Atwood Machines
1 Fixed Pulley
Let’s begin our analysis by studying the fundamental Atwood machine – a fixed pulley with two masses.
Question
Find the accelerations of the masses and the tension in the string.
We solve problems involving Atwood machines by using F=ma equations and an equation of conservation of string.
F=ma equations
Conservation of String
Accelerations
Tension
We can use the idea of effective mass of a fixed pulley to simplify our analysis in fixed pulley combinations.
Effective Mass of a Fixed Pulley
1.1 Fixed pulley combination
Question
Find the accelerations of the masses and the tension in the string.
F=ma equations
We can use the idea of effective acceleration of a fixed pulley to simplify finding the equation for conservation of string in fixed pulley combinations.
Effective Acceleration of Fixed Pulley
Conservation of String
Accelerations
Tension
2 Free Pulley
Now let’s analyze systems with free pulleys in which they are not attached to supports.
Question
Find the accelerations of the masses and the tension in the string.
F=ma equations
Conservation of String
Accelerations
Tension
2.1 Strings
Question
Find the accelerations of the masses and the tension in the string.
F=ma equations
Conservation of String
Accelerations
Tension
2.2 Strings
We can generalize this to strings.
Question
Find the accelerations of the masses and the tension in the string.
F=ma equations
Conservation of String
Accelerations
Tension
2.3 Free Pulley In Between
Question
Find the accelerations of the masses and the tension in the string.
F=ma equations
Conservation of String
Accelerations
Tension
2.4 Massless Free Pulley
Equivalence of Free Pulleys
The subject of massless free pulleys is an interesting topic in Atwood machines. We can use a key idea derived here to solve more interesting problems.
Question
Find the acceleration of the free pulley.
Procedure
Assume the free pulley has mass . This is equivalent to the case with a free pulley derived above.
We have . The force on the pulley must be zero so
. We can use the first equation and third equation to solve for
. Note that we cannot use the second equation.
Question
Find the accelerations of the masses and the tension in the string.
Accelerations
Note that the net force on the left massless pulley must be zero.
Tension
The techniques developed here can be applied to any Atwood Machine problem.
For more information see: Introduction to Classical Mechanics (David Morin)
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