7.4.1 Tension Revisited

FIGURE 7.17 shows a heavy safe hanging from a rope, placing the rope under tension. If you cut the rope, the safe and the lower portion of the rope will fall. Thus there must be a force within the rope by which the upper portion of the rope pulls upward on the lower portion to prevent it from falling.

FIGURE 7.17 Tension forces within the rope are due to stretching the spring-like molecular bonds.

The figure illustrates the zoomed in view of a vertical rope that suspends an object labeled Safe.
FIGURE 7.17 Full Alternative Text

Chapter 5 introduced a ball-and-spring model of solids in which tension is due to the stretching of spring-like molecular bonds. Stretched springs exert pulling forces, and the combined pulling force of billions of stretched molecular springs in a string or rope is what we call tension.

An important aspect of tension is that it pulls equally in both directions. To gain a mental picture, imagine holding your arms outstretched and having two friends pull on them. You’ll remain at rest—but “in tension”—as long as they pull with equal strength in opposite directions. But if one lets go, analogous to the breaking of molecular bonds if a rope breaks or is cut, you’ll fly off in the other direction!

EXAMPLE 7.5 Pulling a rope

FIGURE 7.18a shows a student pulling horizontally with a 100 N force on a rope that is attached to a wall. In FIGURE 7.18b, two students in a tug-of-war pull on opposite ends of a rope with 100 N each. Is the tension in the second rope larger than, smaller than, or the same as that in the first rope?

FIGURE 7.18 Which rope has a larger tension?

The figure labeled ("a") illustrates a person pulling a rope attached to a wall.
The figure labeled (b) illustrates two people pulling a rope in opposite directions.

SOLVE Surely pulling on a rope from both ends causes more tension than pulling on one end. Right? Before jumping to conclusions, let’s analyze the situation carefully.

FIGURE 7.19a shows the first student, the rope, and the wall as separate, interacting objects. Force F⃗S on R is the student pulling on the rope, so it has magnitude 100 N. Forces F⃗S on R and F⃗R on S are an action/reaction pair and must have equal magnitudes. Similarly for forces F⃗W on R and F⃗R on W. Finally, because the rope is in equilibrium, force F⃗W on R has to balance force F⃗S on R. Thus

FR on W=FW on R=FS on R=FR on S=100 N

The first and third equalities are Newton’s third law; the second equality follows from Newton’s second law for the rope with aR=0.

FIGURE 7.19 Analysis of tension forces.

The figure labeled ("a") illustrates three free body diagrams for a student pulling a rope attached to a wall on the right.
The figure labeled (b) illustrates three free body diagrams for two students pulling a rope in opposite directions.

Forces F⃗R on S and F⃗R on W are the pulling forces exerted by the rope and are what we mean by “the tension in the rope.” Thus the tension in the first rope is 100 N.

FIGURE 7.19b repeats the analysis for the rope pulled by two students. Each student pulls with 100 N, so FS1 on R=100 N and FS2 on R=100 N. Just as before, there are two action/reaction pairs and the rope is in equilibrium. Thus

FR on S2=FS2 on R=FS1 on R=FR on S1=100 N

The tension in the rope—the pulling forces F⃗R on S1 and F⃗R on S2—is still 100 N!

You may have assumed that the student on the right in Figure 7.18b is doing something to the rope that the wall in Figure 7.18a does not do. But our analysis finds that the wall, just like the student, pulls to the right with 100 N. The rope doesn’t care whether it’s pulled by a wall or a hand. It experiences the same forces in both cases, so the rope’s tension is the same in both.

REVIEW Ropes and strings exert forces at both ends. The force with which they pull—and thus the force pulling on them at each end—is the tension in the rope. Tension is not the sum of the pulling forces.