Dynamics – Atwood Machines / SBA
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urses/honors/dynamics/Atwood.html
Unit #3 Dynamics

Objectives and Learning Targets
 Resolve a vector into perpendicular components: both
graphically and
algebraically.
 Use vector diagrams to analyze mechanical systems
(equilibrium and non-equilibrium).
 Analyze and solve basic Atwood Machine problems
using Newton’s 2nd Law of Motion.
Unit #3 Dynamics
Atwood Machines

An Atwood Machine is a basic physics laboratory device often used to
demonstrate basic principles of dynamics and acceleration. The machine
typically involves a pulley, a string, and a system of masses. Keys to solving
Atwood Machine problems are recognizing that the force transmitted by a
string or rope, known as tension, is constant throughout the string, and
choosing a consistent direction as positive. Let’s walk through an example
to demonstrate.
Unit #3 Dynamics
Sample Problem #1
Question: Two masses, m1 and m2, are
hanging by a massless string from a
frictionless pulley. If m1 is greater than m2,
determine the acceleration of the two
masses when released from rest.
Unit #3 Dynamics
Sample Problem #1
Question: Two masses, m1 and m2, are hanging by a
massless string from a frictionless pulley. If m1 is
greater than m2, determine the acceleration of the
two masses when released from rest.
Answer: First, identify a direction as positive. Since
you can easily observe that m1 will accelerate
downward and m2 will accelerate upward, since m1 >
m2, call the direction of motion around the pulley and
down toward m1 the positive y direction. Then, you
can create free body diagrams for both object m1 and
m2, as shown below:
Unit #3 Dynamics
Sample Problem #1
Using this diagram, write Newton’s 2nd Law
equations for both objects, taking care to note the
positive y direction:
Next, combine the equations and eliminate T
by solving for T in equation (2) and
substituting in for T in equation (1).
Unit #3 Dynamics
Sample Problem #1
Finally, solve for the acceleration of the system.
Unit #3 Dynamics
Sample Problem #1 (Alternate Solution)
Alternately, you could treat both masses as part of
the same system.
Drawing a dashed line around the system, you can
directly write an appropriate Newton’s 2nd Law
equation for the entire system.
*Note that if the string and pulley were not massless, this
problem would become considerably more involved.
Unit #3 Dynamics
Single Body Analysis (SBA) Steps
General Problem Solving Guidelines (there are exceptions)
1. Draw the situation described including all of the applicable forces.
2. Create Newton’s Second Law expressions for each object in your
diagrams. (ma=winner(s)-loser(s))
3. Solve the expressions for the same variable (i.e. tension) and set
them equal to each other.
4. Solve for the unknown variable.
5. Use your newly found variable to go back and find any others that you
need.
Unit #3 Dynamics
SBA Calculator Tips
Method for using your calculator to find “a” and “T” in a two object
system
1. Follow steps 1-3 listed above
2. Hit [Y=] and enter the first equation on the “Y1” line, hit [ENTER]
3. Enter your second equation on the “Y2” line, hit [2nd] [CALC]
4. Hit [5] for “intersect” hit [ENTER] three times.
5. The value displayed for x is the tension if your equations were
solved for tension, and y is the acceleration. (If your equations were
solved for acceleration the opposite is true)
Unit #3 Dynamics
SBA Sample Problems
***All example problems use a frictionless, massless pulley under ideal conditions. Air
resistance should be considered negligible (0N).
1. A 1 kg mass is suspended by a (ideal) pulley system with a string that is attached to a 9
kg mass. The 9 kg mass is on a frictionless table (mu = 0). Solve for the acceleration (a)
of the system and the tension (T) on each mass.
2. A 2 kg mass is suspended by a (ideal) pulley system with a string that is attached to a 3
kg mass. The 3 kg mass is on a table with a mu value of 0.50. Solve for the acceleration
(a) of the system and the tension (T) on each mass.
3. A 5.00 kg mass is suspended by a (ideal) pulley system with a string that is attached to
a 4.00 kg mass. The 5.00 kg mass is on a table with a mu value of 0.40. Solve for the
acceleration (a) of the system and the tension (T) on each mass.
4. A 2.0 kg mass is suspended by a (ideal) pulley system with a string that is attached to
an 8.0 kg mass. The 8.0 kg mass is on a table with a mu value of 0.200. Solve for the
acceleration (a) of the system and the tension (T) on each mass.
5. A 7.00 kg mass is suspended by a (ideal) pulley system with a string that is attached to
a 5.00 kg mass. The 5.00 kg mass is on a table with a mu value of 0.50. Solve for the
acceleration (a) of the system and the tension (T) on each mass.
Unit #3 Dynamics