Focus on Physics: The Equilibrium Rule—A Personal Discovery

Building an Understanding of Physical Principles

Fig1

Figure 1. Burl and Paul on a scaffold.

Before college, I worked with master sign painter Burl Grey, who, like me, was passionate about science but didn’t study physics in high school. One day Burl asked which of the two ropes holding up our sign-painting scaffold (Figure 1) experienced more of the “stretching force” called tension. Burl twanged the rope near his end of the scaffold—like a guitar string—and I did the same with mine. Burl, who was heavier than me, reasoned that his rope should have more tension because it supported more weight. Hearing his rope twang at a higher pitch than mine reasonably confirmed that his rope experienced more tension.

Fig2

Figure 2. Paul in middle of scaffold.

Would it affect the tensions, we wondered, if I walked to the middle of the scaffold, toward Burl (Figure 2)? Burl’s rope would support more weight and have greater tension, we reasoned, and tension in my rope should decrease accordingly. To exaggerate the point, if we both stood together at one extreme end of the scaffold and leaned outward, the opposite end of the scaffold should rise like a seesaw, its rope going limp with no tension at all (Figure 3).

Figure 3. Burl and Paul at left end, with the right end raised and the rope limp.

Figure 3. Burl and Paul at left end, with the right end raised and the rope limp.

We agreed that my rope’s tension would decrease as I walked toward Burl—but would the decrease be compensated—exactly—by increased tension in Burl’s rope? If so, how would one rope “know” about changes in the other rope? The answer was beyond our understanding. I learned it only after leaving my sign-painting career for prep school, college, and graduate studies that immersed me in the world of physics.

The equilibrium rule
In my first physics class, I learned that things at rest, such as that scaffold, are in mechanical equilibrium. That is, all forces that act on it balance to zero. In mathematical notation, the equilibrium rule is ∑F = 0, with standing for “the sum of” and F for the forces that act on the object. In the case of Burl and me, our weights were 140 and 110 pounds, respectively (we didn’t talk newtons or kilograms back then). The weight of the scaffold was about 100 pounds. If we call the tensions in the ropes positive in direction (upward) and the weights negative (downward), then

∑F = Tension1 + Tension2
– 140 pounds – 110 pounds
– 100 pounds = 0.

Combining the weights of Burl, me, and the scaffold,

Tension1 + Tension2
– 350 pounds = 0.

Solving for tensions of both ropes,

Tension1 + Tension2 = 350 pounds.

Rope tensions must sum to 350 pounds (Figure 4). Can you see that

Figure 4. The upward forces minus the downward forces equals zero; ∑F = 0.

Figure 4. The upward forces minus the downward forces equals zero; ∑F = 0.

a gain in Tension1 by, say, 50 pounds would mean a loss in Tension2 of 50 pounds? To be in equilibrium, it has to be.

Consider another example (Figure 5). A 350-pound bear stands evenly on two weighing scales, each reading 175 pounds (half of 350).

Suppose the bear leans so that one scale reading increases by 50 pounds. This can’t happen unless the other scale reading decreases by

Figure 5. Bear standing on two bathroom scales.

Figure 5. Bear standing on two bathroom scales.

50 pounds. Only then will the combined readings add to 350 pounds. Likewise for the the supporting ropes of the scaffold. A 50-pound gain in one rope can only occur if accompanied by a 50-pound loss in the other. The answer lies in the mathematics: ∑F = 0.

Classroom activity
Place the opposite ends of a long horizontal plank on two bathroom scales on the floor (Figure 6). The sum of the two scale readings equals the weight of the plank. If you move the scales to different positions, still supporting the

Figure 6. Board resting on two bathroom scales.

Figure 6. Board resting on two bathroom scales.

plank, the readings still add to equal the weight of the plank. How nice! Now have two people stand on the plank near each end (Figure 7). The weight readings increase. How much? Enough so that the sum

Figure 7. Same board with people standing at each end.

Figure 7. Same board with people standing at each end.

of the weight readings equal the weights of the people and the plank. Again, the upward support forces of the springs in the scales (like the ropes holding the scaffold) equal the combined downward weights. Or, stated another way, the upward support forces minus the combined downward weights equal zero. The system is in equilibrium—balancing to zero even when the two people assume different positions along the plank.

Interestingly, the equilibrium rule applies not just to objects at rest but whenever any object or system of objects is not accelerating. Hence, a bowling ball rolling at constant velocity is in equilibrium—a state of no change. The ball rolls down the lane without a change in motion until it hits the pins, whereupon a change in its motion disrupts equilibrium. We say that objects at rest are in static equilibrium; objects moving at constant velocity (without acceleration) are in dynamic equilibrium. Whether objects are at rest or steadily traveling in a straight-line path, ∑F = 0.

So, if you’re in an airplane moving at constant velocity, you know from
the equilibrium rule that the thrust of the engines must be equal and opposite to the air resistance that the airplane undergoes as it collides

Figure 8. At constant velocity, thrust minus air resistance equals zero; ΣF = 0.

Figure 8. At constant velocity, thrust minus air resistance equals zero; ΣF = 0.

with air molecules in its path (Figure 8). Only then will the horizontal forces on the plane sum to zero. Dynamic equilibrium occurs only if ∑F = 0. How about that!

When Nellie Newton pushes her desk across the floor at constant velocity, the equilibrium rule tells you that the amount of friction between the bottom of the desk’s legs and the floor exactly equals

Figure 9. Nellie pushing desk across floor.

Figure 9. Nellie pushing desk across floor.

Nellie’s push (Figure 9). Your knowledge of the amount of friction is simply an example of dynamic equilibrium. Cheers to that, for there’s a lot more you know when you know the laws of nature.

The equilibrium rule provides a reasoned way to view all things, whether in static (balancing rocks, steel beams in building construction) or dynamic (airplanes, bowling balls) equilibrium. For both of these types of mechanical equilibrium, all acting forces always balance to zero. In your further study, look for different forms of equilibrium, such as rotational, thermal, and chemical equilibrium. Examples of equilibrium are evident everywhere.

Paul G. Hewitt (pghewitt@aol.com) is the author of the popular textbook Conceptual Physics, 12th edition, and coauthor with his daughter Leslie and nephew John Suchocki of Conceptual Physical Science, 6th edition.

On the web
Related tutorial screencasts from www.hewittdrewit.com: 1. Equilibrium Rule: http://bit.ly/1T8XYZM; 2. Equilibrium Problems: http://bit.ly/27aKofE

Editor’s Note

This article was originally published in the Summer 2016 issue of The Science Teacher journal from the National Science Teachers Association (NSTA).

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One Comment

  1. Deepak Suwalka
    Posted January 17, 2017 at 7:13 am | Permalink

    It’s a nice post about equilibrium. I like the way you have described it. It’s really helpful. Thanks for sharing it.

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