In this paper we summarize the theory of the "falling chimney," which deals
with the breaking of tall structures in mid-air, when they fall to the ground.
We describe how to reproduce these effects using small-scale models built with
toy blocks. We also present an improved and more effective way to perform and
analyze these interesting experiments, by using video capture software together
with a digital video camera.Comment: 6 pages, including 3 figure

Jully, Isabel R.

Varieschi, Gabriele U.

English

arXiv

Have you ever observed a child playing with toy blocks? A favorite game is to build towers and then make them topple like falling trees. To the eye of a trained physicist this should immediately look like an example of the physics of “falling chimneys,” when tall structures bend and break in mid-air while falling to the ground. The game played with toy blocks can actually reproduce well what is usually seen in photographs of falling towers, such as the one that appeared on the cover of the September 1976 issue of The Physics Teacher.1 In this paper we describe how we performed and analyzed these simple but interesting experiments with toy blocks

Loyola Marymount University

Digital Commons@Loyola Marymount Universityand Loyola Law SchoolPhysics Faculty Works Seaver College of Science and Engineering9-1-2005Toy Blocks and Rotational PhysicsGabriele U. VarieschiLoyola Marymount University, gvarieschi@lmu.eduIsabel R. JullyLoyola Marymount University, irjully@lion.lmu.eduThis Article is brought to you for free and open access by the Seaver College of Science and Engineering at Digital Commons @ Loyola MarymountUniversity and Loyola Law School. It has been accepted for inclusion in Physics Faculty Works by an authorized administrator of DigitalCommons@Loyola Marymount University and Loyola Law School. For more information, please contact digitalcommons@lmu.edu.Repository CitationVarieschi, Gabriele U. and Jully, Isabel R., "Toy Blocks and Rotational Physics" (2005). Physics Faculty Works. 17.http://digitalcommons.lmu.edu/phys_fac/17Recommended CitationG. Varieschi and I. Jully, “Toy blocks and rotational physics,” The Physics Teacher, Vol. 43 (6), 360-362, September 2005.Toy Blocks and Rotational PhysicsGabriele U. Varieschi and Isabel R. Jully  Citation: The Physics Teacher 43, 360 (2005); doi: 10.1119/1.2033522 View online: http://dx.doi.org/10.1119/1.2033522 View Table of Contents: http://scitation.aip.org/content/aapt/journal/tpt/43/6?ver=pdfcov Published by the American Association of Physics Teachers  Articles you may be interested in The Gravitational Landscape of the Solar System Phys. Teach. 46, 363 (2008); 10.1119/1.2971221  FALLING APPLE Phys. Teach. 44, 490 (2006); 10.1119/1.2362937  Battle of the Merry-Go-Rounds: An Angular Momentum Demonstration Phys. Teach. 44, 304 (2006); 10.1119/1.2195403  DOING physics -- Physics activities for groups Phys. Teach. 44, 53 (2006); 10.1119/1.2150762  Using Experience to Understand Centripetal Force Phys. Teach. 43, 391 (2005); 10.1119/1.2033532    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:157.242.200.164 On: Fri, 28 Oct 2016 23:53:56Toy Blocks and Rotational PhysicsGabriele U. Varieschi and Isabel R. Jully, Loyola Marymount University, Los Angeles, CA360                                                   DOI: 10.1119/1.2033522 THE PHYSICS TEACHER ◆ Vol. 43, September 2005H ave you ever observed a child playing with toy blocks? A favorite game is to build tow-ers and then make them topple like falling trees. To the eye of a trained physicist this should im-mediately look like an example of the physics of “fall-ing chimneys,” when tall structures bend and break in mid-air while falling to the ground. The game played with toy blocks can actually reproduce well what is usually seen in photographs of falling towers, such as the one that appeared on the cover of the September 1976 issue of The Physics Teacher.1 In this paper we describe how we performed and analyzed these simple but interesting experiments with toy blocks.One of us recently published a detailed paper2 summarizing the physics of the rotational motion of tall structures falling under gravity. In this analysis it is assumed that the structure falls while maintaining a point of contact at the bottom so that its motion is essentially a rotation around this pivot point, and the only external forces are the weight W of the body and a constraint force F at the base (see Fig. 1). We performed experiments using models made with toy blocks and filmed the falling motion with a digital video camera. These video clips and still pictures of the models can be viewed on a related web page.3A classical physics demonstration, usually called the “falling stick” or “hinged stick,” works on the same principle, showing that the acceleration of gravity can be exceeded during the fall by some points of the body (see again the review paper2 and references therein). The difference between a falling stick and a real falling tower (or the toy models we used) is that the latter is not rigid and will usually bend and break during the fall. This can be seen in Fig. 2, and it is well documented by all the photos and videos on our website. 3The Causes of the BreakingThe internal forces causing the tower to bend and break are also shown in Fig. 1. We use polar coordi-nates r and θ to describe the rotational motion of the body around the point of contact with the ground ryxNbPSWF Hθ0Fig. 1. The falling tower or chimney described as a rotat-ing uniform stick. The external forces, the weight W and the constraint force F at the base, are shown in blue. The internal forces (P, S) and the bending moment Nb, at an arbitrary cross section, are shown in red. 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:157.242.200.164 On: Fri, 28 Oct 2016 23:53:56at the origin for a structure of total height H. For an arbitrary cross section of the tower at a distance r from the origin, the internal forces can be modeled as a longitudinal stress force P (either a tension or a compression), a transverse shearing force S, and a “bending moment” Nb, represented by the curved ar-row in Fig. 1. This bending moment is the main cause of the bending and breaking of the structure and can be thought of as originating from a “pair” of forces ap-plied at the leading and trailing edges of the tower.Complete details about the calculation of these forces are given in Ref. 2. Summarizing these results, we observe that the tower can break in two possible ways. In the first case, a particularly strong transverse shear force S can “cut” the tower. This is more likely to happen for real structures (such as falling chimneys, towers, etc.), and the breaking typically occurs near the bottom, where S is larger. In the second case, by far the most common with small-scale models, the structure progressively bends and breaks, due to a combined effect of the bending moment Nb and the longitudinal force P.This combined effect is usually better described by the longitudinal stress at the leading edge L, which can be computed from the previous quantities and de-pends on the tilt angle θ, the height fraction r /H, and the side a of the square cross section of the tower:4THE PHYSICS TEACHER ◆ Vol. 43, September 2005 361      (1)σθLaWrHrHrH2 121 5 3 3 1=− − + − +cos+ −3212HarHrHsin .θ In this equation the leading edge stress L is nor-malized by dividing by the factor W/a2 in order to obtain a dimensionless quantity. For a given value of the ratio H/a (total height H divided by the side a of the tower), Eq. (1) relates L directly to the variables  θ and r /H. By plotting this function of two vari-ables, it is easy to see which values of θ and r /H will maximize L, determining the angle and height frac-tion at which the structure is more likely to break.Toy Block ModelsTo check the theory outlined above, we set up models built with different types of toy blocks. We in-duced their fall, making sure they would start rotating around an appropriate support at the bottom. Details of the blocks we used and the dimensions of the mod-els can be found in Refs. 2 and 3. These experiments are very simple and inexpensive and can be made part of a laboratory class devoted to rotational mechanics or used as physics demonstrations.Fig. 2. A toy model made with wooden blocks. The struc-ture appears to bend and break at r /H ≅ 0.354 and at an angle θ = 20o–25o. VideoPoint software is used to show the progressive bending of the tower. Fig. 3. The normalized longitudinal stress at the leading edge is shown as a function of the height fraction and for several angles. The maxima of the stress curves are marked by solid points. Curves in red are related to the breaking pattern of the structure shown in the previous figure. 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:157.242.200.164 On: Fri, 28 Oct 2016 23:53:56In order to improve on our previous work, we used video capture software (VideoPoint 2.5), together with a digital video camera, to record and analyze the mo-tion. An example of a picture frame taken from one of these video recordings is shown in Fig. 2, while the complete video clips can be viewed on our web page.3  In Fig. 2 we observe the fall of a tower made of 24 cubic wooden blocks (H/a = 24), which appears to bend around the ninth block from the bottom (for  r /H ≅ 0.354). The use of the software allows for a better determination of the breaking angle θ, which is estimated to be around 20–25. As can be seen from the picture, the software can follow the different rota-tions of the top and bottom portions of the structure, showing the point where the tower begins to bend.In Fig. 3 we plot the quantities of Eq. (1) for the case of our toy model with H /a = 24. The normalized stress at the leading edge is shown as a function of the height ratio r /H for different tilt angles. For a given angle, the structure should break at the point where the stress is maximum (corresponding to the solid points in the figure).We notice a good agreement with the behavior of our toy model from Fig. 2. For a breaking angle around 20–25, Fig. 3 predicts a breaking height ratio r /H  ≅ 0.35-0.37 (considering the maxima of the red curves and the related dashed green lines in Fig. 3). Our structure seems in fact to be bending and break-ing at r /H ≅ 0.354, as mentioned above.Similar experiments can be performed with differ-ent types of blocks or varying dimensions of the tow-ers and will typically show results consistent with the theory outlined here. ConclusionToy blocks can easily be used to illustrate some peculiar effects of rotational dynamics. These experi-ments can be effectively integrated into classroom or lab activities at various levels. They can be shown as simple demonstrations of rotational physics in intro-ductory classes, or become more challenging activities for advanced mechanics courses or laboratories.AcknowledgmentsThis research was supported by an award from Research Corp. The authors would like to thank  V. Coletta for the useful discussions regarding this paper.References1. Albert A. Bartlett, “More on the falling chimney,” Phys. Teach. 14, 351–353 (Sept. 1976); and cover photo of the same issue.2. Gabriele Varieschi and Kaoru Kamiya, “Toy models for the falling chimney,” Am. J. Phys. 71 (10), 1025–1031 (Oct. 2003). See also the references in this paper for a complete bibliography on the subject.3. The Falling Chimney Web Page (http://myweb.lmu.edu/gvarieschi/chimney/chimney.html). This site contains all our photos and videos, plus links to other related pages.4. Assuming a square cross section for our structures is useful to simplify the calculations. Results for circular cross-sectional chimneys or similar structures do not change significantly.PACs codes: 01.50M, 01.55, 01.76T, 46.02CGabriele U. Varieschi is an assistant professor in the phys-ics department at Loyola Marymount University. He earned his Ph.D. in theoretical particle physics from the University of California at Los Angeles. His research interests are in the area of astro-particle physics and cosmology.Department of Physics, Loyola Marymount University, 1 LMU Drive, Los Angeles, CA 90045; gvarieschi@lmu.eduIsabel R. Jully is attending Loyola Marymount University in the B.S. engineering physics program.Department of Physics, Loyola Marymount University, 1 LMU Drive, Los Angeles, CA 90045; irjully@lion.lmu.edu362 THE PHYSICS TEACHER ◆ Vol. 43, September 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:157.242.200.164 On: Fri, 28 Oct 2016 23:53:56

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