# Tension forces

#### Prerequisites

Solid matter (and to a lesser extent liquids) tends to hold together even if you try to pull it apart. Phenomenologically, we have come up with an example of a piece of solid matter that responds to being pulled apart by stretching (the spring). Flexible solids, like strings, for example, can be used to transfer forces from one object to another. For example, we might tie a string to a hook on a box and pull the string to pull the box, expecting the string to transfer the force we exert on the string onto the box. To see how this works, let's look at a free body diagram for a string.

If we pull on one end of a string and the string is attached to a box, the box keeps the string taut by pulling on the other end. If the string doesn't change its velocity, we know from Newton's 2nd law that the net force on the string must be 0, so the pulls on either end of the string will be the same.

To understand what's happening inside the string, let's consider a simple model of a string: a chain. (Really, the string is made of wound-together fibers exerting forces on each other by friction, but the chain gives a simpler picture and is qualitatively correct.)

## A simple model of tension: The chain

To see how this works, let's consider a simple model of connected bits of matter — links of a chain being pulled taut by a pair of hands (not shown) pulling in opposite directions — as illustrated schematically with a few links in the figure below.

If the chain is at rest then each link must feel a net force of 0. Consider the free-body diagram of links 1, 2, and 3. Hands are pulling on the chain to the left and the right with forces of magnitude T.

By alternating the use of N2 and N3 we can learn all the forces on each link of the chain.

- The force of the left hand pulling to the right on L1 is T by definition.
- Since L1 is not moving (and not accelerating), by N2 T
_{2→1}must be equal and opposite to T_{LH→1}, so L1 is pulled equally in both directions with a force T. - Moving to link 2, since T
_{2→1 }and T_{1→2 }are the same type of force and have their indices reversed, they are N3 pairs. So they also have the same magnitude. - Since L2 is not moving (and not accelerating), by N2 T
_{3→2}must be equal and opposite to T_{1-->2}, so L2 is pulled equally in both directions with a force T. - Moving to link 3, since T
_{3→2 }and T_{2→3 }are the same type of force and have their indices reversed, they are N3 pairs. So they also have the same magnitude.

We can run down the chain this way, alternating N2 (the forces acting on a particular link) and N3 (the forces that two links exert on each other) to see that every link in the chain is pulled in both directions by the same force, T, as is exerted on the ends. This is how the force on one end is transmitted down the chain.

If we consider a string (fiber) and an interior piece of it as shown in the figure below, we see that the same argument will hold. So every part of the string is pulled equally and oppositely by the same force.

## Vector and scalar tensions

We have been talking about "tension" as a kind of force pulling outward from an object. But in the case of a string or a chain, although there are equal and opposite tension forces, the important thing about a taut string or chain is that *every part of it* is being pulled *in opposite directions* with that same force. For the interior bits of the string or chain, there is no netforce but there is a kind of an "internal directionless force." Since it has no direction it isn't really a force, but it is something significant nonetheless. Imagine having your arms being pulled in two opposite directions. Even though you don't go anywhere, you know about it!

Unfortunately, the tradition is to use the same word — **tension** — to represent an equal interior pull in opposite directions. When you read the word "tension" be careful to see whether a force is meant, with a direction, or this "interior" tension, without direction. Since a quantity without direction is called a **scalar,** as contrasted to a quantity with a direction like a force — a **vector**, these concepts could be distinguished by calling them "scalar tension" and "vector tension", but, alas, this is not standard practice.

## Massless strings

If our string or chain is NOT at rest or moving at a constant velocity, then it has to satisfy Newton's 2^{nd} law: its acceleration has to arise as a result of feeling an unbalanced force. BUT, because we often will connect our strings to objects that are much more massive than they are, the difference in vector tensions needed to accelerate the string is typically much smaller than the tension needed to accelerate the object it is attached to. As a result, we typically choose to ignore the mass of the strings that we use to transfer forces. When we can get away with this we talk about using "massless strings".

In the case that the mass of the string is small and can be ignored (compared to other masses in the problem), the tensions at both ends of the string will have to be equal. Any accelerations that the string needs to do can be accomplished by having a tiny difference in the two tensions. (Since for the string $a = F^{net}/m$, if the mass is very small, you only need a small net force to produce a large acceleration.)

Joe Redish 9/26/11

Last Modified: May 27, 2019