Newton’s Second Law of Motion
Ryan Eagan*, Sean Morton† and Collin Murphy ‡‡
Penn State Worthington Scranton, 120 Ridge View Drive, Dunmore, PA USA
February 19, 2010
Abstract
We analyzed the motion of two objects in a frictionless environment that are
connected by a string. By recording changes in the motion as the tension of the
string acts on each object we use our results to visualize the relationship between
force, mass and acceleration as stated by Newton’s Second Law of Motion.
*
ゆ
ょ
Electronic address: ree5047@psu.edu
Electronic address: som5209@psu.edu
Electronic address: cjm5515@psu.edu
1
Introduction
In this experiment we explore the application of Newton’s second law of motion and in
particular the relationship between force, mass and acceleration. We used a configuration in the
lab similar to an Atwood machine for the acquisition of our results which allowed us to observe
changes in motion that occurred when applying various forces.
Changes in the motion of an object are the result of changes in the forces acting on the
object. Newton’s first law of motion in essence tells us that an object which has no force acting
on it will also have zero acceleration [1]. To change the motion of an object, a force needs to be
applied to the object and naturally when the motion of an object changes there is an associated
acceleration.
Newton’s second law of motion tells us that this acceleration resulting from a change in
motion is directly proportional to the net force acting on the object and it is inversely proportional
to the mass of the object [2]. Mathematically this gives us,
�� � =
�
and is stated as the sum of the net forces �� � are equal to the product of mass
�.
(1)
and acceleration
In 1784 George Atwood published his Treatise on the Rectilinear Motion and Rotation of
Bodies in which he describes a device he conceived and constructed used to demonstrate and
study acceleration under a constant force [3]. He claimed his Atwood Machine had better
accuracy than the inclined plane method being used at the time and was more effective in
conveying concepts to his pupils at Trinity College in Cambridge.
The Atwood Machine is simply two masses connected by a string and hung over a pulley.
One of the benefits of this setup is the ability to model the object as a simple particle and allows
us to ignore the effects of friction and the masses of the strings involved.
Our setup in the lab consisted of a glider on a level flat surface connected to a hanging
mass by a string, which was thus placed over a Smart Pulley. Figure 1 shows a crude sketch of
the configuration along with all associated forces. The glider rides along an air-track allowing us
to simulate a frictionless environment in the lab. The Smart Pulley configuration uses a
photoelectric sensor to record a displacement over a time interval. This information was
collected by Data Studio1 which calculated velocity throughout the interval and plotted it over
time. Taking the slope of that graph provided us with the acceleration of our system. The tension
in the string between the glider and the hanging mass was the force responsible for the change in
motion of the glider and gravitational pull of the earth was the force responsible for the change in
motion of the hanging weight.
1
Data Studio – www.pasco.com
3
FIGURE 1
Newton’s third law tells us that the tension of the string at each object is equal [4]. From
this we get the following mathematical interpretation,
�=
1�
for the value of � at the glider, and for the value of � at the hanging mass we have,
2
−� =
(2)
2�
(3)
which represents the sum of the net forces, � – 2 equal to the product of mass and
acceleration, 2 �. The 2 represents the force of gravity acting on the hanging mass. Algebra
allows us to rewrite Eq. 3 in terms of �,
�=
2� +
�=
2+
2
(4)
and then set Eq. 2 equal to Eq. 4 to solve for acceleration,
2
1
which finally allowed us to then determine the value of � or the net force action on the objects,
�=
1
1+
(5)
2
2
(6)
The glider 1 and hanging mass 2 were augmented varied during the experiment. We
completed two trials, each trial consisted of 4 runs of the glider with a particular configuration of
masses. The data was collected and graphed2 by Data Studio and the slopes of each graph
( = ��� ���� ) for each run in a trial was recorded for comparison to the theoretical
acceleration obtained from the equations above.
2
Data Studio graphs are provided in the appendix.
4
Data Analysis
Our first trial as stated earlier was composed of 4 runs of the glider. The mass for 1 ,
mass of the glider, was 0.2904 kg ± 0.0001. The mass for 2 , mass of the hanger, was 0.0218 kg
± 0.0001. The values were determined using an electronic balance.
From Eq. 5 we determined our theoretical acceleration for the system to be 0.684 m/s2 ±
0.001. Our uncertainty for the theoretical acceleration was based on the measurement of the scale
having an inherent uncertainty of 0.0001 kg along with the value of (9.8 m/s2) which only has
two significant figures and is an approximation because the force of gravity varies from location
to location.
Table 1 provides the tabulated results for our first trial, Trial 1 and Figure 2 shows the
velocity over time plot of all four runs in the trial. Table 1 also lists the difference and percent
difference between the experimental acceleration and theoretical acceleration for each of the runs.
Our experimental acceleration for all four runs showed a range of 0.012. The mean � of the
experimental acceleration values was 0.649 with a standard deviation � at 0.0056 and standard
error � equal to 0.003. These values indicated an acceptable level of precision within our data
accounting for random errors.
TABLE I
Trial 1
Experimental
Acceleration
Theoretical
Acceleration
(kg)
± 0.0001
(m/s2)
(m/s2)
± 0.001
± 0.001
0.3122
0.3122
0.3122
0.3122
0.641
0.650
0.652
0.653
0.684
0.684
0.684
0.684
Total Mass
*
Run #
2
3
4
5
Difference
0.043
0.034
0.032
0.031
Percent
Difference
6.29
4.97
4.68
4.53
*
Run numbers correspond with the labeled Data Studio graphs in the appendix.
Our percent difference triggered some cause for alarm. Values ranging from 4.53 to 6.29
represent a large variance between our experimental values and theoretical values indicating a
possible systematic error within our experiment.
5
FIGURE 2
We forged ahead with our second trial, changing the value of the masses. The mass for
1 was 0.2904 kg ± 0.0001, and 2 was 0.0218 kg ± 0.0001. The value of theoretical
acceleration for trial 2 was determined to be 0.918 m/s2 ± 0.001. Table II shows the results for
our second trial.
TABLE II
Trial 2
Experimental
Acceleration
Theoretical
Acceleration
(kg)
± 0.0001
(m/s2)
(m/s2)
± 0.001
± 0.001
0.3424
0.3424
0.3424
0.3424
0.877
0.877
0.877
0.871
0.918
0.918
0.918
0.918
Total Mass
*
Run #
6
7
8
9
Difference
0.041
0.041
0.041
0.047
Percent
Difference
4.47
4.47
4.47
5.12
*
Run numbers correspond with the labeled Data Studio graphs in the appendix.
Again we found a consistency within our experimental values. Our experimental
acceleration over all four runs showed a range of 0.006. The mean � of the experimental
acceleration values was 0.876 with � at 0.003 and � equal to 0.002. Trial 2 proved to show an
improvement in precision of the experiment in comparison to the statistical information for trial
1, which was also evident by the plot of velocity over time of our second trial, Figure 3.
6
FIGURE 3
The difference and percent difference between the experimental and theoretical
acceleration values for Trial 2 also showed a large deviation between them. Seeing this
consistent pattern in our data, the conclusion of the group was that the cause of this variance was
definitely due to systematic error in our experiment and its execution.
In reviewing our procedure we realized that during the entire execution of our experiment
we failed adjust the pressure or flow of the air track, leading us to theorize that our system was
not frictionless.
To determine if our theory was correct we assumed there was a force of kinetic friction
between 1 and the track. Given this assumption we replaced Eq. 2 with the following equation:
�−
1�
=
(7)
which states the sum of the net forces is the tension � minus the force of friction [5]. This can
now be rearranged and substituted into Eq. 4 and Eq. 5 as done previously. The friction force
is given by:
= � �
and therefore substituting Eq. 8 into Eq. 7 we solved for the coefficient of kinetic friction � as
such:
� =
2
−
1+
1
7
2 �
(8)
(9)
Performing this calculation for each run provided us with a value for both trials. We took the
four values derived for � from Trial 1 and then took the average to be the best determinant of the
coefficient of kinetic friction. We then repeated this for Trial 2 to determine the � value
respectively.
Having determined these values using our experimental acceleration values, we
calculated the force of kinetic friction for both trials and then solved for theoretical acceleration
using the modified form of Eq. 5 that included :
2−�
�=
2+
1
1
(10)
The derived value of � for Trial 1 was 0.004 and for Trial 2 � was 0.005. Using Eq. 10 and the
respective � values we derived a new theoretical acceleration for both trials. The new
theoretical acceleration was 0.647 m/s2 for Trial 1 and Trial 2 was 0.874 m/s2.
TABLE III
Acceleration of All Runs with Consideration of Kinetic Friction
*
Run #
2
3
4
5
6
7
8
9
µk
0.004
0.004
0.004
0.004
0.005
0.005
0.005
0.005
Experimental
Acceleration
Theoretical
Acceleration
(m/s2) ± 0.001
(m/s2) ± 0.001
0.641
0.650
0.652
0.653
0.877
0.877
0.877
0.871
0.647
0.647
0.647
0.647
0.874
0.874
0.874
0.874
Difference
0.006
0.003
0.005
0.006
0.003
0.003
0.003
0.003
Percent
Difference
0.93
0.46
0.77
0.93
0.34
0.34
0.34
0.34
*
Run numbers correspond with the labeled Data Studio graphs in the appendix.
Table III shows our calculated value for the coefficient of kinetic friction, original
experimental acceleration, new theoretical acceleration calculated with force of friction and the
comparative difference. We determined from this data, based on the consistent percent difference
of less than 1, that our proposed theory in regards to a systematic error is supported.
8
Conclusion
As demonstrated we determined by our original experimental results that there is a
relationship between acceleration and the net force acting on an object. The rather large percent
difference between the experimental and theoretical value though does not support Newton’s
second law with much confidence as we had hoped.
Further analysis of our experimental procedure resulted in the discovery of a possible
systematic error based on the consistent precision of our experimental values. Hypothesizing our
systematic error to be the failure to adequately adjust the pressure of the air track to eliminate
friction we decided to test this theory. Using our experimental and theoretical values for
acceleration from our initial experiment we calculated an approximate value for the force of
kinetic friction in each trial, and then used that value to analyze our data as a system affected by
friction. These results proved to be consistent with Newton’s second law as demonstrated by the
drastic reduction in the percent difference to less than 1 percent.
9
References
[1] Raymond A. Serway and John W. Jewett, Jr., Principles of Physics (Brooks/Cole, New York,
2006), p. 99
[2] Raymond A. Serway and John W. Jewett, Jr., Principles of Physics (Brooks/Cole, New York,
2006), p. 101
[3] A. B. Arons, Development of Concepts of Physics : From the Rationalization of Mechanics to
the First Theory of Atomic Structure (Addison-Wesley, Reading, Mass., 1965), p. 147
[4] C. Kittel, W. D. Knight, M. A. Ruderman, A. C. Helmholz, and B. J. Moyer, Mechanics, 2d
ed. (McGraw-Hill, New York, 1973), p. 85
[5] Raymond A. Serway and John W. Jewett, Jr., Principles of Physics (Brooks/Cole, New York,
2006), p. 127
10
APPENDIX: SURVEY OF RESULTS AND RAW LAB DATA
Run #1 Graph – Velocity vs. Time
Run #2 Graph – Velocity vs. Time
Run #3 Graph – Velocity vs. Time
Run #4 Graph – Velocity vs. Time
Run #5 Graph – Velocity vs. Time
Run #6 Graph – Velocity vs. Time
Run #7 Graph – Velocity vs. Time
Run #8 Graph – Velocity vs. Time
Run #9 Graph – Velocity vs. Time
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