Name
Period
Date
5. Atwood's Machine
Structured (S)
Objectives
Build foundational knowledge surrounding the concept of uniform acceleration in closed systems
by:
 Implementing a simple Atwood's machine to experimentally determine the earth's
gravitational acceleration.
 Developing graphical analysis skills and linearization techniques.
Materials and Equipment
For each student or group:
 Data collection system
 Large base and support rod
 Rotary motion sensor
 Extra rod
 Mass and hanger set
 String
 Right angle clamp
 Scissors
Background
We are familiar with drawing free body diagrams and the
pertinent forces acting on common objects, like blocks sliding
down inclines, objects falling through air, and other
projectiles we often treat as point masses. But what about a
free body diagram for an awkward shape? This lab pertains
to a system called an Atwood's machine that can be modeled
with a free body diagram if we treat the entire system as one
accelerating object.
Pulley
Ascending
Descending
An Atwood's machine consists of two masses connected by a
m
1
string over a pulley. If one mass is slightly greater than the
other, the difference in mass creates a net force that
Masses
accelerates the system. If m2 in the diagram at right is
m2
greater than m1, the acceleration of m2 downwards will
equal the acceleration of m1 upwards. If you treat the entire
system of two masses and string as one object that accelerates together (which it does), then the
force accelerating the system down is m2g, and a force of m1g resists this acceleration by pulling
the opposite direction of the actual motion. So, the net force accelerating the system is
(m2 – m1)g. Applying Newton's second law for the entire system then yields the equation:
 m2  m1  g   m2  m1  a
(1)
The sign associated with each term is important, the downward direction is taken as positive for
m2 since it is moving downward and the upward direction is taken as positive for m1 since it is
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Atwood's Machine
moving up. Also note that the tension of the string becomes an internal force and is not a factor
when applying F = ma to the entire "m1 + m2" system.
Solving Eq. 1 for the system acceleration a yields:
ag
 m2  m1 
 m2  m1 
(2)
Note that torque required to cause the pulley to begin spinning is assumed to be negligible, as is
the mass, friction, and rotational inertia of the pulley. These factors are left to future lessons for
a more detailed analysis.
Relevant Equations
F  ma
(3)
 m2  m1  g   m2  m1  a
ag
 m2  m1 
 m2  m1 
Safety
Add this important safety precaution to your normal laboratory procedures:
 If the difference between masses on each hanger is too great, the masses may fly off the pulley
and hanger when released. Use reasonable amounts of mass on each hanger to avoid flying
masses.
Procedure
After you complete a step (or answer a question), place a check mark in the box () next to that step.
Note: When you see the symbol "�" with a superscripted number following a step, refer to the numbered Tech
Tips listed in the Tech Tips appendix that corresponds to your PASCO data collection system. There you will find
detailed technical instructions for performing that step. Your teacher will provide you with a copy of the
instructions for these operations.
Set Up
1.  Assemble the large base and support rod on the lab table. Mount the extra rod to the rod
and base using the right angle clamp so that the extra rod hangs over the edge of the lab
table.
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PS-2899A
Student Worksheet (S)
Rotary motion
sensor
Mass hangers
2.  Mount the rotary motion sensor to the end of the extra rod using the rod clamp attached
to the top of the rotary motion sensor.
3.  Use a length of string just longer than the distance from the pulley to the floor. Tie two
mass hangers to the string, one on each end. Now drape the string over the large groove
on the pulley with the two mass hangers hanging freely.
Note: make certain that the two mass hangers will not make contact with each other as they descend and
ascend. Re-adjust if necessary.
4.  Place a total of 58 g of mass on each of the two mass hangers. Do this by using two 20 g
masses, one 10 g mass, one 5 g mass, one 2 g mass, and one 1 g mass.
Note: The mass hangers themselves have a mass of 5 g, making the total mass equal to 63 g.
5.  Connect the rotary motion sensor to the data collection system
Acceleration on the y-axis of a graph with Time on the x-axis.
�(2.1)
and display Linear
�(7.1.1)
Collect Data
6.  Remove one of the 1 g masses from one hanger (m1) and place it on the other hanger (m2).
There should now be a difference in mass of 2 grams.
7.  Pull the m1 mass hanger down near the ground so that the m2 mass hanger is near the
pulley and ready to fall.
8.  Start data recording, �(6.2) and then release the masses to accelerate freely.
9.  Catch the ascending mass hanger before it hits the pulley, or the descending mass hits
the floor.
10.  Stop data recording. �(6.2) Adjust the scale of the graph to show all data.
�(7.1.2)
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Atwood's Machine
11.  Find the average acceleration of the system while the masses were in motion.
�(9.4)
Note: In your average, be sure to exclude any points at the beginning or end of your data set that are
not consistent (these may occur because you began data recoding before the system accelerated, or
because you caught the masses before you stopped recording data).
12.  Record the average acceleration and the difference in mass between the mass hangers
∆m (kg) in Table 1.
13.  Repeat the Collect Data steps four more times, moving 1 gram of mass from the m1 mass
hanger to the m2 mass hanger each time. Because there is only one 1 g mass on each
hanger, you will need to use combinations of masses to move only 1 g at a time. For
example, move a 2 g mass from m1 to m2 and a 1 g mass from m2 to m1 which equals 1 g
from m1 to m2.
Note: Moving 1 g of mass from one mass hanger to the other creates a 2 g mass difference between
the hanging masses.
Data Analysis
Table 1: Average acceleration and difference in mass
Acceleration (m/s2)
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Δm (kg)
PS-2899A
Student Worksheet (S)
1.  Plot a graph of Acceleration versus ∆m data from Table 1. Label the overall graph, the
x-axis, the y- axis, and include units on your axes.
2.  Apply a line of best fit to the graph. Record the resulting equation here.
Best fit line equation:
.
Analysis Questions
1. ∆m represents the difference in mass that led to the acceleration of the system.
Draw a free body diagram with the forces acting on the system.
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Atwood's Machine
2. Use your graph of acceleration versus ∆m to determine an experimental value for
the earth's gravitational acceleration constant g. Explain the procedure you used to
determine this value.
3. Calculate the % error in your experimental value for g.
4. How would the accelerations you measured in this lab change if you were still
moving the same small masses from one hanger to the other for each trial, but the
total mass of the system was much larger? The measured accelerations would all be:
Greater ______
Lower ______
Unchanged ______
Explain your answer.
________________________________________________________________________________________
5. For low acceleration values, how would the mass and friction of the pulley affect
the values for g? Would they be erroneously high or low? Explain your answer.
________________________________________________________________________________________
________________________________________________________________________________________
6. For high values of acceleration, the string may actually slip on the pulley wheel.
How would this error affect the observed values for system acceleration?
________________________________________________________________________________________
________________________________________________________________________________________
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PS-2899A
Student Worksheet (S)
Synthesis Questions
Use available resources to help you answer the following questions.
1. A mass of 1.2 kg hangs from one end of an Atwood's machine while a mass of 1.0 kg
hangs on the other end. Find the acceleration of the system.
2. Astronauts used an Atwood's machine to determine the acceleration due to gravity
on a new planet. They placed 250 g on one pan and 750 g on the other. If the Atwood's
machine accelerated at 3.0 m/s2, what is the acceleration due to gravity on the planet?
(Show your work.)
3. Jack built an Atwood's machine that has two seats for people to sit on. When
Jonathan and Emily sat down their acceleration was 4.0 m/s 2. What is the ratio of
Jonathan's weight (m2) to Emily's (m1)? Assume Jonathan weighs more than Emily.
(Use g = 10 m/s2 in your calculations.) Show your work.
Multiple Choice Questions
Select the best answer or completion to each of the questions or incomplete statements below.
1. Which of the following expressions is a correct
expression of Newton's second law for the forces acting
on m1?
A.
T  m1 g  m1a
B.
T  m1 g  m1a
C.
m1 g  T  m1a
D.
m1 g  T  m1a
E.
m1 g  T  m1a
m2 > m1
T
m1
T
Ascending
m1g
m2
Descending
m2g
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Atwood's Machine
2. Using the figure from the previous question, if m2 equals 10 g and m1 equals 5 g
what is the acceleration of the system? (Use g = 10 m/s2 in your calculations.)
A. 3.33 m/s2
B. 50.0 m/s2
C. 10.0 m/s2
D. 5.0 m/s2
E. 2.0 m/s2
3. The acceleration of the masses doubles when an unknown mass is added to m2.
What is the unknown mass? (m1 = 5 g and m2 = 10 g)
A. 5 g
B. 10 g
C. 15 g
D. 20 g
E. 25 g
4. In this lab, which of the following quantities needed to be constant throughout the
experiment?
A. m1
B. m2
C. m1 – m2
D. m1 + m2
E. m1/m2
5. Which combination of masses shown below would result in the greatest
acceleration?
A. m1 = 5 g; m2 = 10 g
B. m1 = 10 g; m2 = 15 g
C. m1 = 15 g; m2 = 20 g
D. m1 = 20 g; m2 = 25g
E. All of these masses would result in the same acceleration.
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PS-2899A