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.

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 1. See http://www.backpacking.net/packwate.html and http://www.wildbackpacker.com/backpacking-gear/backpacks/how-to-pack-a-backpack/ 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 thesuggests that as the weight of a healthy adult increases, the largest backpack weightalso increases and should be about 25–30% of a person's body weight for a reasonably fit adult.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.

Observations by the author over the last 20 years of students carrying backpacks on extended (> seven days) Outward Bound wilderness trips suggests that healthy lower weight adults can carry at least as much weight as higher weight adults. In particular, students of mass over 100 kg (220 lb) who are not overweight according to their BMI (Body Mass Index) struggle to carry the same amount of weight as 60-kg (132-lb) students. These observations suggest that strength, which is expected to increase with human frame size, is not the only determining factor in how heavy a backpack a hiker can carry. This got the author thinking about what other factors might be important. An important missing ingredient in the above discussion is that the hiker carries his own body weight (in addition to his backpack), and clearly this is larger for the heavier hiker.

2,3 Looking for scaling laws, or physics with nuts and shells ,” Phys. Teach. 37, 376– 378 (Sept. 1999). 2. H. David Sheets and James. C. Lauffenburger, “,” Phys. Teach., 376–(Sept. 1999). https://doi.org/10.1119/1.880354 Godzilla versus scalinglaws of physics ,” Phys. Teach. 43, 530– 532 (Nov. 2005). 3. Thomas R. Tretter, “,” Phys. Teach., 530–(Nov. 2005). https://doi.org/10.1119/1.2120383 3–6 Godzilla versus scalinglaws of physics ,” Phys. Teach. 43, 530– 532 (Nov. 2005). 3. Thomas R. Tretter, “,” Phys. Teach., 530–(Nov. 2005). https://doi.org/10.1119/1.2120383 Physics and size in biological systems ,” Phys. Teach. 27, 234– 253 (April 1989). 4. George Barnes “,” Phys. Teach., 234–(April 1989). https://doi.org/10.1119/1.2342746 5. David A. Winter, Biomechanics and Motor Control of Human Movement, 4th ed. ( Wiley , New York , 2009), see Table 4.1. Size and shape in biology ,” Sci. 179, 1201– 1204 (1973). 6. Thomas McMahon, “,” Sci., 1201–(1973). https://doi.org/10.1126/science.179.4079.1201 W h , scales as the volume of the hiker: W h ∝ V ∝ L 3 , where L is the linear size of the hiker. 7 7. The quantity L is any linear dimension of the hiker and could for example be the height. muscle, which will scale as L2. Thus, since L 2 = ( L 3 ) 2 / 3 ∝ ( W h ) 2 / 3 , we expect the strength of the hiker to scale as W h 2 / 3 , and as the size of the hiker (or any animal for that matter) is scaled up, strength increases more slowly than body weight. This result is well known and has wide applications in biological systems. For example, it gives a rather straightforward explanation of why small biological specimens, e.g., ants, have large strength compared to their body weight while large biological specimens, e.g., elephants, have small strength compared to their body weight. Similar scaling arguments have also been made for respiratory rate, basal metabolic rate, heart rate, and energy cost of running as a function of animal weight. 8 8. Irving P. Herman, Physics of the Human Body ( Springer-Verlag , Berlin , 2007), see Table 1.13 and references therein. We create a model based on a scaling of the hiker's frame that incorporates the hiker's weight as well as the backpack weight. Such a scaling model describes how the properties of a system change with size. Recent work has introduced the idea of scaling in an elementary way,and several works elaborate on some of the beautiful scaling models in biological systems.The basic idea is that if one scales up the human frame in 3D, then the weight of the hiker,, scales as the volume of the hiker:, whereis the linear size of the hiker.The strength of the hiker is expected to scale as the cross-sectional area of thewhich will scale as. Thus, since, we expect the strength of the hiker to scale as, and as the size of the hiker (or any animal for that matter) is scaled up, strength increases more slowly than body weight. This result is well known and has wide applications in biological systems. For example, it gives a rather straightforward explanation of why small biological specimens, e.g., ants, have large strength compared to their body weight while large biological specimens, e.g., elephants, have small strength compared to their body weight. Similar scaling arguments have also been made for respiratory rate, basal metabolic rate,rate, and energy cost of running as a function of animal weight.

W h wearing a well-fitted backpack. For most people the body structure limiting the largest backpack weight, W bp , is some part of the lower body, possibly the quadriceps muscles, the knee structure, or calf muscle, depending on the person and the type of terrain she is hiking on. The part of the human body that limits the backpack weight will actually support a weight or carrying capacity W C = α W h + W bp , where α is the fraction of the body weight supported by the limiting body structure (see Fig. 1 muscle, one can estimate α by assuming that the quadriceps support all the body weight with the exception of the lower leg and half the upper leg. Using standard body part masses, 5 5. David A. Winter, Biomechanics and Motor Control of Human Movement, 4th ed. ( Wiley , New York , 2009), see Table 4.1. We consider a healthy hiker of weightwearing a well-fitted backpack. For most people the body structure limiting the largest backpack weight,, is some part of the lower body, possibly the quadricepsthe knee structure, or calfdepending on the person and the type of terrain she is hiking on. The part of the human body that limits the backpack weight will actually support a weight or carrying capacity, where α is the fraction of the body weight supported by the limiting body structure (see Fig.). For example, if the limiting body structure is the quadricepone can estimate α by assuming that the quadriceps support all the body weight with the exception of the lower leg and half the upper leg. Using standard body part masses,we obtain an α of ∼0.90 for this case.

W C of the human body scales with increasing body weight W h assuming a healthy adult. The weight that the limiting body structure can support will scale with the cross-sectional area of muscle or bone. 3,5,6,8,9 Godzilla versus scalinglaws of physics ,” Phys. Teach. 43, 530– 532 (Nov. 2005). 3. Thomas R. Tretter, “,” Phys. Teach., 530–(Nov. 2005). https://doi.org/10.1119/1.2120383 5. David A. Winter, Biomechanics and Motor Control of Human Movement, 4th ed. ( Wiley , New York , 2009), see Table 4.1. Size and shape in biology ,” Sci. 179, 1201– 1204 (1973). 6. Thomas McMahon, “,” Sci., 1201–(1973). https://doi.org/10.1126/science.179.4079.1201 8. Irving P. Herman, Physics of the Human Body ( Springer-Verlag , Berlin , 2007), see Table 1.13 and references therein. 9. R. J. Maughan, J. S. Watson, and J. Weir, “ Strength and cross-sectional area of human skeletal muscle ,” J. Physiol. 338, 37– 49 (1983). muscle, the relationship between cross-sectional area and strength has been examined in some detail for a particular muscle group and is consistent with a linear relationship but with significant scatter. 9 9. R. J. Maughan, J. S. Watson, and J. Weir, “ Strength and cross-sectional area of human skeletal muscle ,” J. Physiol. 338, 37– 49 (1983). W h increases, the human body grows equally along the x-, y-, and z-directions, then any cross-sectional areas including those of muscle and bone will scale as W h 2 / 3 as discussed above. 10 10. We have assumed that when the human frame is scaled up, muscle and bone are scaled up by the same linear dimensions as the human frame. Evidence that this is approximately true comes from the fact that (muscle mass)∝(body mass)1.0 and (skeletal mass) ∝ (bodymass)1.08, with both exponents being close to or equal to 1.0. See Ref. 8 , Table 1.13. W C of the person will scale with his weight as W h 2 / 3 . We next consider how the carrying capacityof the human body scales with increasing body weightassuming a healthy adult. The weight that the limiting body structure can support will scale with the cross-sectional area ofor bone.In the case ofthe relationship between cross-sectional area and strength has been examined in some detail for a particulargroup and is consistent with a linear relationship but with significant scatter.If, asincreases, the human body grows equally along the-,-, and-directions, then any cross-sectional areas including those ofand bone will scale asas discussed above.Thus the strength and carrying capacityof the person will scale with his weight as

W h 2 / 3 is written as a proportionality rather than an equality. To create a model of a hiker with a backpack and to estimate this proportionality constant, we consider W bp0 to be the largest backpack weight that a hiker of a particular weight W h0 can carry. Now ( W h ) 2 / 3 ∞ W C = α W h + W bp and ( W h0 ) 2 / 3 ∞ W C0 = α W h0 + W bp0 . Dividing the first equation by the second, we obtain an equality from our original proportionality: W h W h0 2 / 3 = α W h + W bp α W h0 + W bp0 , 1 and we refer to this as a scaling result. Solving for W bp , this equation yields the largest backpack weight as: W bp = α W h 0 + W bp 0 W h W h0 2 / 3 − α W h . 2 Figure 2 W bp as a function of W h . We have estimated that for a hiker of weight W h0 of 700 N, a maximum backpack weight W bp0 of 210 N (30% of body weight at this particular weight) is possible and we take α to be 0.90. Clearly W bp is far from linear. The vertical dashed lines indicate a reasonable range of healthy adult weights with lower and upper limits of 400 N and 1080 N, respectively. W bp has a maximum, and setting d W bp / d W h = 0 using Eq. W h ∣ ( W bp = W bp,max ) = 8 27 α ( α W h0 + W bp0 ) 3 1 α W h0 2 . 3 Substituting for W b in Eq. W bp is: 11 11. A note on notation: the largest backpack weight for a particular weight ( W h ) is denoted by W bp . The maximum value of W bp as a function of W h is denoted by W bp,max . W bp,max = 4 27 ( α W h0 + W bp0 ) 3 1 α W h0 2 . 4 We note that the relationship between strength of the hiker andis written as a proportionality rather than an equality. To create a model of a hiker with a backpack and to estimate this proportionality constant, we considerto be the largest backpack weight that a hiker of a particular weightcan carry. Nowand. Dividing the first equation by the second, we obtain an equality from our original proportionality:and we refer to this as a scaling result. Solving for, this equation yields the largest backpack weight as:Figureshowsas a function of. We have estimated that for a hiker of weightof 700 N, a maximum backpack weightof 210 N (30% of body weight at this particular weight) is possible and we take α to be 0.90. Clearlyis far from linear. The vertical dashed lines indicate a reasonable range of healthy adult weights with lower and upper limits of 400 N and 1080 N, respectively.has a maximum, and settingusing Eq. (2) yields the position of this maximum as:Substituting forin Eq. (2) the maximum value ofis:

Physically it is clear why this maximum is obtained. While the hiker s strength increases with her weight, tending to increase W bp , the hiker's weight increase itself takes up a larger part of the carrying capacity, tending to decrease W bp . These two factors conspire together to give this maximum. Using the previous values of W h0 , W bp0 , and α, we obtain W bp,max = 221 N for a hiker of weight of 492 N, and W bp decreases to 150 N for a hiker weight of 1080 N.

W bp0 is likely to be larger for groups of hikers who have trained specifically for backpacking and somewhat lower for untrained groups of hikers. Figure 2 W bp versus W h for the selected values of W bp0 of 175 N, 210 N, and 280 N, corresponding to 25%, 30%, and 40% of W h0 . It is clear that for larger W bp0 , corresponding to well-trained hikers, the variation of W bp with W h is reduced. For the case W bp0 of 280 N, the average value of W bp over the hiker weight range between the dashed lines is 260 N, with a variation of about 8%. The value ofis likely to be larger for groups of hikers who have trained specifically for backpacking and somewhat lower for untrained groups of hikers. Figurealso showsversusfor the selected values ofof 175 N, 210 N, and 280 N, corresponding to 25%, 30%, and 40% of. It is clear that for larger, corresponding to well-trained hikers, the variation ofwithis reduced. For the caseof 280 N, the average value ofover the hiker weight range between the dashed lines is 260 N, with a variation of about 8%.

It is important to know the assumptions and limits of any model created. This model i) assumes that the limiting body structure (and therefore the parameter α) does not change with the hiker s weight, and ii) an increase in hiker weight leads to an increase in cross-sectional area of muscle and bone. The BMI gives a measure (albeit imperfectly) of how overweight a person is and can be thought of as measure of adipose tissue content. If the increase in weight comes from an increase in BMI, then the scaling of muscle or bone cross-sections with weight may not hold. For these reasons we would consider this model as a good starting point to develop a more sophisticated model of a backpacker.

• carrying capacity depends on cross-sectional area of the limiting body structure, and • the carrying capacity has contributions from both backpack weight W bp and the weight of the body α W h that is borne by the limiting body structure. We have incorporated two important features to create a model of a hiker carrying a backpack:

This model is much improved over the simple assumption that the largest backpack weight W bp a hiker can manage increases with hiker weight and is simply a percentage of the weight W h of the person hiking. This model allows the largest backpack weight W bp to be estimated from the weight W h of a hiker and predicts a maximum in W bp as a function of W h .

In summary, this work began with observations of the performance of student backpackers and the realization that the lower mass (and less strong) but otherwise healthy students seemed to perform at least as well as heavier (and stronger) students. This suggested that strength by itself was not the only important ingredient in determining the weight of backpack that could be carried. This brought about the realization that incorporation of the backpackers weight as part of the carrying capacity was essential to model the backpacker.