Module 3 -- Tension Force

From PER wiki

Ropes, wires, strings, etc. are commonly used to provide force in everyday situations. A force provided by a rope or string is generally called a tension force.

Learning Goals

After completing this module, you should be able to:

State the two important properties of the tension forces exerted by a segment of a massless rope.
Describe the tension in a rope held at more than two points.
Describe the tension in a rope that passes over a massless pulley.
Correctly include tension forces arising from massless ropes in free body diagrams.

Idealized Ropes

When discussing ropes, strings, etc. in this course, it will generally be assumed that they have zero mass and do not stretch. In this case, their behavior is fairly simple. The important aspects can be summarized with two simple rules:

A segment of a massless rope can only exert a tension force if it is secured between two points of contact on different objects.
If a massless rope is stretched between two points of contact with other objects, the tension force exerted by a given segment of the rope on the objects on either side will be equal in size and will point directly along the rope segment.

Tension Force as a Constraint Force

Consider a box hanging from an ideal rope attached to the ceiling of an elevator. The elevator is accelerating upwards with an acceleration of magnitude a. The circumstance here is that the rope constrains the box to move upwards with the same acceleration as the elevator. We therefore know that the force of tension must be enough to cause this motion. Therefore we'll typically apply Netwon's second law to the box which will enable us to find an equation for the tension.

A box hanging from the ceiling of an elevator
The box of mas m in the figure is interacting with two objects: the rope and the Earth. As a result of these two interactions there are two forces acting on the box (see the free body diagram): $\vec{F}_{bE}$ is the gravitational force on the box exerted by the Earth which is pointing down and has a magnitude of $F b E = m g$ . $\vec{F}_{bR}$ is the force of tension exerted on the box by the rope which is pointing up. In this problem, the forces on the box are along the vertical direction, therefore we can write Newton's second law as: $\sum F_{y} = F_{bR} - mg = ma$ where the signs in the expression above are consistent with our choice of coordinate system (+y - axis pointing up). We obtain: $\mathcal{}F_{bR} = m(a+g)$ Note that the magnitude of the force of tension is obtained after applying Newton's second law and depends on the box's acceleration (or motion), while the gravitational force on the box is given by the gravitational Force Law.

Illustrative Examples

Rope Held at Both Ends
A massless rope held at both ends will exert a force of the same size (size T in the FBD's drawn here) on the object at each end. The forces will point directly along the rope as shown in the free body diagrams of the two people pulling the rope in the example below:

Multiple Contact Points
A massless rope held at a point in the middle of its length will be treated as if it were divided into two ropes at that midpoint. Thus, the two situations drawn below are equivalent for the purposes of analyzing the motion* of the people involved* and will each produce the FBD's shown below.

Idealized Massless Pulleys

Pulleys are often used with ropes or strings. For now, when you see a pulley in a picture you should assume it is massless. This assumption implies that it will require no work to rotate or translate the pulley. In this approximation, the pulley only changes the direction of the rope. Later in the course, when we have learned about rotational motion, we will be able to discuss the effects of massive pulleys. Pulleys are commonly encountered in two qualitatively different types of situation.

Example 1: Pulley Fixed to a Surface
In the figure, a box (1) connected to another box (2) by a rope (R) passing through a pulley, is moving along a horizontal and smooth table (t). The idealized massless pulley fixed to a surface acts only to change the direction of the rope, and therefore also changes the direction of the tension force applied to the objects at either end of the rope.

Example 2: Pulley Fixed to an Object
An idealized massless pulley fixed to an object whose motion we are analyzing not only changes the direction of the tension in the rope, but also increases the number of effective contact points the object has with the rope. Every point in the rope at which it transitions from contact with the pulley to noncontact or at which it transitions from noncontact to contact with the pulley counts as a separate contact point between the object and the rope for the purposes of constructing a free body diagram of the situation.