JoinFocus on Physics: The Delightful Catenary Curve
By sstuckey
Posted on 2017-05-02
Figure 1. A. Notre Dame Cathedral, Paris. B. St. Paul’s Cathedral, London.
When teaching how tension and compression relate to geometrical structures such as bridges, arches, and domes, I show a picture of the Notre Dame Cathedral in Paris (Figure 1A), completed in the 14th century. I point out the elaborate buttresses that keep the walls from pushing outward while supporting its weight. Architects of the day had not yet learned how to hold up a very large, massive building without external propping. This was accomplished in the 17th century in the construction of St. Paul’s Cathedral in London (Figure 1B).
Why, I ask, is St. Paul’s Cathedral free of such buttresses? Aha, inside its famous dome is an inner “secret dome” that provides structural support. To understand this, let’s first investigate the roles of tension and compression in structures.
Tension
I stretch a length of rope taut, explaining that the stretching force we call tension acts in a direction parallel to the direction of the rope. When I let the rope sag between my hands, tension vectors within the sagging rope continue to align with the rope. The curved shape of the sag is determined by this alignment of tension vectors. Likewise for a sagging chain or sagging cable.
A rope, chain, or cable supported at its ends and hanging only by its own weight takes the shape of a special curve called a catenary. I sketch a sagging
Figure 2. Tension between links in the chain align with (are parallel to) the curve of the chain. The curve is a catenary.
chain on the board and show that tension vectors between links of the chain are everywhere parallel to the curve with no components of tension perpendicular to the curve (Figure 2). The chain ends can be held at different distances apart, making the curve deep or shallow. As long as the chain supports only its own weight, it’s a catenary.
If a sagging chain or cable supports weight that is distributed uniformly in a horizontal direction, as is approximately true in a suspension bridge, then the shape of the curve is a parabola, the same curve followed by a tossed ball. The curved cables of a suspension bridge or suspended roadway are approximately parabolas. Only if the cable supports only its own weight—such as sagging clotheslines, power lines, and strands of spider webs—is the shape a catenary.
Compression—and the inverted catenary (an arch)
Of particular interest is an inverted catenary, where internal forces are of compression rather than tension. When a free-standing arch takes the shape
Figure 3. Curves of the sagging chain held by Manuel and of the Gateway Arch in St. Louis are catenaries.
of an inverted catenary, the weight of the arch is supported by compression forces pressing along the arch’s curve. There are then no compressive forces perpendicular to the curve. My grandson Manuel delightfully shows two catenaries in Figure 3, one of a suspended chain and in the background the Gateway Arch in St. Louis, Missouri.
I sketch the Gateway Arch, showing that compression vectors between adjacent slabs that make up the arch are everywhere parallel to the curve (Figure 4). I tell students that they could make a stable mini-arch out of slippery
Figure 4. Compression vectors between slabs are everywhere parallel to the Gateway Arch.
blocks of ice if the shape of the arch is a catenary! But if the shape were any other, such as a semicircle, blocks of ice would squeeze free, and the arch would collapse. Where strength is important, modern arches are usually catenaries.
The three-dimensional catenary: a dome
I ask students to imagine rotating an arch through a complete circle. I then help them to reason that the result would be a dome. Just as for an arch, the weight of any dome produces compression, which tends to collapse the dome unless the compression forces are aligned with (parallel to) the dome’s curve—in other words, a catenary.
Christopher Wren, the brilliant architect who designed St. Paul’s Cathedral, believed that a hemispheric dome (half of a sphere) would be the most
Figure 5. The nesting domes of St. Paul’s Cathedral.
beautiful but knew it would be too weak by itself to hold the extremely heavy lantern structure he planned to place atop it. So, he cleverly designed three nesting domes: a hemispheric outer dome, a steeper inner dome, and—hidden in the middle—a dome that approximated a catenary and provided the necessary structural support (Figure 5).
Modern domes that span vast areas without the interruption of supporting columns are most often three-dimensional catenaries. There are shallow domes (the Jefferson Memorial) and tall ones (the United States Capitol). The catenary principle is employed in the Houston Astrodome, the roof of Washington Dulles International Airport, and the igloos in the Arctic.
The catenary egg
To relate this physics concept to the natural world, I introduce students to the shape of a common egg. People have always wondered why crushing an egg by
Figure 6. Why does squeezing an egg along its long axis fail to break the egg?
squeezing along its long axis is so difficult (Figure 6), while a chickie can easily poke its way out from the inside. Penetrating the shell from inside deals with tension rather than compression. Only the weaker shell tension must be overcome. But what makes the compressive forces on the outside so strong? Can you guess? I ask. (And of course they can.)
A double catenary
I direct student attention to the two most curved surfaces of an egg: its two ends. With an egg held so that first one end, then the other, is on top, they see that a chain follows the contour of the egg—at each end (Figure 7). Aha! Students are delighted to see that both ends of the egg, one more strongly curved than the other, are catenaries. Nature has not overlooked the catenary!
Figure 7. A chain follows the curves of an egg—catenaries at both ends.
Paul G. Hewitt (pghewitt@aol.com) is the author of the popular textbook Conceptual Physics 12th edition and coauthor with his daughter Leslie Hewitt and nephew John Suchocki of Conceptual Physical Science 6th edition, Pearson Education.
On the web
For more on the catenary: http://bit.ly/catenaries. Tutorial screencast lessons by the author are on www.HewittDrewIt.com and www.ConceptualAcademy.com.
Editor’s Note
This article was originally published in the April/May 2017 issue of The
Science Teacher journal from the National Science Teachers Association (NSTA).
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Figure 1. A. Notre Dame Cathedral, Paris. B. St. Paul’s Cathedral, London.
Early Childhood and Lower Elementary Teachers Need to Attend the 6th Annual STEM Forum & Expo
By Korei Martin
Posted on 2017-05-01
Come join us at the 6th Annual STEM Forum & Expo. It is the event where early childhood and lower elementary teachers need to be. It is a unique event where you will find a great deal of focused ideas on how to engage, nurture, and teach what our little ones are already so curious about. Science, technology, engineering, and math is all around our students and they have a natural love for figuring out how things work and why. The STEM Forum is a wonderful opportunity for educators in the early years to find resources that will help build your knowledge on integrating STEM into your lessons as well as give you some great fun ideas you can take immediately back to the classroom and use the next day.
As early childhood and lower elementary teachers we tend to focus on making sure our students are first learning and practicing their foundational skills like, the alphabet, numbers, reading, and writing and perhaps forget the importance of STEM to our students’ futures. We need to teach them how to be those innovative thinkers, problem solvers, and creators of the future and it all starts with us! With a strong understanding of STEM and how to go about presenting it to young learners that is exactly what you as a teacher can do for your students. Attending the conference will give you many opportunities to find ideas that you can use in planning your literacy blocks, but also effectively integrating science or engineering into some of those already great lessons. There are so many books that lend themselves to science and engineering. Some of the sessions at this year’s STEM Forum will focus on how to use science and engineering with literature as well as give you some activities to try with your own students. Presenters are educators, like myself, that have come to share what has effectively worked for them. They bring along great resources and knowledge that they have used or found helpful in developing their lessons that they are more than happy to share with other educators.
Our young students deserve to explore and allow their natural curiosity to guide their learning. As early childhood and lower elementary teachers our job is to focus that natural curiosity and assist them in making important connections about the world and their place in it. So please, come and join us at the 6th Annual STEM Forum & Expo. You will not believe how many excellent and focused sessions for young learners you will find that will assist you in becoming a well-rounded STEM educator. We hope to see you there!
Adriana Guerra began her teaching career at E.P. Foster STEM Academy 12 years ago. She was a presenter at the 5th Annual STEM Forum & Expo in Denver, CO and is currently on the steering committee as the early childhood/lower elementary strand leader for the 6th Annual STEM Forum & Expo. She received my bachelor’s degree from University of Southern California, her Master’s degree from Bradley University and her teaching credentials from California Lutheran University.
The mission of NSTA is to promote excellence and innovation in science teaching and learning for all.
Future NSTA Conferences
2017 STEM Forum & Expo
Kissimmee/Orlando, July 12–14
2017 Area Conferences
Baltimore, October 5–7
Milwaukee, November 9–11
New Orleans, Nov. 30–Dec. 2
Come join us at the 6th Annual STEM Forum & Expo. It is the event where early childhood and lower elementary teachers need to be. It is a unique event where you will find a great deal of focused ideas on how to engage, nurture, and teach what our little ones are already so curious about. Science, technology, engineering, and math is all around our students and they have a natural love for figuring out how things work and why.
Legislative Update
FY2017 Funding, Vouchers and More
By Jodi Peterson
Posted on 2017-05-01
First up—guidance you do not want to miss: Last week, the U.S. Department of Education issued a seven page memo detailing possible uses of federal funding through ESSA to support STEM education. The must-see document has helpful examples of how states, schools and districts can leverage ESEA, IDEA, and Perkins Funds for STEM education. The letter can be found here.
Congress and the FY2017 Budget
The budget grabbed all the attention last week, as the Continuing Resolution (CR) for FY2017 programs official ran out on April 28 amid the ensuring drama over whether to shut down the federal government, actually fund FY2017 programs (we are now halfway through FY2017), extend the CR for another week, fund the wall, and finish a health care bill (Congress choose to extend the existing CR for another week).
Late Sunday night (April 30) POLITICO reported that a funding agreement was in place through the end of September. It appears that funding for Student Support and Academic Enrichment (SSAE) Title IV block grant program for fiscal year 2017, authorized by the Every Student Succeeds Act, will be at $400 million this year, a fraction of the ESSA authorization level of $1.65 billion. With the low funding level, Congress changed the distribution for this program: money will go directly to the states and will be a competitive grant program lead by the states (this year only), instead of a formula program to all schools. More on the Title IV program here.
The Trump <skinny> budget for FY2018 proposes $59 billion for the Department of Education, a $9.2 billion cut to the Education Department’s $68 billion budget, which would cut agency spending by 13 percent below the 2017 CR level. Read more here.
Update on Every Student Succeeds Act
Several states have submitted their plans to implement the Every Student Succeeds Act, and more are working to meet the September deadline for ESSA plans. Right now 10 states have plans out in some form. The U.S. Department of Education (ED) has 120 days to review the plans; if it makes no comment during that period, the plans are considered approved.
What’s not in the state plans? Expanded school choice programs, reports U.S. News & World Report. Many states are proposing to track student scores with new methods beyond just test scores, reports PBS.
In an interview last week Education Secretary Betsy DeVos told Fox News that the Every Student Succeeds Act does away with the whole argument about Common Core and “it leaves up to the states” to make decisions about which standards to choose. “There really isn’t any Common Core anymore,” stated DeVos. President Trump and his advisors have repeatedly said the Administration would repeal Common Core.
Spotlight on School Vouchers
There has been a lot of press attention on school vouchers lately. Many advocates believe that the national school voucher program President Trump has called for could be fashioned after the tax credit scholarship program in Florida—created by former Governor Jeb Bush—that now pays private school tuition for almost 100,000 low income students so that they can attend private, mostly religious schools.
A national tax credit based on the Florida model to expand school vouchers nationwide could be inserted into a tax reform package and could be passed through a simple majority in the Senate. Sen. Marco Rubio (R-Fla.), and Rep. Todd Rokita (R-Ind.) have proposed legislation that would create a program similar to the Florida tax credit scholarship at the federal level. Read more here.
In addition to Florida, currently about 17 states also offer some type of tax credit scholarships. These programs provide tax credits to individuals or corporations that donate to a state-approved organization, which then awards funds to working class families for private school tuition.
In anticipation of school voucher legislation, a group of Senate Democrats has requested that the Government Accountability Office (GAO) conduct a study to investigate the structure and accountability of tax credit scholarship programs in the states and determine whether state voucher programs have resulted in mismanagement of public funds.
Education Dept. Needs to Improve Oversight of 21st Century Program
A GAO report last week found that the Department of Education oversight of afterschool programs funded via the 21st Century Program was lacking.
From the report: “Relevant research we reviewed that compared program participants to those of non-participants suggests that the 21st Century program is effective in improving students’ behavioral outcomes, such as school-day attendance and reduced disciplinary incidents, more often than their academic outcomes. However, because Education’s current 21st Century performance measures primarily focus on students’ reading and math scores on state tests, Education lacks useful data about whether the program is achieving its objectives to improve students’ behavioral outcomes such as attendance and discipline—the areas where the program most frequently has a positive effect. Education officials have not substantially revised the program’s performance measures since 1998, in part because its authorization lapsed from fiscal years 2008 through 2016. Leading practices in performance measurement call for federal agencies to align performance measures with program objectives.”
President Trump proposed zeroing out this program in the FY2018 budget, after OMB Director Mick Mulvaney told reporters of “no demonstrable evidence” that after-school programs help students. The GAO is considered the watchdog of government.
Executive Order to Rid Schools of “Federal Overreach”
And, finally, President Trump last Wednesday signed an executive order that directs Education Secretary Betsy DeVos to determine where the “federal government has unlawfully overstepped state and local control.”
Administration officials told USA Today that “In recent years too many in Washington have advanced top-down mandates that take away autonomy and limit the options available to educators, administrators, and parents,” and the executive order puts an end to this overreach, yet admitted to reporter Greg Toppo that DeVos already was empowered to do this. Read more.
Stay tuned, and watch for more updates in future issues of NSTA Express.
Jodi Peterson is the Assistant Executive Director of Communication, Legislative & Public Affairs for the National Science Teachers Association (NSTA) and Chair of the STEM Education Coalition. Reach her via e-mail at jpeterson@nsta.org or via Twitter at @stemedadvocate.
The mission of NSTA is to promote excellence and innovation in science teaching and learning for all.
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First up—guidance you do not want to miss: Last week, the U.S. Department of Education issued a seven page memo detailing possible uses of federal funding through ESSA to support STEM education. The must-see document has helpful examples of how states, schools and districts can leverage ESEA, IDEA, and Perkins Funds for STEM education. The letter can be found here.
Congress and the FY2017 Budget