Focus on Physics: The Equilibrium Rule—A Personal Discovery

Building an Understanding of Physical Principles


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.


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 ( 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 1. Equilibrium Rule:; 2. Equilibrium Problems:

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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Ideas and information from NSTA’s Summer K-12 journals

These issues are great additions to your summer reading list! Most of the lessons in these journals include a detailed chart connecting the lesson to the NGSS.

Science and Children – From Molecules to Organisms

The featured articles focus on developing a progression of learning for younger students.

  • Native Plants and Seeds, Oh My! – Using a plant found in the school garden (milkweed), this lesson includes several parts on the basics of plants and investigations with native plants. Photographs show students at work.
  • Who Is Your Champion? – With a focus on designs and models, students consider the question “What can we learn from plants and animals to help solve the problems we face in our lives?”
  • Stalk It Up to Integrated Learning – Plant parts as food is the basis for this set of learning activities.
  • Elementary Anatomy – Young students enjoy learning about themselves. This lesson for preschool students helps students learn about body parts they can’t see.

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Soaring in a Digital Ecosystem

This column regularly describes digital tools to help teachers make learning more personal and effective for all students. When these tools converge, they create a sort of digital ecosystem designed to make students more collaborative and innovative, skills essential for success in today’s world. But are your students truly using digital technology to its maximum benefit?

The SAMR model
Our efforts toward digital convergence are based on the Substitution SAMR-box2Augmentation Modification Redefinition (SAMR) model ( (see box), which leads to higher-order technology in the classroom. Used at a low level, technology merely serves as a substitution—for example, using a word processor instead of paper and pencil to write a conclusion.

The next level is augmentation, in which technology improves on a learning task similar to what students could do without the technology, such as using the formatting tools in a word processor to highlight areas of interest. Much of classroom technology falls into these two categories, including scientific probes and graphing calculators ( Our goal is to move on to the next levels of technology use: modification and redefinition of student work to demonstrate understanding.

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Place-Based Learning in Middle School: Putting Scientific Principles to Work in your Community

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“When we try to pick out anything by itself, we find it hitched to everything else in the Universe.” -John Muir, My First Summer in the Sierra, 1911.

We hope that you are enjoying your summer!  As teachers, we realize that your mind is never far from your classroom, even if your body is lounging on a chair next to *insert appropriate body of water here*. As science teachers, especially, even the sounds of waves and splashing children have entirely different meaning to us than to those in other walks of life.  You might hear water hitting the beach and start pondering frequency, wavelength, and longshore drift and before you know it your mind starts generating lesson plans.  Teachers are constantly mining personal experiences for ideas to help students connect what they learn to the world around them.

Making these connections is infinitely easier for our students if we are able to take them beyond the confines of the schoolroom. While the majority of us would hesitate to invite our students on summer vacation with us, we work hard to provide real-world, authentic learning opportunities for them. When students embark on a nature walk around the school grounds, enjoy a guest speaker from the local community, experience a well-planned outdoor education trip, or gather data for citizen-scientist programs science concepts come alive in a way that even the best textbooks can never match.

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Changing grade levels

5229139935_f4b54c053c_mNext year there will be an opening in the high school science department. Although I love teaching middle school, I’m tempted by the opportunity to try something different and use more of what I majored in (chemistry). What advantages and disadvantages should I consider?—C., New Jersey

Taking on new subjects or grade levels can be exciting and professionally rejuvenating. It can also be a lot of work, almost like starting over.

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NSTA Legislative Update: Update on ESSA; Good News for STEM and FY2017 Appropriations


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July 14, 2016: Congress is set to adjourn for the summer and will return after Labor Day. Before leaving town though there was a flurry of activity around appropriations for FY2017 programs and career and technical education. And the political drama continues as Education Secretary King answers questions from key Congressional Republicans over implementation of the Every Student Succeeds Act.

The good news for STEM: The House of Representatives Appropriations Committee has approved a FY2017 Labor HHS and Education spending bill that includes $1 billion for the new Every Student Succeeds Act Title IV block grants.  This amount is $500 million above the President’s budget request and $700 million above the Senate funding ($300m).  The program is authorized at $1.65 billion in ESSA.

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3D Brings Science to Life

Middle school children are inquisitive and enjoy classroom opportunities to learn visually. Subsequently, an option worth consideration is an application of technology known as 3D. It’s similar to the 3D technology that is used in movie theaters and is designed to enhance visualization of pairs of images and gives users a greater sense of depth perception.

For nearly 150 years, stereoscopes have been used for looking at images that depict left-eye and right-eye views of the same object; culminating into a single three-dimensional image.  Subsequently, when viewing the image with special projection hardware and eyewear, a typical stereoscope provides each eye with a lens that makes the image seen through it appear larger and more distant, resulting in the illusion of depth.


Recent Advances in technology have led to much more sophisticated ways of projecting the third dimension.  For example, Data Light Processing (DLP) technology creates a stunning picture and is used in contemporary projectors. DLP technology is extremely fast, and projects two images on the screen at the same time, i.e., one for each eye. As a tool for conceiving the image, 3D glasses are used to combine the two images into 3D and can be purchased from a variety of projector manufacturers, e.g., InFocus, Texas Instruments, etc. Continue reading …

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Write Grant Proposals That Win

A successful grant application can provide you with the funding you need to do exciting new activities with your students. The only problem is that grant writing is an art form of its own. There’s a new NSTA Press book that can help.

Write Grant Proposals That WinBe a Winner! A Science Teacher’s Guide to Writing Successful Grant Proposals by Patty McGinnis and Kitchka Petrova offers practical tips and strategies to help you write winning proposals.

 “As a science educator, you are concerned with the state of science education in your K-12 schools, and you understand the importance of facilitating your students’ science learning through the science and engineering practices identified in the Next Generation Science Standards (NGSS). Unfortunately, funds for purchasing materials are not always available in schools, thus requiring you to seek outside funding opportunities. Given the economic situation of many school districts, it is more imperative than ever to master the art of grant proposal writing to secure funds for innovative classroom projects,” write McGinnis and Petrova.

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Water play…exploration…science inquiry

Children at a water table outsideWater explorations are a popular in early childhood programs during the summer. Exuberant water explorations can happen outdoors. The experience of wetness is enjoyable and clothes that get wet accidentally can dry on the child rather than having to be changed. Natural materials such as leaves and twigs can be incorporated into the exploration.

To keep the water experiences enjoyable, meaningful and a powerful learning opportunity, take a look at these resources.

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Managing Communication Channels

3172841858_4f317b12f7_mLast year, I tried improving my communications with students and parents via electronic media. I had lots of responses, but I was being texted, tweeted, emailed, and called on the phone at all times of the day and night. While I want to encourage these communications, I’m looking for ideas to manage them and keep my sanity! —G., Colorado

It sounds like you have a case of “be careful what you wish for….” Many teachers would love to have parents and students contacting them, but I can understand how this can become overwhelming.

In a recent article in Educational Leadership (May 2016)*, Catlin Tucker, an English teacher from California, shared her ideas on “avoiding technology overload.” You may find them helpful as you try to manage communications with the many other responsibilities of a science teacher:

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