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Backpack Weight and the Scaling of the Human Frame
Abstract
Modeling real-life situations is an important part of introductory physics. Here we consider the question “What is the largest weight of backpack a hiker can manage?” A quick perusal of the Internet suggests that as the weight of a healthy adult increases, the largest backpack weight W bp also increases and should be about 25–30% of a person's body weight for a reasonably fit adult.1 We show here that a careful modeling of the hiker and backpack leads to a somewhat different result, with hikers of sufficiently large (but otherwise healthy) weight not being able to carry as much backpack weight as hikers of smaller weight.
The backpack weight a hiker is capable of carrying depends on several factors. These include the hiker’s weight, body mass index (BMI), fitness, and training. It can also depend on the terrain on which the hiker travels, e.g., off-trail or on-trail. However, online advice has tended to focus on the hiker’s weight and suggests that pack weight should be a certain percentage of the hiker’s weight. We developed a model of how the ability of a hiker to carry a backpack depends on the weight of the hiker. Modeling such real-life situations helps grab the attention of students in introductory physics. We have expanded on our earlier idea and present a more sophisticated model of an adult backpacker. In particular, we generalize our model for scaling of the human frame, and investigate how the ability to carry a backpack (within our model) varies with the form of scaling used.
Arguments based on elastic stability and flexure, as opposed to the more conventional ones based on yield strength, require that living organisms adopt forms whereby lengths increase as the ⅔ power of diameter. The somatic dimensions of several species of animals and of a wide variety of trees fit this rule well.
It is a simple matter to show that energy metabolism during maximal sustained work depends on body cross-sectional area, not total body surface area as proposed by Rubner ( 1 ) and many after him. This result and the result requiring animal proportions to change with size amount to a derivation of Kleiber's law, a statement only empirical until now, correlating the metabolically related variables with body weight raised to the ¾ power. In the present model, biological frequencies are predicted to go inversely as body weight to the ¼; power, and total body surface areas should correlate with body weight to the ⅝ power. All predictions of the proposed model are tested by comparison with existing data, and the fit is considered satisfactory.
In The Fire of Life, Kleiber ( 5 ) wrote "When the concepts concerned with the relation of body size and metabolic rate are clarified, . . . then compartive physiology of metabolism will be of great help in solving one of the most intricate and interesting problems in biology, namely the regulation of the rate of cell metabolism." Although Hill ( 23 ) realized that "the essential point about a large animal is that its structure should be capable of bearing its own weight and this leaves less play for other factors," he was forced to use an oversimplified "geometric similarity" hypothesis in his important work on animal locomotion and muscular dynamics. It is my hope that the model proposed here promises useful answers in comparisons of living things on both the microscopic and the gross scale, as part of the growing science of form, which asks precisely how organisms are diverse and yet again how they are alike.
Scaling laws relating the volume of a class of objects to a characteristic dimension of the object appear commonly in physics, chemistry, and biology. In this laboratory exercise for an introductory physics course scaling laws are derived for machine nuts and clam shells. In addition to covering a standard problem in physics, determining volume of the object by measuring the buoyant force on it, the biologically interesting idea of scaling laws are incorporated into the same lab.
The concept of how scale affects systems and organisms is central to
many science disciplines and serves as a unifying theme identified 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.''
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 appropriate to use
with beginning physics students in high school or nonmajor college
students.
Described is the subject of biological scaling for physics teachers including examples and in-depth reading. Topics are elements of scaling, terminal velocities, Lilliputian and Brobdingnagian, brain evolution, dolphin echolocation, surface tension, gravity change, food and oxygen, and seeing. Ten references on physics and size, and ten questions are appended. (YP)
The classic book on human movement in biomechanics, newly updated. Widely used and referenced, David Winter's Biomechanics and Motor Control of Human Movement is a classic examination of techniques used to measure and analyze all body movements as mechanical systems, including such everyday movements as walking. It fills the gap in human movement science area where modern science and technology are integrated with anatomy, muscle physiology, and electromyography to assess and understand human movement. In light of the explosive growth of the field, this new edition updates and enhances the text with: Expanded coverage of 3D kinematics and kinetics. New materials on biomechanical movement synergies and signal processing, including auto and cross correlation, frequency analysis, analog and digital filtering, and ensemble averaging techniques. Presentation of a wide spectrum of measurement and analysis techniques. Updates to all existing chapters. Basic physical and physiological principles in capsule form for quick reference. An essential resource for researchers and student in kinesiology, bioengineering (rehabilitation engineering), physical education, ergonomics, and physical and occupational therapy, this text will also provide valuable to professionals in orthopedics, muscle physiology, and rehabilitation medicine. In response to many requests, the extensive numerical tables contained in Appendix A: "Kinematic, Kinetic, and Energy Data" can also be found at the following Web site: www.wiley.com/go/biomechanics.
Arguments based on elastic stability and flexure, as opposed to the more conventional ones based on yield strength, require that living organisms adopt forms whereby lengths increase as the (2/3) power of diameter. The somatic dimensions of several species of animals and of a wide variety of trees fit this rule well. It is a simple matter to show that energy metabolism during maximal sustained work depends on body cross-sectional area, not total body surface area as proposed by Rubner (1) and many after him. This result and the result requiring animal proportions to change with size amount to a derivation of Kleiber's law, a statement only empirical until now, correlating the metabolically related variables with body weight raised to the (3/4) power. In the present model, biological frequencies are predicted to go inversely as body weight to the (1/4) power, and total body surface areas should correlate with body weight to the (5/8) power. All predictions of the proposed model are tested by comparison with existing data, and the fit is considered satisfactory. In The Fire of Life, Kleiber (5) wrote "When the concepts concerned with the relation of body size and metabolic rate are clarified, . . . then compartive physiology of metabolism will be of great help in solving one of the most intricate and interesting problems in biology, namely the regulation of the rate of cell metabolism." Although Hill (23) realized that "the essential point about a large animal is that its structure should be capable of bearing its own weight and this leaves less play for other factors," he was forced to use an oversimplified "geometric similarity" hypothesis in his important work on animal locomotion and muscular dynamics. It is my hope that the model proposed here promises useful answers in comparisons of living things on both the microscopic and the gross scale, as part of the growing science of form, which asks precisely how organisms are diverse and yet again how they are alike.
The maximum voluntary force (strength) which could be produced by the knee-extensor muscles, with the knee held at a right angle, was measured in a group of healthy young subjects comprising twenty-five males and twenty-five females. Both legs were tested: data from the stronger leg only for each subject were used in the present study. Computed tomography was used to obtain a cross-sectional image of the subjects' legs at mid-thigh level, measured as the mid-point between the greater trochanter and upper border of the patella. The cross-sectional area of the knee-extensor muscles was determined from the image obtained by computer-based planimetry. The subjects' height and weight were measured. An estimate of body fat content was obtained from measurements of skinfold thicknesses and used to calculate lean body mass. Male subjects were taller (P less than 0.001), heavier (P less than 0.001), leaner (P less than 0.001) and stronger (P less than 0.001) than the female subjects. No significant correlation was found to exist between strength of the knee-extensor muscles and body weight in the male or in the female subjects. In the male subjects, but not in the female group, there was a positive correlation (r = 0.50; P less than 0.01) between strength and lean body mass. Muscle cross-sectional area of the male subjects was greater than that of the female subjects (P less than 0.001). The ratio of strength to cross-sectional area for the male was 9.49 +/- 1.34 (mean +/- S.D.). This is greater but not significantly so, than that for females (8.92 +/- 1.11). In both male and female groups, there was a significant (P less than 0.01) positive correlation between muscle strength and cross-sectional area. A wide variation in the ratio of strength to muscle cross-sectional area was observed. This variability may be a result of anatomical differences between subjects or may result from differences in the proportions of different fibre types in the muscles. The variation between subjects is such that strength is not a useful predictive index of muscle cross-sectional area.
- Irving P Herman
Irving P. Herman, Physics of the Human Body (Springer-Verlag,
Berlin, 2007), see Table 1.13 and references therein.
- P Irving
- Herman
Irving P. Herman, Physics of the Human Body (Springer-Verlag,
Berlin, 2007), see Table 1.13 and references therein.