration uncertainty is the larger of the fit uncertainty and scale uncertainty for each slope.
3.6 Constraints and Generalizability Constraints in this experiment stem from assumptions made in the analysis: the string and pully are effectively massless, the air track is perfectly level, and friction and any other resistance forces are negligible. The simplest analysis, which assumes that the force acting upon the system is equal to the weight of the hanging mass, is only valid when the stated conditions are met. Otherwise, contributions of the additional forces and inertias will have to be considered as part of the net force and inertia of the system. However, a comparison of this net force and the system’s acceleration will still comprise a valid test of Newton’s second law as Equation 3-1 describes it. Carrying out these experiments will be good tests of the generalizability of Newton’s laws of motion to more complex systems.
4 References [1] University of Cincinnati PHYS 1051L, "Lab 05: Newton's Laws Part I," [Online]. [Accessed 24 September 2018]. [2] R. D. Knight, B. Jones and F. Stuart, "Newton's Second Law," in College Physics: A Strategic Approach, 3rd ed., Boston, Pearson Education, Inc., 2015, pp. 109-111. [3] R. D. Knight, B. Jones and F. Stuart, "Motion and Forces," in College Physics: A Strategic Approach, 3rd ed., Boston, Pearson Education, Inc., 2015, pp. 99-101.
10 | E i b e n , e t a l . [4] A. Eiben, J. Blasing and H. Leach, "Finding and Modeling the Effects of Mass, Length, and Release Angle on the Period of a Pendulum for Small Swings," University of Cincinnati PHYS 1051L, Cincinnati, 2018. [5] University of Cincinnati PHYS 1051L, "Pre-Lab Lab 04," [Online]. [Accessed 4 September 2018]. [6] C. Allen, V. Coy and S. Smith, "Newton’s Laws: Applied Force (N) vs. Acceleration (m/s^2)," University of Cincinnati PHYS 1051L-015, Cincinnati, 2018.