PSE Lab 3: Proving Newton’s Second Law with
Dynamic Carts and Hanging Weights
Archie Wheeler
January 20, 2013
Abstract
I set out to demonstrate Newton’s Second Law by using Pasco Dynamic Carts and Hanging Masses. The data presented in this lab were
not sufficient in and of themselves to demonstrate Newton’s Law to be
true. The lab was done in two phases, the first of which being more successful than the second, however both parts demonstrated a least-squares
regression fit of 5% error.
Contents
1 Objectives
2
2 Preliminaries
2
3 Methods
2
4 Experimental Setup
2
5 Data
5.1 Part 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Part 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
7
8
6 Analysis
6.1 Part 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
12
7 Conclusion
14
8 Manufacturer Websites
16
1
1
Objectives
I seek to demonstrate the relationship between force and acceleration per Newton’s Second Law through several different approaches. Hopefully the results
of this lab will demonstrate to the reader that even though a law may be described by one equation, there are more than one direct applications of that
equation. I also seek to demonstrate that I can perform Advanced-Lab-quality
error analysis.
2
Preliminaries
dv
dp
=m
= ma
(1)
dt
dt
Newton’s second law gives the physicist a definition of force, namely, a force
is a change in momentum per time, or an acceleration of a mass. If there is a
force acting on a group of mass, the entire mass will change accelerate (changing
its velocity and momentum) at a rate proportional to the force and inversely
proportional to the mass. Likewise, if we know the mass of an entity and its
acceleration, we can also determine the net force acting on it.
F=
3
Methods
This lab will be divided into two portions.
In the first part of the lab, I will calibrate a force sensor and attach it to
a dynamic collision cart using a force sensor adaptor plate. The force sensor
will measure the tension of a string that runs horizontal to the table, off the
edge, over a smart pulley, and down towards a hanging mass. The weight of
the hanging mass will provide the horizontal force needed to accelerate the cart,
as shown in figure 1. The average force as read by the force sensor, and the
average acceleration as read by the smart pulley will be measured in table 1 for
various masses. The force will be plotted against acceleration, and the slope
should reveal the mass of the cart, per Newton’s second law.
In the second part of the lab, we use the same basic setup, except we implement a free pulley that hangs off the table and holds the hanging mass. The
other end of the string is attached to a force sensor, held by a frame built using
metal rods. The tension caused by the hanging mass is split into two equal
tensions on the two strings, as shown in figure 3. The force and acceleration
are again recorded in a table (table 2) for various masses, and these data are
plotted in a graph.
4
Experimental Setup
1. Track
2. Cart
2
3. Weights
4. Pasco Smart Pulley (Part No. 514-06266)
5. Free pulley
6. Vernier Dual-Range Force Sensor
7. Scale
8. Science workshop interface and Data Studio Software
9. Graphical Analysis software
Figure 1: A free body diagram of the experimental setup for part 1
3
Figure 2: My experimental setup for part 1. See free body diagram figure 1 for
labels.
Ftension = T = mcart · a
(2)
Fweight = mg = mweight · g
(3)
These two equations give us a theoretical values for the forces we expect in
part 1.
4
Figure 3: A free body diagram of the experimental setup for part 2
T1 = T2 = mcart · acart
(4)
Fweight = mweight · g = T1 + T2
(5)
These two equations give us theoretical values for the forces we expect in
part 2.
5
Figure 4: My experimental setup for part 2. See free body diagram figure 3 for
labels.
6
5
5.1
Data
Part 1
Data collected 20:00, 17 January 2013
mcart+f orcesensor = 0.640kg ± 0.006kg
Mass
Mean
0.0100
0.0200
0.0300
0.0400
0.0500
0.0600
0.0700
(kg)
σ
0.0001
0.0004
0.0004
0.0005
0.0005
0.0006
0.0007
Force
Mean
0.0890
0.1759
0.2702
0.3573
0.4212
0.4687
0.3607
(N)
σ
0.0070
0.0101
0.0154
0.0077
0.0171
0.0099
0.0015
Acceleration (ms-2 )
Mean
σ
0.0541
0.0082
0.2198
0.0341
0.3528
0.0229
0.4919
0.0145
0.5807
0.0199
0.6710
0.0193
0.5004
0.0126
Table 1: Force and acceleration as read by the force sensor and smart pulley
respecively. The masses were compared with the scale in room 219 for accuracy.
Figure 5: Graph of data run for m = 0.050kg. Note that the highlighted sections
are the data where the cart was moving. The data to the left was when I was
still holding the cart. The spike in the acceleration graph is when the cart hit
the stop table. The rest of the data is the cart at rest. This graph is a good
representative from the other graphs which were produced in this lab.
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5.2
Part 2
Data collected 20:45, 17 January 2013
mcart = 0.504kg ± 0.0001kg
mpulley = 0.0142kg ± 0.0001kg
Mass
Mean
0.0242
0.0342
0.0442
0.0542
0.0642
0.0742
0.0842
(kg)
σ
0.0001
0.0004
0.0004
0.0005
0.0005
0.0006
0.0007
Force
Mean
0.2866
0.3759
0.3835
0.4670
0.5004
0.5146
0.5407
(N)
σ
0.0028
0.0092
0.0022
0.0309
0.0152
0.0401
0.0191
Acceleration (ms-2 )
Mean
σ
0.1611
0.0342
0.2936
0.0254
0.3436
0.0450
0.4728
0.0636
0.5404
0.0367
0.5799
0.0523
0.6426
0.0424
Table 2: Force and acceleration as read by the force sensor and smart pulley
respectively. The masses were compared with the scale in room 219 for accuracy.
Figure 6: Graph of data run for m = 0.0342kg. Note that the highlighted
sections are the data where the cart was moving. The data to the left was when
I was still holding the cart. The spike in the acceleration graph is when the cart
hit the stop table. The rest of the data is the cart at rest. This graph is a good
representative from the other graphs which were produced in this lab.
8
6
Analysis
Analysis for both parts began at 21:30 17 January 2013
Solving Newton’s Second equation for mass yields the following relation:
F
(6)
a
We can perform an analysis for each individual row on our data via the
propagation of error technique:
m=
2
df =
X ∂f 2
∂xi
i
2
(∆xi )
(7)
Or, for the relation in equation 6:
2
2
−F
1
2
2
(∆F ) +
(∆a)
dm =
a
a3
2
(8)
Which simplifies to
r
dm =
∆F 2
F 2 ∆a2
+
2
a
a6
9
(9)
Figure 7: A plot of the force verses the acceleration. By Newton’s law, the
slope should be the mass of the cart. The slope of this line is 0.6299kg and the
y-intercept is 0.047766N
The measured value of our cart was 0.640 ± 0.006kg. The slope of our
least-squares regression fit was 1.7% below our accepted value, but all of our
calculated masses were well above the accepted value, (even their error bars did
not enclose the accepted value in the more accurate cases).
This is likely due to a flaw in our system setup. Light weights were used
that did not produce much force. This had the advantage of a relatively slow
acceleration, lending itself well to pretty graphs. However, the force sensor
was attached to the PASCO interface by a wire which dragged on the table as
the cart was pulled along. This added friction is likely the cause behind the
y-intercept of 0.047N and the reason why all of the estimated data points are
above the actual. If you tied a really long rope to the back of a dog-sled, the
sled dogs would think that whatever they were pulling was really heavy based
on their harnesses. The same principle applies here.
10
Mass
Mean
0.0100
0.0200
0.0300
0.0400
0.0500
0.0600
0.0700
(kg)
σ
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
Force
Mean
0.0890
0.1759
0.2702
0.3573
0.4212
0.4687
0.3607
(N)
σ
0.0170
0.0201
0.0254
0.0177
0.0271
0.0199
0.0115
Acceleration (ms-2 )
Mean
σ
0.0541
0.0082
0.2198
0.0341
0.3528
0.0229
0.4919
0.0145
0.5807
0.0199
0.6710
0.0193
0.5004
0.0126
Mass of Cart (kg)
Mean
σ
1.6448
4.6396
0.8003
0.5730
0.7658
0.1582
0.7263
0.0564
0.7252
0.0634
0.6986
0.0421
0.7208
0.0429
Table 3: Calculated mass of cart, with propagation of error per equation 9.
Note in this table, error for the Pasco Force sensor was added in pessimistically
to the data presented in table 1 (±0.01N) The accuracy for a smart pulley’s
timer was limited by the ADC, at about 0.1ms, well beneath the precision of
these tables when the acceleration data points were averaged.
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6.1
Part 2
Figure 8: A plot of the force verses the acceleration. By Newton’s law, the
slope should be the mass of the cart. The slope of this line is .5297kg and the
intercept is 0.2088N
This graph striked me as odd, and I wanted to figure out why. I created a
plot of the force verses the mass, and discovered that the relationship was not
directly proportional (nor linear, if one were to do all of the work to sub in for
a in equation 10).
12
If we take a moment to look at Newton’s Second Law, we see why:
F = T1 = ma
where a is the acceleration measured by the smart pulley. However, the mass
is not just the mass of the cart in this system.
a
a
(10)
T1 = mcart a + mfree pulley + mhanging mass
2
2
Solving this for the mass of the cart gives us a different relationship than we
saw in equation 6.
mfree pulley
mhanging mass
T1
−
+
mcart =
a
2
2
(11)
and
r
∆F 2
F 2 ∆a2
∆m2
+
+
(12)
a2
a6
4
Unfortunately, this relation as well did not lead to satisfying results, as
the following table shows the calculated values for m based on the previous
equations.
dm =
13
Mass
Mean
0.0242
0.0342
0.0442
0.0542
0.0642
0.0742
0.0842
(kg)
σ
0.0001
0.0004
0.0004
0.0005
0.0005
0.0006
0.0007
Force
Mean
0.2866
0.3759
0.3835
0.4670
0.5004
0.5146
0.5407
(N)
σ
0.0128
0.0192
0.0122
0.0409
0.0252
0.0501
0.0291
Acceleration (ms-2 )
Mean
σ
0.1611
0.0342
0.2936
0.0254
0.3436
0.0450
0.4728
0.0636
0.5404
0.0367
0.5799
0.0523
0.6426
0.0424
Mass of Cart
Mean
σ
1.7596 2.3462
1.2561 0.3823
1.0866 0.4270
0.9536 0.2941
0.8867 0.1254
0.8431 0.1628
0.7923 0.0975
Table 4: Even with the corrected equation, the mean still lies a great deal off
from the accepted value. Note in this table, error for the Pasco Force sensor was
added in pessimistically to the data presented in table 1 (±0.01N) The accuracy
for a smart pulley’s timer was limited by the ADC, at about 0.1ms, well beneath
the precision of these tables when the acceleration data points were averaged.
Even though I could not find an equation that could accurately explain how
to find the mass, a linear fit to the graph over the domain of m = (0.0242kg, 0.842kg)
did seem to explain the data well, and the slope of the line was remarkably close
(5.1% high) to the measured value of .504kg, considering the large y-intercept.
The cause of this discrepancy is a bit more puzzling in this case. In part one,
there was a source of friction that was more or less constant across each trial run
which lended itself well to a linear fit of the form ax + b. However, in this case,
the relationship is a bit more complex. I could not locate the source of error
either experimentally or through equations that could explain the discrepancy
in the data.
7
Conclusion
Unfortunately, the objectives of this lab were not completely met. Although
there were some parts of the analysis which did successfully come close to finding
the mass of the cart, the set of estimations overall left much to be desired.
The following list is made up of questions asked on the AU Physics Wiki for
this lab that prompt the student to evelauate the lab.
• Discuss the percent error (relative error) that you calculated for both
graphs.
In both graphs, the percent error was positive, and decreased for individual
rows in the table as mass increased. This implies that as the forces grow,
the effect of the y-intercept (the friction in the system) becomes less and
less important. This intuitively makes since. If you want to open a door,
you will pull hard on it to open it so that the friction of the door is
insignificant.
• Interpret the value of the intercept.
14
The intercept is the combination of all the factors in the system which
drag back on the cart. These could be air resistance, friction in the tire
of the cart or in the pulley, but most likely the friction of the cord being
dragged along. In part 2, I was not able to determine where the large
intercept came from.
• Examine how the presence of constant frictional force would affect the
results of the experiment.
A constant frictional force will change what would be a proportional relationship (y = ax) into a linear relationship (y = ax + b) where the friction
can be expressed in terms of some constant b.
• Speculate on the origin(s) of error.
As stated in analyses sections and previously in this conclusion, sources of
error include dragging of electric cords, but also the inertia of the hanging
mass. Some of the force is “used up” in being applied to other components
in the system. The larger the mass, the closer the systems’ acceleration
approaches to free fall. So in reality, neither of these equations are linear
as mhanging weight mcart
• Discuss the difference between the measured and calculated tensions.
The measured tension was always less than what the calculated tension
was. This is primarily because some of the force was being applied on the
hanging mass, which would relieve a small amount of tension off of the
string.
• Mention what you learned in this experiment.
LATEX, Matlab, and refresher on other technologies that make life easier.
But as far as practical usage goes, I learned how to identify what things
need to be set up in a system, when measurements should be taken in the
flow of performing a lab, and how to research manufacturer information
in order to get error results.
• State which experimental method you prefer, and give justification for
your preference.
Both methods are flawed, primarily because the hanging masses have inertia as well. If I could work out the reason why part 2 was off, then I
feel that the second part would be better, because there is no dragging of
a cord along with an instrument. Furthermore, the rate of acceleration is
slower, so there is a longer period in each trial, and more data points can
be collected.
• Include any additional comments that you think are important.
White boards as tables are the greatest thing ever, because the entire surface is scratch paper for annotating, labeling, and working out equations!
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8
Manufacturer Websites
• www2.vernier.com/booklets/dfs.pdf
• http://www.pasco.com/prodCatalog/ME/
ME-6838 photogate-pulley-system/index.cfm#specificationsTab
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