Godzilla Versus Scaling Laws of PhysicsThomas R. Tretter

Citation: The Physics Teacher 43, 530 (2005); doi: 10.1119/1.2120383 View online: http://dx.doi.org/10.1119/1.2120383 View Table of Contents: http://scitation.aip.org/content/aapt/journal/tpt/43/8?ver=pdfcov Published by the American Association of Physics Teachers Articles you may be interested in Newton's Zeroth Law: Learning from Listening to Our Students Phys. Teach. 43, 41 (2005); 10.1119/1.1845990 Speed of Sound Using Lissajous Figures Phys. Teach. 43, 36 (2005); 10.1119/1.1845989 Educational Reform Versus Classical Approach Phys. Teach. 40, 451 (2002); 10.1119/1.1526608 Case-study experiments in the introductory physics curriculum Phys. Teach. 38, 373 (2000); 10.1119/1.1321825 Looking for scaling laws, or physics with nuts and shells Phys. Teach. 37, 376 (1999); 10.1119/1.880354

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Godzilla Versus Scaling Laws of PhysicsThomas R. Tretter, University of Louisville, Louisville, KY

T he concept of how scale affects systems and organisms is central to many science disci-plines and serves as a unifying theme identi-

fied by Project 2061 as important for all students.1 This science education reform document indicates that by the end of 12th grade, “Students should know that because different properties are not affected to the same degree by changes in scale, large changes in scale typically change the way that things work in physical, biological, or social systems.”2 The focus of this paper is to provide a detailed description of a way to actively involve students in discovering a scaling effect in an interesting context. The activity described is most ap-propriate to use with beginning physics students in high school or nonmajor college students.

As early as the 17th century the importance of scal-ing effects on structural strength were noted by Galileo in a discussion of differences between a horse and a dog.3 The discussion compares a horse falling eight feet and breaking bones to a dog falling eight feet and walking away, and continues to note that a dog could carry two or three similarly sized dogs on its back, whereas a horse could not carry even one similarly sized horse. Other examples of some implications of scaling effects are explored in classic readings such as Haldane’s On Being the Right Size4 or Thompson’s On Growth and Form.5 The teaching of scaling concepts to physics students can be challenging. Scaling was in-cluded in the first chapter of the first Physical Science Study Committee (PSSC) book in the early 1960s, but was later dropped because teachers reported that stu-

dents found it too difficult.6 Some physics books have sections on scaling,7-9 but many texts present only a brief introduction to the topic. There are published examples detailing pedagogical techniques to bring scaling issues to the fore of students’ attention, such as investigations of scaling laws that relate one measur-able variable to another.10 George Barnes outlines a wide variety of interesting situations that can be incor-porated into the teaching of the physics of scale and includes an extensive reference list for additional read-ing.11 The present paper describes an exercise in which students themselves make measurements and rough calculations to discover an interesting and significant result having to do with the concept of scale. Follow-ing an activity such as this, an instructor could choose to follow up and delve more deeply into other aspects of scaling effects as suggested by some of the literature cited above.

On screen (measured with ruler):

case length = 2 cm, foot length = 28 cm, foot width = 23 cm

Real life:

Length of back = 52 cm = case length, so 1 cm on screen = 26 cm real life.Length of foot = 26 * 28 cm = 728 cmWidth of foot = 26 * 23 cm = 598 cm

Approximating her feet as rectangular, area of Godzilla's two feet = (728 cm)(598 cm)(2 of them) = 870,688 cm2

23 cm

28 cm

Fig. 1. Godzilla’s footprint.

530 DOI: 10.1119/1.2120383 THE PHYSICS TEACHER ◆ Vol. 43, November 2005 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

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THE PHYSICS TEACHER ◆ Vol. 43, November 2005 531

Context for Godzilla‘s Scaling Problem

Before this activity, students will have been taught that surface area-to-volume ratios vary according to scale. I begin the exercise by showing a clip from the 1998 movie Godzilla, starring Matthew Broderick. With the VCR counter set to 0 at the beginning of the film (bypassing any previews), at about the 10:15 mark is a scene showing Broderick standing in the footprint of Godzilla (see Fig. 1). Students measure the size of this footprint using a ruler while the scene is paused. The scale of the image is estimated from the size of a case that Broderick is carrying. A few seconds later in the film it is clear that the case is ap-proximately the same length as his back from waist to shoulder. See Fig. 1 for sample computations of the area of Godzilla’s two feet.

Having captured students’ interest, I detour away from the video for a while to discuss the concept that bone strength is related to the cross-sectional area of the bone, and that a major source of stress on the leg bones comes primarily from the weight being sup-ported. To have students get a sense of how much weight per unit cross-sectional area a bone typically supports, I start by having them compute the pressure (stress) exerted on their own leg bones. Students mea-sure the circumference of one of their ankles and from that compute the cross-sectional area of their two ankles together. It’s true that this measurement isn’t all bone, but we use this value as a rough approximation for the leg bone cross-sectional area. A typical compu-tation is as follows:

Ankle circumference = 25 cm, so r = 25 cm /(2π) = 4.0 cmArea = πr2 = 50 cm2. For two ankles, area = 100 cm2

Weight = 165 lb = 734 NPressure = (734 N)/(100 cm2) = 7.3 N/cm2

This procedure is then extended to elephants. Data for a specific adult elephant12 showing a foot circum-ference of 52 in (130 cm) and a weight of 9300 lb (4.1 x 104 N) results in a value of 7.5 N/cm2 for the pres-sure exerted by each foot. This result emphasizes that these pressures are similar across species regardless of size. Students are led to the conclusion that the pres-sures are related to the compressive strength of bone

material and can’t be greatly exceeded regardless of how large the animal may be.

Godzilla’s Size ProblemAt this point, the lesson returns to Godzilla with

the goal of estimating the pressure her ankles would have to support. Godzilla is shown on the video stomping her way through New York City begin-ning at about counter mark 26:00. As students view this clip, I ask them to estimate the area of Godzilla’s ankles as compared to the area of her feet (computed earlier). Estimates vary, but a generous estimate would be that the ankle is approximately one-quarter the area of the foot, and using the data from Fig. 1 gives a value of 2.2 x 105 cm2 for the total cross-sectional area of the two ankles.

Students next estimate Godzilla’s weight. At about counter mark 34:00, the movie shows post-rampage scenes including a building with a huge hole through it (see Fig. 2). Ignoring the impracticality of such a building being punctured this way and left standing, the size of this hole is used to estimate that Godzilla’s volume is 8.5 x 104 m3 (see Fig. 2). Assuming that Godzilla’s density is approximately that of water, 1000 kg/m3, results in a weight of 8.3 x 108 N. This value together with the ankles’ cross-sectional area gives a pressure of about 3.8 x 103 N/cm2! This is about 500 times greater pressure on the bone than for other crea-tures.

To help students get a sense of the magnitude of this

120 m

about 1/4 of height(7.5 stories)

30 m

30 storiescounted

Fig. 2. Estimating Godzilla’s volume. Using the hole in the MetLife building, count the floors and estimate each floor to be about 4 meters tall, which includes the structural supports between each floor. Assume Godzilla is roughly a cylinder the size of the hole to compute her volume.

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532 THE PHYSICS TEACHER ◆ Vol. 43, November 2005

number, I have them compute the weight human an-kles would have to support if they were subjected to the same pressure. This turns out to be about 380,000 N (85,000 lb)! Because Godzilla is merely an overgrown (due to radioactive exposure—according to the movie) lizard, her bone material should differ little from ours, and so her leg bones would never be able to support her weight. Physics wins, Godzilla loses.

I often have students calculate how big Godzilla’s ankles would have to be to support her weight, as-suming that the bones could only safely support the same pressure as that of humans. The result is that each ankle would have a diameter of about 85 m i.e., nearly three times the diameter of her whole body (see Fig. 2)! This is a good example of the ridiculousness of such a creature ever existing and an illustration of size limits due to the physics of scaling. Of course, I point out to students that the weight of this version of Godzilla would be significantly more than what we computed earlier because of the huge legs, which means the ankles would have to be even bigger to sup-port the extra weight due to the enlarged legs.

CommentsMany physics students have little difficulty un-

derstanding the mathematical development of the variable surface area-to-volume ratio at different sizes, typically employing either cubes or spheres to use a simple mathematical formula to show this variable ratio. However, I often find that the consequences of such a mathematical result don’t really hit home con-ceptually until doing an investigation such as this one. I’ve found this lesson to be an effective way to help students understand the broad concept of the phys-ics of scaling, and invariably students will come back to class with further ideas for similar explorations of other movies, such as those with house-sized spiders, gigantic ants from outer space, or giants of one sort or another. I take these opportunities to ask them how such a size would impact the need for oxygen acquisition and lung capacity or perhaps heat dissipa-tion from the skin. I then ask students to consider the other end of the spectrum, the very small. With nano-technology tools and processes being improved rap-

idly, working at a very small scale is quickly becoming a reality for more and more applications. At this point, students are often ready to explore such questions independently, and even generate other scale-related questions of their own. This demonstrates that they truly have come to understand that size does matter.

References1. American Association for the Advancement of Science

(AAAS), Benchmarks for Science Literacy (Oxford Uni-versity Press, New York, 1993).

2. Ref. 1, p. 279.

3. G. Galileo, Dialogues Concerning Two New Sciences, translated by H. Crew & A. de Salvio (Northwestern University Press, Evanston, IL, 1950, Original work published 1638).

4. J.B.S. Haldane, On Being the Right Size and Other Es-says, edited by J. M. Smith (Oxford University Press, Oxford, 1985, Original work published 1927).

5. D.A.W. Thompson, On Growth and Form, edited by J. T. Bonner (Cambridge University Press, London, 1961, Original work published 1917).

6. C.E. Swartz and T. Miner, Teaching Introductory Physics: A Sourcebook (AIP Press, New York, 1998), p. 82.

7. P.G. Hewitt, Conceptual Physics, 3rd ed. (Scott Fores-man /Addison Wesley, Menlo Park, CA, 1999).

8. A.B. Arons, Teaching Introductory Physics (Wiley, New York, 1997), Chap. 1.

9. Ref. 6, Chap. 3.

10. H.D. Sheets and J.C. Lauffenburger, “Looking for scal-ing laws, or physics with nuts and shells,” Phys. Teach. 37, 376–378 (Sept. 1999).

11. G. Barnes, “Physics and size in biological systems,” Phys. Teach. 27, 234–253 (April 1989).

12. Retrieved from http://www.indyzoo.com/pdf/what_is_an_elephant_a.pdf on April 18, 2005.

PACS codes: 01.50Fa, 01.90, 89.02

Tom Tretter has taught high school mathematics and physics for 12 years. He earned his doctorate in science education at the University of North Carolina at Chapel Hill and is currently an assistant professor of science educa-tion at the University of Louisville.

Department of Teaching and Learning, University of Louisville, Louisville, KY 40292; tom.tretter@louisville.edu

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