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Physics 101/111---Lab 7: Atwood’s Machine
1990-2007, James J. DeHaven Ph.D.
Atwood’s machine is a physical device used to measure the acceleration due to gravity, and to study the
dependence of acceleration on
€ net force as well as on the total mass of a system. In this lab you will construct an
Atwood’s machine using the “smart pulley” apparatus and study the acceleration of a system using the computer
to analyze your results.
Introduction and Theory
The Atwood’s machine consists simply of two masses draped over a pulley. Sometimes, in order that the
masses not collide with each other a second pulley is employed, and this will be our configuration:
Fig. 1 Smart pulley Atwood’s machine schematic
This arrangement consists of two masses m1 and m2 , which can either balance each other or exert a net
unbalanced force and set the system in motion.
Fig. 2 Schematic of a typical Atwood’s experiment
m1
m2
The net force on this system will be given by[1]:
[1]
Fnet = m 2 g
_
m1g
_
f
-2 In equation 1, f represents the aggregate frictional forces. Since Fnet = Ma = (m1 + m2 )a, we can write equation [2]
which incorporates this information:
[2]
( m 1+ m2 ) a = m 2 g
_
m1 g
_
f
and then solve the equation as shown in equation [3]:
[3]
m 2 g _ m1 g
a =
( m1 + m 2)
_
f
In the special case where the velocity of the system is constant, the frictional force can be measured:
[4]
f = m2 g
_
m1 g
In this experiment, you will first evaluate the frictional force using the case of constant velocity. You will
then find the acceleration for various combinations of mass and net force and compare your results with the
theoretically predicted values.
There are two perspectives from which you can view the significance of this experiment. You can assume
that the value for g the gravitational acceleration near the earth’s surface, can be determined independently, and that
the Atwood’s experiment is a test of Newton’s laws of motion, in particular the second law.
On the other hand, if you assume the validity of the second law, then the Atwood’s experiment provides a
method for determining g, the gravitational acceleration near the earth’s surface. Inverting equation 3 yields an
expression for g in terms of the acceleration, a, of the system:
[5]
g =
(m1 + m 2 ) a + f
( m 2 – m1 )
Equipment
Smart pulley
auxiliary pulley
Knurled nut
short post
universal clamp
paper clips
string
2 weight hangers
2 sets weights
gray connectors where needed
extenders
-3 You will use one smart pulley and one simple pulley (without a photogate and electrical hook-up) to
construct the Atwood’s apparatus depicted schematically in figure 1, and then mount this on the edge of the lab
table using the universal clamp.
Experimental
1. Measurement of frictional force
Begin with ascending and descending masses of 50g each in equilibrium
as illustrated in figure 3.
Add small masses (paperclips) to m2 until a small push causes m2 to
descend with constant velocity. Use the computer to see whether m2 is
accelerating, decelerating or moving with a constant speed. Once you have found
the proper mass, you can leave it in place because it exactly balances the force of
friction. Unbalance the machine by adding ten extra grams to m2, and measure the
acceleration. Use EX7Atwoodpulley2007.cmbl to extract values for the
acceleration from the smart pulley apparatus.
50 g
50 g
Fig. 3 Initial Atwood’s
machine configuration
2. Varying the total mass: dependence of acceleration on mass
Repeat this measurement for three additional total masses as shown in figure 4. It is essential that you
repeat the frictional force measurement for each different weight.
100 g
100 g
150 g
150 g
200 g 2 0 0 g
Fig. 4 Additional Atwood’s machine configurations
3. Total mass constant: dependence of acceleration on force
Balance the machine with 160 g on each side (make the last 10 g up with small masses). Find the frictional
force as above and correct for it.
Now transfer 1g from m1 to m2 and measure the acceleration. Then transfer 2 g, another 2g, and, finally, 5g,
for a total of 10g in all.
4. Unsolicited But Helpful Advice
1) Use paper clips as custom masses
2) Make sure masses are known to within 0.5g
3) Put a foam rubber “catcher” under the weight hangers.
-4 Data
For each experiment you will need to record the values for the hanging masses and the accelerations
reported by the computer. You will also need to know the mass of the paper clips needed to offset the effects of
friction.
Calculations
The idea is to compare the experimentally observed acceleration to that predicted by theory. You need to be
careful, in this calculation, about how you take into account the frictional force. If you have left the paper clips in
place on m2 in each experiment, then there is no need to include the frictional force in your calculations. However,
the total mass accelerated is now the sum of m1 , m2, and the paper clips.
Use a spreadsheet (Microsoft Excel) to do your calculations. This lab has a second purpose: to provide you
with an introduction to spreadsheets, if you have not yet begun to use them. Therefore you must do your
calculations for this experiment using Microsoft Excel, and hand in a printout of your spreadsheet with you lab
report.
To assist you in learning Excel, we have constructed a spreadsheet document in advance which is
preprogrammed with most of the calculations you need to do. This is located on your desktop under the file name
Ex7Atwoodsheet2007.xls. The opening screen for this spreadsheet is reproduced in figure 5.
The basic function of Ex7Atwoodsheet2007.xls is to apply equation [3] to the data from the experiment.
Because this will be, for some students, their first experience with spreadsheets, we have filled in some of the
columns in the spreadsheet.
The spreadsheet is divided into two sections corresponding to the two experimental regimes used in this
experiment
1) Variation of Masses: Four different mass combinations are used, but the applied force (the mass
imbalance) is kept constant at 10 grams (0.098 Newton's).
2) Variation of Applied Force: One mass combination is used but the imbalance between the two mass
hangers is varied.
Variation of Masses
We have entered the values for m1 , m2 and mI into the spreadsheet. The entry for mI takes into account
the moment of inertia of the rotating pulley. In applying our expression for the net force (equation [2]) we initially
assumed that the system undergoing acceleration consisted simply of the two weight hangers. In doing this, we
neglected to take into account that the pulleys themselves would require an applied net force to set them into
motion. The pulleys therefore act as though they were a masses undergoing an acceleration, and can be treated as
such. The effective contribution of the pulleys to the inertia of the system has been experimentally determined by
your instructors to be 10 grams and this has been entered into the appropriate column in the spreadsheet. This
aspect of the experiment is more fully explained in Appendix A.
The column labeled acc-theo represents the theoretical value for the acceleration of the system based on
equation [3]. It is calculated automatically by the spreadsheet because a formula has been entered into that column
prior to the lab. It utilizes the data in the first four columns (m1 , m2 , mI, mclips ). You will have to enter your values
for the masses of the paper clips (used to compensate for friction) into the cells in column mclips . This will alter
the entries for acc-theo from the values they initially had when you opened the spreadsheet.
Since one of the goals of this experiment is to compare the values for acceleration calculated on the basis of
Newton’s second law with those observed experimentally, there is a column for your experimental data labeled
actual value .
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Figure 5 Opening screen for Atwood’s machine spreadsheet
In order to determine the extent to which your experimental results diverge from the theoretical predications,
you will want to measure the difference between the two. In the column labeled deviation , you will enter the
following formula into the cell just under the heading (cell H4). Note that formulas in EXCEL always begin with
an equals (=) sign:
[6]
=G4 - F4
Use the “drag and fill” feature (which should have been demonstrated to you by your instructor) to extend
this formula to the other three cells in this column (H5, H6, and H7). In the next column, abs deviation , you will
calculate the absolute value of all the values in the deviation column. Into the cell just under the heading (cell H5),
enter the following formula[7]:
[7]
=Abs(H4)
Again drag and fill to fill in the three cells directly under the one into which you just entered the absolute
value formula. Finally, express the absolute deviation as a percent deviation from the theoretical value by entering
formula[8] into cell J4, and dragging and filling as above:
[8]
=I4/F4
-6 This last column of cells has been formatted so that the numerical results of the calculations will be
expressed as percentages.
There should now be a table in which all 4 rows of all the columns (from m1 on the left to %deviation on
the right) have been filled in. You now need to find the average of the deviations and the average of the absolute
values of these deviations.
Begin by calculating the average of the values in the deviation column. To do this, go to cell H9 in the
Atwood spreadsheet and type in formula [9]:
[9]
=AVERAGE(H4:H7)
Enter in the cell immediately below it the formula for the standard deviation (formula [10]):
[10]
=STDEV(H4:H7)
In formulas 10 and 11, the colon symbol between the cell addresses is used to tell the computer to do the
calculation for a range of cells. The expression, (H4:H7), is really shorthand for cells H4, H5, H6, and H7.
Now use the drag and fill feature to extend the calculations for average and standard deviation two columns
to the right, so that you have calculated these quantities for the abs deviation and the %deviation columns. Please
note the difference in meaning for the deviation column and the standard deviation cell underneath it. The deviation
column gives the difference between the predicted and observed values. The average cell at the bottom of the
column gives the average of these differences and the standard deviation cell contains a measure of their
uncertainty. It is the “deviation of a deviation”--we are stuck with a language that doesn’t have a good substitute
for the word “deviation” and so we use it twice here. When interpreting your data, don’t be confused by this
linguistic ambiguity.
Variation of Applied Force
You should now repeat this process for the data in which you have varied the applied force, and enter your
formulas and numerical values into the bottom half of the spreadsheet. Since the total mass on the hangers remains
the same, while its distribution between the two weight hangers is varied, you will have needed to adjust for the
force of friction only once, and so all four values for mclips ought to be the same.
When you are finished, your table ought to look like the one shown in Figure 6. Note, however, that many of your
values will, in most cases, differ from those shown the figure inasmuch as your experimental data has been
recorded independently.
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Fig. 6 Completed Atwood’s machine spreadsheet. Your data will be different
and so it is unlikely that your results will coincide with those in this figure
Things to Discuss
This section provides questions to stimulate your thinking. There may be issues you wish to raise in your
discussion section in addition to those outlined here. In addition, you will probably not want to address every
issue raised here. In any case, your discussion should be a self-contained, coherent, logical essay, and not be a
series of responses to or restatements of the talking points raised below.
The basic question to ask in this lab is something like this: Does experiment confirm theory? In other words,
within experimental limits, does this experiment confirm Newton’s second law? Newton’s law says that if identical
unbalanced net forces are applied to two or more objects, the accelerations will be in inverse ratio to the objects’
masses. Thus if you apply the same force to four different objects, and plot the acceleration versus the
1/ mass, you should get a straight line whose slope is numerically equal to the applied force. The first set of
experiments addresses this situation directly.
Likewise, Newton’s second law states that, if the force applied to a particular object is varied, then the acceleration
of that object will vary linearly with the applied force, with the object’s mass serving as a proportionality constant.
In other words, if you plot acceleration versus applied force, you will obtain a line whose slope is the reciprocal of
the mass. The second set of experiments addresses this problem.
-8 Appendix A
The expression for the net force on the hanging masses in the Atwood’s machine has been given above in equation
[1]:
[1]
Fnet = m 2 g
_
m1g
_
f
From this we evaluated the acceleration (equation [2]) by using Newton’s second law to analyze the motion of m1
and m2:
[2]
(m 1 + m 2 ) a = m 1 g
_
m2g
_
f
This expression neglects the contribution of the rotational motion of the pulley and is therefore inexact. If rotation
is neglected, an error will be introduced into our predictions, one that is increasingly important for smaller values of
m1 and m2 . To take this into account, we must analyze the torque exerted upon the pulley by the unbalanced
masses, transmitted to the pulley by the string. Since, by the time you do this lab, it is unlikely that you will have
begun to study rotational dynamics, the analysis of this effect has been relegated to an appendix. Although it is
included at the moment for completeness, it will complement your study of rotational dynamics when you take the
topic up, and should be retained for study at that time.
The torque is related to the applied force according to equation [A1]:
[A1]
τ =F r
where τ is the torque, F is the component of the applied force applied perpendicular to a line drawn
from the point of application to the axis of rotation, and r is the distance from the axis of rotation to the point where
the force is a applied. The torque causes an angular acceleration, according to equation [A2]:
[A2]
τ = Iα
where I is the moment of inertia. The moment of inertia is a quantity which represents the resistance to angular
acceleration of a rotatable body and is the rotational analog of mass. The moment of inertia of an object in general
increases as the mass of the object increases, and also as that mass is distributed more towards the periphery of the
object. The angular acceleration of an object is related to the tangential acceleration of a point at a distance r from
the axis of rotation by equation [A3]:
[A3]
a =αr
In our experiment, F is the same as the force applied to the pulley by the string, since this force is always
tangent to the pulley’s rim, and hence always perpendicular to a line segment running from the rim to the center of
the pulley. The radius of the pulley is the same as r .
-9 When we take into account the perpendicular force which induces a torque in the pulley, equation[2] becomes
equation [A4]:
[A4]
( m1+ m 2 ) a + F = m 2 g _ m1 g _ f
which, upon substitution of the relationship between F and
[A5]
τ
in equation [A1], yields equation [A5:
( m 1 + m 2 ) a + τr = m 2 g _ m 1 g _ f
Further substitution of the relationship between the torque and the angular acceleration in equation [A2]
yields equation [A6]:
[A6]
( m 1 + m 2 ) a + Irα = m 2 g _ m 1 g _ f
Using equation [A3], the angular acceleration is expressed in terms of the acceleration of a point on the rim
of the pulley, which is identical to the acceleration of the string, and, consequently, the acceleration of our system of
hanging masses. This relationship is expressed in equation [A7]:
[A7]
( m 1 + m 2 ) a + I a2 = m 2 g _ m 1 g _ f
r
The moment of inertia of the pulley can be expressed as the sum of the moment of inertia of its spokes plus
that of its rim. These can be calculated assuming that the rim is a thin hoop of plastic of mass mrim and that the
flat spokes are slices of a disk connecting the rim to the pulley’s hub of mass msp . Substitution of the standard
formulas for objects with these configurations leads to equation [A8]:
[A8]
( m1+ m 2 ) a +
( 12
m sp r 2 + m rim r 2) a
= m 2 g _ m1 g _ f
2
r
which in turn can be simplified to equation [A9]:
[A9]
( m 1 + m 2 ) a + ( 1 m sp + m rim) a = m g _ m g _ f
2
2
1
-10 The contribution of the pulley masses can be summed into an effective mass, meff , leading to equation [A10]:
[A10]
( m 1 + m 2 + m eff ) a = m 2 g _ m 1 g _ f
Finally, equation [A10] is solved for a , the acceleration, yielding equation [A11]:
[A11]
m 2 g _ m1 g _ f
a =
( m 1 + m 2 + m eff )
Equation [A11] is nearly the same as equation [3] except that it now contains the correction for the
spinning pulley. In fact, it is really equation [A11] that we solve in the Atwood spreadsheet. The quantity mI in the
spreadsheet is really meff in equation [A11].
The value of meff could be determined by directly measuring the masses of the pulley parts, but this would
require destruction of a pulley. An alternative is to measure the acceleration for small values of the hanging mass
and extrapolating to zero grams of hanging mass.
Report:
Introduction: Write a brief introduction stating the objectives of the experiment, and a concise summary of the
methods that will be used.
Experimental: Describe the experimental apparatus and precisely what variables will be measured and how they will
be measured.
Results: Summarize the results of the experiment. Show sample calculations. If you are attaching computer
generated tables or graphs, briefly explain them here. Remember to attach your spreadsheet.
Discussion: Explain the significance of your results and their connection with more general physical principles.
Where it is possible, compare your numbers with accepted values. Explain any sources of error.