Tension in a String: Atwood's Machine

A string or a cord can be a rather confusing element in applications of Newtons laws. however if you are to become an orthopedic surgeon or if you need to understand how a tendon works you need to learn how to deal with a cord in Newtons laws.

The reason strings, cords, and the like are difficult is that we have a distinct sense of a direction when a rope is being pulled. In fact for massless cords passing over frictionless pulleys or surfaces the whole rope is characterized by a single tension, which we usually denote , T. If a rope is in tension, then at any cross section I might choose, the left part pulls on the right by a force T and the right side pulls on the left by a force, T. So all I have to remember is that

1. There is a single tension, T characterizing an ''ideal'' cord.
2. A rope can only pull long its length. It never pushes and it never exerts a force perpendicular to its length.

Rule 1) sets the magnitude of the forces produced by a cord and rule 2) determines the direction of the force produced on an object in contact with the cord. We have several amusing examples involving the tension in an ideal cord:

The Atwood machine was invented by George Atwood in 1784 as a laboratory experiment to prove Newtons laws. Since then it has haunted students of mechanics and made life easier for elevator motors. It consists of two masses, and connected by a massless cord over an ideal massless pulley. When the apparatus is in a stable equilibrium irrespective of the position chosen. This is the idea behind the counter-balance of an elevator: to approximate the loaded elevator system to a stable Atwood machine so as to minimize the force which must be supplied by the elevator motor to raise and lower the elevator. If we make there is a slight acceleration towards the heavier mass. Lets calculate this acceleration.

We write Newton's laws for introducing the acceleration positive for clockwise rotation of the pulley and the tension, T, in the string. At this stage I do not know the magnitude of either of these. Newton's second law becomes:

For I have a similar equation where I note that because of the properties of the ideal cord the magnitude of T and a must be identical. We get

We have ended up with two equations and two unknowns. Adding the equations we get the following

In other words the gravitational force of the difference mass acts to accelerate the total mass and this is why the acceleration tends to be so disappointingly small.

We can test our equations with a little experiment. We choose the difference mass to be

and the total mass to be

This yields an acceleration

which amusingly is about the free fall acceleration on the moon. The time taken to fall 1 m is then

We perform the experiment and find agreement with this number.