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Agility and body sturdiness of primitive man in comparison to contemporary man
V. Capecchi
Agility and Body Sturdiness of Primitive Man in Comparison to Contemporary Man
Institute qf AnthropologT, University of Sio~a, Siena, Italy
Demonstration that a hypothetical primitive man, smaller in size than a contemporary man but with the same anthropometrical proportions, is not only intrinsically stronger but is structurally more robust.
M. Rigato h~stitute of Physics, University of Siena, Siena, Italy
Received 14 November 1980 and accepted 25 .January 1982 Ke~wor&": Paleoanthropology,
biomechanics, human scale models.
1. I n t r o d u c t i o n It is admitted that primitive man was favored by his small body size (Hofstetter, 1972; Genet-Varcin, 1979). The aim of this study is not only to prove mathematically that advantage, but also to evaluate it quantitatively. With reference to the small sizes found in the paleoanthropological remains of H a d a r in Ethiopia and of Laetolil in Tanzania (Johanson & White, 1979), a previous work set out the basis for a comparison between a contemporary man U of body height h = 180 cm and a hypothetical primitive man U ' of body height h' = 112.5 cm, both having identical anthropometrical proportions, same body density and being capable of the same maximum strength per unit of muscular section (Capecchi & Rigato, 1980). With the introduction of the scale factor L = h/h' = 1.6, it was shown that U' turns out to possess a relative strength (strength/weight ratio) 60 ~ better than U's, and therefore U' is capable of springing and climbing with such rapidity and boldness that would be utterly impossible to U. Proceeding now in the study of comparisons, other consequences of the differences in the physique are determined here below. 2. J u m p Consider U in the act of jumping, from a standing position, suddenly extending his legs: if So is the elevation that the barycenter achieves in the process ofextention of the subject's legs, and F,~ is the relevant average value of the applied force, the muscular work is the product FmSo. The equivalent increase in the potential energy is given by the product of the weight mg by the elevation s achieved by the barycenter at the maximum height of the jump, hence F,,,So = m g s, s --
Fm mg
So = f,~ So
where evidently fro is the relative force. Journal of Human Evolution (1982) U, 441445
0047-2484/82/060441+05 $03.00/0
9 1982 Academic Press Inc. (London) Limited
442
v.
CAPECCHI
A N D M. R I G A T O
I n the previous work, referred to above, it was shown that f , , cx: L 1. Now, as so is found to be a length proportional to the person's height and therefore so oc L, the result can be seen as s c < L - 1 L , which is s c < L ~ that is to say that s is found to be invariable as regards the scale factor. Theretbre, if U' does the same movement, the total elevation obtained by his barycenter is again s. I n conclusion, the height of the j u m p is the same. It follows that in as m u c h as j u m p i n g over obstacles is concerned, U' is not at all at a disadvantage because of his smaller size.
3.
Velochy
Consider U running at a speed v, in first approximation to be taken as constant. Such speed is somewhat limited by air resistance, which at present stage is disregarded; to a larger extent, it is limited by the vertical movements of the b a r y c e n t e r - - a g a i n , these m a y be neglected because they do not affect the comparison between U and U', as was seen. I n the main, however, speed is limited by the necessity of constantly accelerating and decelerating the legs. So that the terms of the problem m a y be established, it is noted here that the velocity of U is the velocity of his barycenter referred to the one foot touching the ground, equal to the speed of the other foot referred to the barycenter, i.e. referred to the relevant articulation. Hence, treating a leg as a system b o u n d to an axis and maintained in forced oscillation, the m a x i m u m kinetic energy Iw2/2 (I being the m o m e n t of inertia and co the m a x i m u m angular velocity) is equivalent to the work done by the muscles in action, comparable for our present purposes to the work done by a muscular force F,~ for a shift d of its point of application :
89
m=N/~F'~d I
Since the velocity v that we w a n t to determine is given by the product of a) by the length / of the leg, we obtain
v = co l = N / 2 F~ d l2 I T h e m o m e n t of inertia is given by the expression I::[
t"
J
x 2 d i n = m Tv."
where symbols are selfoexplanatory. Hence, having noted that F~ is proportional to the muscular section and therefore to L 2, that ~, d, l are lengths proportional to L and that the mass m is proportional to the volume and therefore to L a, we find vc
HOMINID'S AGILITY
443
However, this result has been reached neglecting the air resistance R, which is now considered. For the purpose of the present comparison, it is sufficient to note that R is proportional to the surface (of the body image projected on a plane perpendicular to the direction of the motion) and to the square of the speed, and that, therefore, it is proportional to L ~. I t follows that also the ratio F ~ / R between motive power and air resistance does not vary as regards the scale factor. Accordingly, it m a y be concluded that U', in spite of his smaller size, is capable of running at the same speed as U.
4. Axial Compression and Traction Adding now the further hypothesis that corresponding elements of the two skeletons be made of bone tissues having the same mechanical properties, we show other effects relevant to the strength of the bone structure. Consider U standing. Take S to be the section of his body, with an arbitrary horizontal plane, and m to be the mass above it. The pressure p on S is mg
p = -~-, whence it can be seen that p oc L. Calling p' the pressure on the corresponding section S' of U', we have p=Lp',
whence p' -- ~P ~0"6 p.
It follows that the strain of axial compression due to his own weight is for U' 60 ~ of U's. IrL particular, if we identify S as the sole of a foot, it can be seen that walking in soft ground U' tends to sink much less than U or, conversely, that he encounters the same difficulties only when carrying a load equal to 60 ~ of his own weight. Analogous considerations m a y be made regarding traction strains, where p is taken to represent the relevant unit strain. Further, as S may be taken to symbolize just the section of the bone tissue, the conclusions reached become applicable to the whole structure of the skeleton.
5. Flexion, Shearing and Torsion Even with regard to flexion, shearing and torsion strains we must reach the conclusion that U' is always in more advantageous conditions than U, because any structure subject to a stress proportional to its mass presents a resistance which increases with the decreasing of the scale factor. I n fact, if we consider a bar of rectangular section subjected to elastic flexion, the bending angle /3 (that is, in the case of small deformations, the ratio between deflection s and length l) is given by the formula s
1 Fl ~
444
V. C A P E C C I I I AND M. R I G A T O
where J is the flexion modulus, F the bending force, l the length, b the width, a the bar thickness. As J depends exclusively on the material ( J o e l ~ and F is in the hypothesis proportional to the mass and therefore to the volume (FooLS), and as l, b, a are values proportional to L, we have
]~ c< L, that is to say, the flexion is proportional to the scale factor, and as regards the relative strength the opposite property applies. I f we now consider a parallelepipedon subjected to elastic shear strain, the sliding angle is given by the formula 1F --" G S "
where G is the shear modulus, F the force applied and S the base area. Like in the above, we have G o c L ~ and hence 4 ~ L Therefore, also shear deformation is proportional to tile scale factor, and the opposite property applies as concerns the relevant resistance. Last, a cylindrical bar subjected to elastic torsion is considered. The torsion angle 0~ is given by the formula 1Ml K r4
where K is the torsion modulus, M the torsion moment, l the length and r the bar radius. As M, being the moment of a couple, is given by the product of a force and its lever a r m (which in the hypothesis are proportional to the mass and to linear dimensions of the bar respectively), being K oc L~ we find ~ocL which shows that even the resistance to torsion is inversely proportional to the scale factor.
6. C o n c l u s i o n s It is pointed out that the above results stem exclusively from a comparison of dimensions, hence such results are the consequence of general physical properties. Moreover, our observations do not at all entail that biological tissues be assimilable to homogeneous and/or isotropous materials. Accordingly, the actual mechanical characteristics of the tissues under consideration are unimportant for the purposes of the present study, as the surmise is merely a comparison of structures subjected to identical conditions. The results so far obtained indicate that the hypothetical primitive man, thanks to his smaller size and therefore to his greater relative force, is not only capable of springing and climbing much quicker than the contemporary man, but of running at the same speed and of j u m p i n g over the same obstacles.
HOMINID'S AGILITY
445
Furthermore, if one takes into account that the paleoanthropological remains above referred to show prints of muscular insertions m u c h deeper a n d wider than in today's man, it must be admitted that the primitive m a n had an actual d y n a m i c advantage even greater t h a n that determined through the hypothesis o f a n t h r o p o m e t r i c a l similarity. Moreover, within the limits imposed by the hypothesis and the approximations adopted here, w e find, with particular reference to the bone framework but also and for the same reasons to the entire b o d y structure, that a smalt b o d y size adds to the other advantages a better mechanical strength, which entails an improved protection against the consequences of collisions in general and of falls in particular. T h e previous conclusion, according to which the ascertained d y n a m i c characteristics do constitute conditions of remarkable advantage for the needs--essential for s u r v i v a l - of hunting for food and avoiding environmental dangers, is therefore reaffirmed.
References
Gapecchi, V. & Rigato, M. (1980). The strerlgth/weight ratio in primitive man as compared to that of contemporary man. Journal of Haman Evolution 9, 325-327. Genet-Varcin, E. (1979). Les I-Iommes Fossiles. Paris: Soe. Edit. nouv Boubde. Hofstetter, R. (1972). Les caractdres lcs plus 61evds, leur origine et leur signification biologique. Act. X V H Congrgs Zoologie Montecarlo, p. 24. Johanson, D. C. & White, T. D. (1979). A systematic assessment of early African hominids. Science 203, 321-330.
Agility and Body Sturdiness of Primitive Man in Comparison to Contemporary Man
Institute qf AnthropologT, University of Sio~a, Siena, Italy
Demonstration that a hypothetical primitive man, smaller in size than a contemporary man but with the same anthropometrical proportions, is not only intrinsically stronger but is structurally more robust.
M. Rigato h~stitute of Physics, University of Siena, Siena, Italy
Received 14 November 1980 and accepted 25 .January 1982 Ke~wor&": Paleoanthropology,
biomechanics, human scale models.
1. I n t r o d u c t i o n It is admitted that primitive man was favored by his small body size (Hofstetter, 1972; Genet-Varcin, 1979). The aim of this study is not only to prove mathematically that advantage, but also to evaluate it quantitatively. With reference to the small sizes found in the paleoanthropological remains of H a d a r in Ethiopia and of Laetolil in Tanzania (Johanson & White, 1979), a previous work set out the basis for a comparison between a contemporary man U of body height h = 180 cm and a hypothetical primitive man U ' of body height h' = 112.5 cm, both having identical anthropometrical proportions, same body density and being capable of the same maximum strength per unit of muscular section (Capecchi & Rigato, 1980). With the introduction of the scale factor L = h/h' = 1.6, it was shown that U' turns out to possess a relative strength (strength/weight ratio) 60 ~ better than U's, and therefore U' is capable of springing and climbing with such rapidity and boldness that would be utterly impossible to U. Proceeding now in the study of comparisons, other consequences of the differences in the physique are determined here below. 2. J u m p Consider U in the act of jumping, from a standing position, suddenly extending his legs: if So is the elevation that the barycenter achieves in the process ofextention of the subject's legs, and F,~ is the relevant average value of the applied force, the muscular work is the product FmSo. The equivalent increase in the potential energy is given by the product of the weight mg by the elevation s achieved by the barycenter at the maximum height of the jump, hence F,,,So = m g s, s --
Fm mg
So = f,~ So
where evidently fro is the relative force. Journal of Human Evolution (1982) U, 441445
0047-2484/82/060441+05 $03.00/0
9 1982 Academic Press Inc. (London) Limited
442
v.
CAPECCHI
A N D M. R I G A T O
I n the previous work, referred to above, it was shown that f , , cx: L 1. Now, as so is found to be a length proportional to the person's height and therefore so oc L, the result can be seen as s c < L - 1 L , which is s c < L ~ that is to say that s is found to be invariable as regards the scale factor. Theretbre, if U' does the same movement, the total elevation obtained by his barycenter is again s. I n conclusion, the height of the j u m p is the same. It follows that in as m u c h as j u m p i n g over obstacles is concerned, U' is not at all at a disadvantage because of his smaller size.
3.
Velochy
Consider U running at a speed v, in first approximation to be taken as constant. Such speed is somewhat limited by air resistance, which at present stage is disregarded; to a larger extent, it is limited by the vertical movements of the b a r y c e n t e r - - a g a i n , these m a y be neglected because they do not affect the comparison between U and U', as was seen. I n the main, however, speed is limited by the necessity of constantly accelerating and decelerating the legs. So that the terms of the problem m a y be established, it is noted here that the velocity of U is the velocity of his barycenter referred to the one foot touching the ground, equal to the speed of the other foot referred to the barycenter, i.e. referred to the relevant articulation. Hence, treating a leg as a system b o u n d to an axis and maintained in forced oscillation, the m a x i m u m kinetic energy Iw2/2 (I being the m o m e n t of inertia and co the m a x i m u m angular velocity) is equivalent to the work done by the muscles in action, comparable for our present purposes to the work done by a muscular force F,~ for a shift d of its point of application :
89
m=N/~F'~d I
Since the velocity v that we w a n t to determine is given by the product of a) by the length / of the leg, we obtain
v = co l = N / 2 F~ d l2 I T h e m o m e n t of inertia is given by the expression I::[
t"
J
x 2 d i n = m Tv."
where symbols are selfoexplanatory. Hence, having noted that F~ is proportional to the muscular section and therefore to L 2, that ~, d, l are lengths proportional to L and that the mass m is proportional to the volume and therefore to L a, we find vc
HOMINID'S AGILITY
443
However, this result has been reached neglecting the air resistance R, which is now considered. For the purpose of the present comparison, it is sufficient to note that R is proportional to the surface (of the body image projected on a plane perpendicular to the direction of the motion) and to the square of the speed, and that, therefore, it is proportional to L ~. I t follows that also the ratio F ~ / R between motive power and air resistance does not vary as regards the scale factor. Accordingly, it m a y be concluded that U', in spite of his smaller size, is capable of running at the same speed as U.
4. Axial Compression and Traction Adding now the further hypothesis that corresponding elements of the two skeletons be made of bone tissues having the same mechanical properties, we show other effects relevant to the strength of the bone structure. Consider U standing. Take S to be the section of his body, with an arbitrary horizontal plane, and m to be the mass above it. The pressure p on S is mg
p = -~-, whence it can be seen that p oc L. Calling p' the pressure on the corresponding section S' of U', we have p=Lp',
whence p' -- ~P ~0"6 p.
It follows that the strain of axial compression due to his own weight is for U' 60 ~ of U's. IrL particular, if we identify S as the sole of a foot, it can be seen that walking in soft ground U' tends to sink much less than U or, conversely, that he encounters the same difficulties only when carrying a load equal to 60 ~ of his own weight. Analogous considerations m a y be made regarding traction strains, where p is taken to represent the relevant unit strain. Further, as S may be taken to symbolize just the section of the bone tissue, the conclusions reached become applicable to the whole structure of the skeleton.
5. Flexion, Shearing and Torsion Even with regard to flexion, shearing and torsion strains we must reach the conclusion that U' is always in more advantageous conditions than U, because any structure subject to a stress proportional to its mass presents a resistance which increases with the decreasing of the scale factor. I n fact, if we consider a bar of rectangular section subjected to elastic flexion, the bending angle /3 (that is, in the case of small deformations, the ratio between deflection s and length l) is given by the formula s
1 Fl ~
444
V. C A P E C C I I I AND M. R I G A T O
where J is the flexion modulus, F the bending force, l the length, b the width, a the bar thickness. As J depends exclusively on the material ( J o e l ~ and F is in the hypothesis proportional to the mass and therefore to the volume (FooLS), and as l, b, a are values proportional to L, we have
]~ c< L, that is to say, the flexion is proportional to the scale factor, and as regards the relative strength the opposite property applies. I f we now consider a parallelepipedon subjected to elastic shear strain, the sliding angle is given by the formula 1F --" G S "
where G is the shear modulus, F the force applied and S the base area. Like in the above, we have G o c L ~ and hence 4 ~ L Therefore, also shear deformation is proportional to tile scale factor, and the opposite property applies as concerns the relevant resistance. Last, a cylindrical bar subjected to elastic torsion is considered. The torsion angle 0~ is given by the formula 1Ml K r4
where K is the torsion modulus, M the torsion moment, l the length and r the bar radius. As M, being the moment of a couple, is given by the product of a force and its lever a r m (which in the hypothesis are proportional to the mass and to linear dimensions of the bar respectively), being K oc L~ we find ~ocL which shows that even the resistance to torsion is inversely proportional to the scale factor.
6. C o n c l u s i o n s It is pointed out that the above results stem exclusively from a comparison of dimensions, hence such results are the consequence of general physical properties. Moreover, our observations do not at all entail that biological tissues be assimilable to homogeneous and/or isotropous materials. Accordingly, the actual mechanical characteristics of the tissues under consideration are unimportant for the purposes of the present study, as the surmise is merely a comparison of structures subjected to identical conditions. The results so far obtained indicate that the hypothetical primitive man, thanks to his smaller size and therefore to his greater relative force, is not only capable of springing and climbing much quicker than the contemporary man, but of running at the same speed and of j u m p i n g over the same obstacles.
HOMINID'S AGILITY
445
Furthermore, if one takes into account that the paleoanthropological remains above referred to show prints of muscular insertions m u c h deeper a n d wider than in today's man, it must be admitted that the primitive m a n had an actual d y n a m i c advantage even greater t h a n that determined through the hypothesis o f a n t h r o p o m e t r i c a l similarity. Moreover, within the limits imposed by the hypothesis and the approximations adopted here, w e find, with particular reference to the bone framework but also and for the same reasons to the entire b o d y structure, that a smalt b o d y size adds to the other advantages a better mechanical strength, which entails an improved protection against the consequences of collisions in general and of falls in particular. T h e previous conclusion, according to which the ascertained d y n a m i c characteristics do constitute conditions of remarkable advantage for the needs--essential for s u r v i v a l - of hunting for food and avoiding environmental dangers, is therefore reaffirmed.
References
Gapecchi, V. & Rigato, M. (1980). The strerlgth/weight ratio in primitive man as compared to that of contemporary man. Journal of Haman Evolution 9, 325-327. Genet-Varcin, E. (1979). Les I-Iommes Fossiles. Paris: Soe. Edit. nouv Boubde. Hofstetter, R. (1972). Les caractdres lcs plus 61evds, leur origine et leur signification biologique. Act. X V H Congrgs Zoologie Montecarlo, p. 24. Johanson, D. C. & White, T. D. (1979). A systematic assessment of early African hominids. Science 203, 321-330.