text:
Many researchers have suggested that cursorial mammals concentrate limb muscle mass proximally to reduce energy costs during locomotion. Although supported by experiments where mass is added to an individual's limbs, mammals with naturally occurring distally heavy limbs such as primates have similar energy costs compared with other mammals. This study presents a new hypothesis to explain how animals with distally heavy limbs maintain low energy costs. Since distal mass should increase energy costs due to higher amounts of muscular power outputs, this hypothesis is based on the divergent effects of stride frequency on internal and external power outputs (the power output to move the limbs and the body center of mass, respectively). The use of low stride frequencies reduces limb velocities and therefore decreases internal power, while associated long strides increase the vertical displacement of the body center of mass and therefore increase external power. Total power (the sum of internal and external power) may therefore not differ among mammals with different limb mass distributions. To test this hypothesis, I examined a sample of infant baboons (Papio cynocephalus) during ontogeny and compared them with more cursorial mammals. Limb mass distribution changes with age (from distal to more proximally concentrated) in baboons, and the infants used shorter strides and higher stride frequencies when limb mass was most proximally concentrated. Compared with non-primates who have more proximally concentrated limb mass, the infants used longer strides and lower stride frequencies. Relatively low internal power was associated with low stride frequencies in both the intra- and inter-specific samples. However, only in the inter-specific comparison were relatively long strides associated with high external power outputs. In both the intra-specific and the inter-specific samples, total power did not differ between groups who differed in limb mass distribution. The results of this study suggest that a trade-off mechanism is available to quadrupeds with distally heavy limbs allowing them to maintain similar total power outputs (and likely similar energy costs) compared with mammals with more proximally concentrated limb mass.

Reducing the energy costs of locomotion may be an important consideration in mammalian limb design (see Hildebrand,1985; Hildebrand and Hurley,1985; Myers and Steudel,1985; Steudel,1990). For example, cursorial mammals tend to concentrate limb mass proximally, leaving their distal limb elements slender and light(Hildebrand, 1985; Lieberman et al., 2003). Theoretically, distally heavy limbs should increase energetic costs due to increases in internal power - the amount of muscular power (the rate that muscles do work) required to move limb segments relative to the body (see Hildebrand and Hurley, 1985; Dellanini et al., 2003). Although the relationship between power outputs and energy costs of locomotion remains unclear (see Heglund et al.,1982a; Kram and Taylor,1990; Minetti et al.,1999), recent studies suggest that the cost of swinging the limbs may comprise up to 25% of the cost of locomotion(Marsh et al., 2004; Modica and Kram, 2005; Pontzer et al., 2005). Despite the possible energetic constraints on limb design,several mammals, such as primates, have evolved distally heavy limbs(Preuschoft and Günther,1994; Raichlen,2004). This study examines locomotor ontogeny in infant baboons to determine how ontogenetic changes in limb mass distribution impact power outputs.

Experimental studies have generally confirmed that higher costs are associated with distal limb loading (Myers and Steudel, 1985; Steudel,1990). However, in the only comparative experimental study of the energetic cost of naturally occurring distal limb mass, Taylor et al.(1974) showed that three mammals who differed greatly in their limb mass distributions (cheetah,gazelle and goat) did not differ significantly in their energetic costs of locomotion at a given velocity. Quadrupedal primates offer another example that confounds predicted higher costs due to heavy distal limb elements(Taylor et al., 1982; Heglund, 1985; Steudel-Numbers, 2003). Primates have distally heavy limb muscles that control grasping hands and feet, yet their energetic costs do not differ from those of similarly sized mammals (Taylor et al., 1982; Heglund, 1985; Steudel-Numbers, 2003). Although the results of Taylor et al.(1974) have been called into question because of methodological concerns (see Lieberman et al., 2003), their study, combined with the primate data, suggests that limb mass distributions do not necessarily determine energy costs.

The purpose of the present study is to explore mechanisms used by quadrupeds with distally heavy limbs that allow them to maintain similar energy costs compared with mammals with more proximal limb mass concentrations. Without a mechanism, the evolution of distally heavy limbs could have come with a great energetic price. Any mechanism must begin by reducing the impacts of limb mass distribution on internal power. The total power output during quadrupedalism is, however, generally divided into two parts: external power is the power required to lift and accelerate the body center of mass, and internal power is the power required to move the limbs relative to the body center of mass(Heglund et al., 1982b; Minetti et al., 1999). Taxa with distally heavy limbs must find a way to minimize the impacts of their limb mass on internal power and therefore maintain similar total power outputs compared to mammals with more proximally concentrated limb mass.

Adjustments in kinematics may represent a mechanism that could mitigate the added energy costs of distally heavy limbs, since locomotor kinematics can have a strong impact on power outputs during locomotion(Hildebrand and Hurley, 1985; Cavagna and Franzetti, 1986; Cavagna et al., 1991; Minetti et al., 1995; Schepens et al., 2001; Heglund and Schepens, 2003; Schepens et al., 2004). For example, several researchers have shown that a change in stride frequency at a given velocity has divergent effects on internal and external power(Cavagna and Franzetti, 1986; Cavagna et al., 1991; Minetti et al., 1995). Relatively low stride frequencies reduce the velocity at which muscles and tendons must move a limb, thereby reducing internal power(Cavagna and Franzetti, 1986; Cavagna et al., 1991; Minetti et al., 1995; Schepens et al., 2001; Heglund and Schepens, 2003; Schepens et al., 2004). Relatively low stride frequencies are associated with relatively long strides(since velocity is the product of stride frequency and stride length), and long strides increase external power due to larger vertical displacements of the body center of mass (Cavagna and Franzetti, 1986; Cavagna et al., 1991; Minetti et al.,1995; Schepens et al.,2001; Heglund and Schepens,2003; Schepens et al.,2004). Interestingly, although humans walk and run with stride frequencies that nearly minimize total power, they deviate slightly from optimum stride frequencies (Cavagna and Franzetti, 1986; Cavagna et al., 1991; Minetti et al.,1995).

Although freely chosen stride frequencies do not completely minimize total power, metabolic energy expenditure is minimized at freely chosen stride frequencies (Zarrugh and Radcliffe,1978). There may be other constraints on stride frequency that allow individuals to minimize energy expenditure despite slightly higher power outputs. Humans' freely chosen stride frequencies closely match those predicted by the Force Driven Harmonic Oscillator model (FDHO) described by Holt et al. (1990). The FDHO models the lower limb as a harmonic oscillator whose period is dependent on limb inertial properties, but the limb also requires a periodic driving function (muscle activity) to maintain its oscillation amplitude(Holt et al., 1990; Holt et al., 1991). Freely chosen combinations of stride frequencies and lengths are well predicted by the FDHO, suggesting an optimal combination of kinematics that is related to limb mass distribution. Any deviations from the freely chosen frequency,either above or below, result in increased energy expenditure due to increased muscle activity (Holt et al.,1991). It is likely then, that stride frequencies are constrained by limb inertial properties so that a frequency is chosen to minimize the driving function (Holt et al.,1991).

This same tuning of kinematics to inertial properties may apply to quadrupeds as well. Stride lengths and stride frequencies appear to vary in a predictable way with differences in limb mass distribution(Raichlen, 2004; Raichlen, 2005a). Taxa with distally heavier limbs tend to walk with lower stride frequencies and longer strides (Preuschoft and Günther,1994; Myers and Steudel,1997; Raichlen,2004; Raichlen,2005a), and this trend appears valid both across mammalian taxa(Preuschoft and Günther,1994; Myers and Steudel,1997) and within ontogenetic samples where limb mass distributions change with age (Raichlen,2005a). Additionally, lower stride frequencies are brought about by a combination of longer swing durations and longer stance durations. Thus, quadrupeds with distally heavy limbs could take advantage of the trade-offs between kinematics and power outputs simply by virtue of the fact that their stride frequencies are tuned to their limb inertial properties following the FDHO model. As a by-product of this tuning, kinematics may mitigate the energetic consequences of distally heavy limbs. If true then,compared with more cursorial taxa, quadrupeds with distally heavy limbs using lower stride frequencies would reduce internal power outputs (due to slower limb velocities during both swing and stance phase), while their longer strides would increase external power outputs so that total power outputs would not differ among individuals regardless of limb mass distribution.

### Hypothesis testing

This study tests the hypothesis that quadrupedal kinematics associated with distally heavy limbs allows individuals to benefit from the trade-off mechanism described above and, therefore, they can maintain similar total power outputs compared with quadrupeds with distally lighter limbs. Two samples are used to test this hypothesis. First, a sample of infant baboons(Papio cynocephalus) was examined during development. Primate ontogeny offers a natural experiment because infant primate limb mass distributions change with age (Grand,1977; Turnquist and Wells,1994; Raichlen,2005b). At young ages, infant primates have distally heavy limb muscles, used for strong grasping of their mothers fur, and limb mass becomes more proximally concentrated with age(Turnquist and Wells, 1994; Raichlen, 2005b). As they age,and mass becomes more proximally concentrated, the infant baboons use relatively higher stride frequencies and shorter strides(Raichlen, 2005a). Therefore,power outputs can be examined in this sample as both limb mass distributions and kinematics change. In the second test of the trade-off mechanism, infant baboons are compared with a sample of non-primate cursorial quadrupeds who have more proximal limb mass concentrations. This sample of infant baboons uses lower stride frequencies and longer strides than other mammals(Raichlen, 2005a) and therefore may use these kinematic differences to maintain similar total power outputs compared with other mammals.

Hypothesis 1. Infant baboons with distally heavy limbs (younger individuals) should have lower internal and higher external power compared with older individuals because of their lower stride frequencies and longer strides. However, age-related differences in internal and external power should lead to similar total power at all ages.

Hypothesis 2. Because of their more distally heavy limbs compared with non-primates, the infant baboons use lower stride frequencies and longer strides than more cursorial non-primates(Raichlen, 2005a). Due to these kinematic differences, internal power should be lower, external power should be higher, and total power should not differ significantly between the baboons and the non-primates.

### Sample

Three infant baboons (Papio cynocephalus Linnaeus 1758; two females and one male; Table 1)were obtained from the Southwest Foundation for Biomedical Research (SFBR, San Antonio, TX, USA) from a population rejected by their mothers and placed in the SFBR nursery. Infant baboons were housed at the University of Texas Animal Resource Center using methods approved by both the University of Texas Institutional Animal Care and Use Committee (IACUC) and the SFBR IACUC. The study period lasted for approximately nine months, corresponding to the time during which the infants could be handled safely without administering anesthesia.

Table 1.

Description of sample and number of strides analyzed at each age

SubjectAge (months)n
Infant 1 (17194) 3.33
4.27 10
4.73 10
5.87 16
6.10 32
7.50 26
Infant 2 (18573) <3.0 21
3.67 19
4.67 38
4.97 31
5.53 18
7.40 13
8.10 22
Infant 3 (18572) <4.7
4.70 20
5.00 11
5.57 12
6.23 23
7.43 33
8.13 29
8.97 69
SubjectAge (months)n
Infant 1 (17194) 3.33
4.27 10
4.73 10
5.87 16
6.10 32
7.50 26
Infant 2 (18573) <3.0 21
3.67 19
4.67 38
4.97 31
5.53 18
7.40 13
8.10 22
Infant 3 (18572) <4.7
4.70 20
5.00 11
5.57 12
6.23 23
7.43 33
8.13 29
8.97 69

Subject describes the infant number that is used in the text; the SFBR identification number is listed in parentheses. Infant 1 is male; infants 2 and 3 are female. n is the number of accepted strides analyzed and is included here for each sampled age.

### Inertial properties

Body segment inertial properties of the infant baboons were calculated using a geometric modeling technique(Raichlen, 2004; Raichlen, 2005b). Briefly,each body segment was modeled as a column with a circular cross section. Model shapes were constructed using external circumference measurements taken at three locations on each limb segment. From these measurements, column models were constructed, and segment inertial properties (mass, center of mass, mass moments of inertia) were calculated(Raichlen, 2004; Raichlen, 2005b). The natural pendular period (NPP) of the limbs was also calculated. The NPP is the duration of one complete oscillation of the limb if it was swinging as a pure pendulum and is calculated according to the following equation:
$\ NPP=2{\pi}\sqrt{\frac{I}{Mg(D)}},$
(1)
where I is the limb's mass moment of inertia about the proximal joint (the shoulder in the forelimb or the hip in the hindlimb), M is the limb's mass, g is acceleration due to gravitational forces(9.8 m s-2) and D is the distance of the center of mass from the proximal joint. The NPP represents an important link between limb mass distribution and kinematics because swing phase is often modeled as an approximation of a pendulum (see Mochon and McMahon, 1980; Mochon and McMahon, 1981; Holt et al.,1990; Holt et al.,1991). Larger NPPs due to more distal limb mass imply longer swing durations and therefore longer stride durations. Raichlen(2005a) has also shown that relatively large NPPs lead to relatively long stance durations. The use of NPP in this study is not meant to imply that swing phase is purely passive, with no muscular action involved (see Marsh et al., 2004; Modica and Kram, 2005), but that limb motion during swing phase is tuned to the NPP (see Holt et al., 1990; Holt et al., 1991). NPPs were made dimensionless (dNPP) (according to Hof, 1996) to compare values among individuals who differed in body size and therefore limb length(h):
$\ dNPP=\frac{NPP}{\sqrt{\frac{h}{g}}}.$
(2)

### Grouping

Individuals were grouped based on their limb inertial properties, and these groupings did not conform to arbitrary age classes. Specifically, changes in dNPP were used to group infants according to limb mass distributions. Researchers have shown that limb NPPs change during swing phase due to flexion and extension of limb joints and that this change should be taken into account when comparing the effects of inertial properties on locomotion(Myers and Steudel, 1997; Raichlen, 2004; Raichlen, 2005a). When swing-phase inertial properties are compared [by calculating the minimum dNPP during swing phase (dNPPmin)], the infant baboons do not show a gradual change with age(Raichlen, 2005a). In fact,each individual falls into two groups: Group 1 comprises the young infants with distally heavy limbs and dNPPmins that do not differ significantly, and Group 2 comprises older infants who have more proximal concentrations of limb mass and dNPPmins that do not differ significantly (Raichlen,2005a). Between the two groups, dNPPmins do differ significantly (Raichlen,2005a). These groupings proved robust in an examination of the effects of inertial properties on kinematics, with Group 1 infant baboons(more distal limb mass) using relatively lower stride frequencies and longer strides than Group 2 infants (Raichlen,2005a).

### Mechanical power

Three-dimensional kinematic data were obtained from each infant baboon at regular intervals during development (Table 1). Infant baboons were allowed to walk and run at freely chosen velocities through a Lexan tunnel made up of three removable sections(0.61×0.91×1.22 m each). Prior to tunnel entry, spherical reflective markers (14 mm; Oxford Metrics Inc., Oxford, UK) were glued to the major joints of the forelimbs and hindlimbs of each infant baboon (hip, knee,ankle, shoulder, elbow, knee). The placement of each marker was consistent with the segment definitions used for inertial property data collection (see Raichlen, 2005b).

Three-dimensional marker trajectories were captured for one side of the body during each locomotor trial using a five-camera 60 Hz Vicon 250 data acquisition system (Oxford Metrics Inc.). In addition to the Vicon system,video data of each trial were collected using a digital video camera(JVC-GRDVL9800E; Wayne, NJ, USA) at 60 frames s-1. Video data were used to determine touchdown and toe-off events.

The total positive work that must be supplied by muscles and tendons during locomotion (Wtot) can be divided into two parts(Fenn, 1930). The first part is the mechanical work required to move body segments relative to the whole-body center of mass (internal work; Wint). Wint is calculated from changes in each segment's energy over an entire stride (after Fedak et al.,1982; Willems et al.,1995). The second part of Wtot is external work (Wext); the work that must be supplied to lift and accelerate the whole body center of mass(Cavagna et al., 1977). Power(int and ext) is simply the rate at which internal or external work is done. Alternative approaches to calculating ext will be discussed in a later section.

The kinetic energy of each body segment relative to the body center of mass was calculated from the 3-D marker positions captured by the Vicon motion analysis system. For each stride, one side of the body (side facing cameras or ipsilateral side) was divided into five segments (trunk, arm, forearm, thigh,leg) that were defined by passive reflective markers at each of the major joints. Affixing markers to define the hand and foot segments was problematic both because of the small size of the individuals' hands and feet and because the infant baboons were more likely to remove those markers. For this analysis, the hands and feet were considered point masses at the distal ends of the forearms and lower legs, respectively, and the head was considered a point mass at the cranial end of the trunk. The positions and velocities for segments on the contralateral side of the body were estimated assuming the movements of the contralateral side segments during half a stride were the same as the movements of the ipsilateral segments during the other half of the stride (see Fedak et al.,1982; Willems et al.,1995).

Mass-specific Wint was calculated as the sum of the positive changes in each limb's kinetic energy per stride divided by body mass(after Willems et al., 1995). Dividing mass-specific Wint by stride duration gives int. This method of calculating internal work allows for transfers of energy between segments of a single limb but not between limbs. To calculate Wext,potential and kinetic energies (both horizontal and vertical) of the body center of mass were summed at each instant in time to obtain the total energy of the center of mass (Ecm). Center of mass position and velocity were reconstructed based on the positions of limb segment centers of mass (after Minetti et al.,1999). Mass-specific external mechanical work(Wext) was calculated as the sum of the positive increments in the Ecm curve over an entire stride divided by body mass. Mass-specific external power(ext) was calculated as(Wext/stride duration).

### Manipulation of internal power calculations

A manipulation of inertial properties was performed to examine the sensitivity of int to an individual's limb mass distribution. This manipulation consisted of scaling the segment inertial properties of all subjects with distally light limbs to the inertial properties of the individual with the heaviest distal limb elements. int was recalculated using these scaled values for limb segment inertial properties. The following equations were used to scale segment inertial properties of older individuals (individual 2) to those of the subject with the heaviest distal limb elements (individual 1):
$\ B_{\mathrm{scaled}}=B_{2}(M_{1}{/}B_{1}),$
(3)
$\ D_{\mathrm{scaled}}=h_{2}(D_{1}{/}h_{1}),$
(4)
$\ I_{\mathrm{scaled}}=M_{2}h_{2}(I_{1}{/}M_{1}h_{1}^{2}),$
(5)

where B1 and B2 are the body masses of individuals 1 and 2, respectively, M1 and M2 are the segment masses of individuals 1 and 2,respectively, h1 and h2 are the segment lengths of individuals 1 and 2, D1 is the segment center of mass position from the proximal end of the segment for individual 1,and I1 is the segment mass moment of inertia about its center of mass for individual 1.

### Data analysis

Comparisons in power outputs between Group 1 and Group 2 infant baboons were performed in two ways. First, analyses of covariance (ANCOVAs) were used to compare variables between groups, with velocity as the covariate since power outputs are correlated with velocity in infant baboons. Since Group 1 and Group 2 individuals differ in size, ANCOVAs were also performed with dimensionless velocity [velocity/(gh)0.5] as the covariate. Significant differences were determined using Tukey-Kramer post-hoc tests to account for multiple comparisons. In addition to ANCOVAs, a residuals analysis was performed. Specifically, residuals were calculated from the least-squares regression line relating dimensionless stride frequency [stride frequency/(g/h)0.5] and dimensionless velocity as well as mass-specific internal power and dimensionless velocity. Residual internal power outputs were then regressed on residual stride frequencies. These residuals were calculated from the entire sample (Group 1 and Group 2 combined). A significant positive correlation indicates that those individuals who use high stride frequencies relative to velocity also have high internal power outputs. A similar analysis was performed for external power. Residuals of external power and dimensionless velocity were regressed on residuals of dimensionless stride length (stride length/h) and dimensionless velocity. A significant positive correlation indicates that those individuals using relatively long strides would also have relatively high external power outputs. All analyses were performed on log-transformed data since the relationship between power outputs and velocity is not linear.

Recently, Raichlen has shown that this sample of infant baboons follows the predicted relationships between limb mass distribution and kinematics(Raichlen, 2005a). The infant baboons show a gradual shift from more distal to more proximal limb mass distribution patterns (Raichlen,2005b) but, because of limb flexion during swing phase, the infants fall into two main groups, within which dNPPmins calculated during swing phase do not differ significantly. Group 1 includes the youngest ages of infant baboons and has significantly more distal limb mass than Group 2, which includes the older ages. Group 1 infants use significantly lower stride frequencies and significantly longer strides than Group 2 individuals (Table 2). Based on these results, a trade-off between internal and external power is possible.

Table 2.

Between-group comparisons of kinematics and power outputs in the combined sample

VariableLSM1 (95% C.I.)LSM2 (95% C.I.)H0
Stride frequency1 −0.632 (0.008) −0.580 (0.004) <0.0001
Stride length1 0.277 (0.008) 0.223 (0.004) <0.0001
Internal power2 −0.854 (0.098) −0.795 (0.042) 0.711
External power2 −0.142 (0.090) −0.190 (0.040) 0.022
Total power2 −0.060 (0.091) −0.090 (0.040) 0.014
Internal power3 −0.923 (0.022) −0.811 (0.011) <0.0001
External power3 −0.203 (0.028) −0.194 (0.015) 0.594
Total power3 −0.151 (0.031) −0.145 (0.016) 0.741
VariableLSM1 (95% C.I.)LSM2 (95% C.I.)H0
Stride frequency1 −0.632 (0.008) −0.580 (0.004) <0.0001
Stride length1 0.277 (0.008) 0.223 (0.004) <0.0001
Internal power2 −0.854 (0.098) −0.795 (0.042) 0.711
External power2 −0.142 (0.090) −0.190 (0.040) 0.022
Total power2 −0.060 (0.091) −0.090 (0.040) 0.014
Internal power3 −0.923 (0.022) −0.811 (0.011) <0.0001
External power3 −0.203 (0.028) −0.194 (0.015) 0.594
Total power3 −0.151 (0.031) −0.145 (0.016) 0.741

LSM1 is the least-squares mean for Group 1 for each variable after dimensionless velocity has been taken into account. LSM2 is the least-squares mean for Group 2 for each variable after dimensionless velocity has been taken into account. 95% confidence intervals are given in parentheses. H0is the probability that LSM1 is equal to LSM2.

1

Stride frequencies and stride lengths were made dimensionless, and dimensionless velocity was the covariate.

2

Velocity was used as the covariate.

3

Dimensionless velocity was used as the covariate.

### Hypothesis 1: ontogenetic analysis

int does not show significant between-group differences when analyzed using velocity as the covariate but is significantly lower in Group 1 individuals compared with Group 2 over the dimensionless velocity range(Fig. 1A,B; Table 2). int was recalculated for each stride of all Group 2 individuals using scaled segment inertial properties (see Materials and methods). If older individuals had the limb mass distributions of younger individuals, but did not reduce their stride frequencies, they would have had approximately 20% higher int values at a given velocity (Fig. 1C). So, by using low stride frequencies when their mass is most distal, the infant baboons save approximately 20% of the mechanical power they would otherwise have had to output to move their limbs relative to their body.

Group 1 individuals have significantly higher ext over the velocity range, but do not show significant between group differences in ext at a given dimensionless velocity (Fig. 1D,E; Table 2). Finally, tot is larger in Group 2 individuals at a given velocity but does not show significant between-group differences in the combined sample of infant baboons over the dimensionless velocity range (Fig. 1F,G;Table 2). Between-group differences in body size must be taken into account because the same velocity has different impacts on individuals with different limb lengths(Alexander and Jayes, 1983), so the results of the dimensionless velocity comparisons hold greater significance in this study. Although the between-group comparisons of int, ext and tot do not support a trade-off mechanism when examined over the range of dimensionless velocities because ext did not differ between groups, they do support the hypothesis that differences in kinematics between individuals who differ in limb mass distributions will be associated with similar total power outputs.

Fig. 1.

Between-group comparisons of mass-specific power outputs in the ontogenetic sample of infant baboons. In all panels, Group 1 infants (filled circles) have significantly heavier distal limb segments than Group 2 infants (open circles). (A) Mass-specific internal power(int) compared over the range of raw velocities. (B) Mass-specific int compared over the range of dimensionless velocities. (C) Manipulation of int calculations for Group 2 (see text for further explanation). (D) Mass-specific external power(ext) compared over the range of raw velocities. (E) Mass-specific ext compared over the range of dimensionless velocities. (F) Mass-specific total power(tot) compared over the range of raw velocities. (G) Mass-specific tot compared over the range of dimensionless velocities. Statistical between-group differences using ANCOVAs were found for B, D and F. These results can be found in Table 2.

Fig. 1.

Between-group comparisons of mass-specific power outputs in the ontogenetic sample of infant baboons. In all panels, Group 1 infants (filled circles) have significantly heavier distal limb segments than Group 2 infants (open circles). (A) Mass-specific internal power(int) compared over the range of raw velocities. (B) Mass-specific int compared over the range of dimensionless velocities. (C) Manipulation of int calculations for Group 2 (see text for further explanation). (D) Mass-specific external power(ext) compared over the range of raw velocities. (E) Mass-specific ext compared over the range of dimensionless velocities. (F) Mass-specific total power(tot) compared over the range of raw velocities. (G) Mass-specific tot compared over the range of dimensionless velocities. Statistical between-group differences using ANCOVAs were found for B, D and F. These results can be found in Table 2.

Individual baboon between-group comparisons generally follow the trends of the combined sample analysis, although the differences are not always significant (Table 3). Only Infant 1 shows significantly lower Group 1 values of int. Although least-squares mean int is lower in Group 1 for the other two individuals, these differences were not significant. For all other variables, the individuals follow the combined-sample results.

Table 3.

Between-group comparisons of power outputs in infants 1-3

VariableGroupLSM (95% C.I.)H0
int Infant 1 Group 1 −1.195 (0.114) <0.0001
Infant 1 Group 2 −0.873 (0.071)
Infant 2 Group 1 −0.735 (0.047) 0.234
Infant 2 Group 2 −0.703 (0.024)
Infant 3 Group 1 −0.822 (0.037) 0.076
Infant 3 Group 2 −0.864 (0.014)
ext Infant 1 Group 1 −0.304 (0.105) 0.269
Infant 1 Group 2 −0.235 (0.065)
Infant 2 Group 1 −0.098 (0.039) 0.531
Infant 2 Group 2 −0.084 (0.020)
Infant 3 Group 1 −0.259 (0.051) 0.745
Infant 3 Group 2 −0.250 (0.018)
tot Infant 1 Group 1 −0.307 (0.109) 0.094
Infant 1 Group 2 −0.197 (0.068)
Infant 2 Group 1 −0.002 (0.036) 0.444
Infant 2 Group 2 0.014 (0.018)
Infant 3 Group 1 −0.261 (0.051) 0.947
Infant 3 Group 2 −0.262 (0.018)
VariableGroupLSM (95% C.I.)H0
int Infant 1 Group 1 −1.195 (0.114) <0.0001
Infant 1 Group 2 −0.873 (0.071)
Infant 2 Group 1 −0.735 (0.047) 0.234
Infant 2 Group 2 −0.703 (0.024)
Infant 3 Group 1 −0.822 (0.037) 0.076
Infant 3 Group 2 −0.864 (0.014)
ext Infant 1 Group 1 −0.304 (0.105) 0.269
Infant 1 Group 2 −0.235 (0.065)
Infant 2 Group 1 −0.098 (0.039) 0.531
Infant 2 Group 2 −0.084 (0.020)
Infant 3 Group 1 −0.259 (0.051) 0.745
Infant 3 Group 2 −0.250 (0.018)
tot Infant 1 Group 1 −0.307 (0.109) 0.094
Infant 1 Group 2 −0.197 (0.068)
Infant 2 Group 1 −0.002 (0.036) 0.444
Infant 2 Group 2 0.014 (0.018)
Infant 3 Group 1 −0.261 (0.051) 0.947
Infant 3 Group 2 −0.262 (0.018)

All ANCOVAs were performed with dimensionless velocity as the covariate.

LSM, least-squares mean; int, internal power; ext, external power; tot, total power.

H0 is the probability that the LSM for Group 1 is equal to the LSM for Group 2.

Because ANCOVA results may have been compromised by only slight differences in kinematics and power outputs, a residuals analysis was performed. Residuals of dimensionless stride frequency and dimensionless velocity were regressed against residuals of intand dimensionless velocity (Fig. 2A). A significant positive correlation (r=0.48; P<0.001) indicates that those individuals who have relatively low dimensionless stride frequencies at a given dimensionless velocity have relatively low int at a given dimensionless velocity. Note also that groups cluster together, with Group 1 clustering in negative residual space (both negative dimensionless stride frequency residuals and negative int residuals). The same analysis of residual stride lengths and ext also shows a slight,but significant, positive correlation between regressed residuals(Fig. 2B; r=0.21; P<0.001). This relationship is, however, quite weak, suggesting that ext may not be sensitive to slight differences in stride length. Infant baboons show the same patterns when examined individually (Table 4). The residuals for int and stride frequency are more highly correlated than those of ext and stride length,although all correlations are significant.

Table 4.

Residuals analysis of kinematics and power outputs in the individual infant baboons

InfantResidual regressionrP
intvs SF 0.37 <0.0001
intvs SF 0.59 <0.0001
intvs SF 0.29 <0.0001
extvs SL 0.36 <0.0001
extvs SL 0.23 0.003
extvs SL 0.19 0.009
InfantResidual regressionrP
intvs SF 0.37 <0.0001
intvs SF 0.59 <0.0001
intvs SF 0.29 <0.0001
extvs SL 0.36 <0.0001
extvs SL 0.23 0.003
extvs SL 0.19 0.009

Residuals were calculated from least-squares regression lines relating each variable with dimensionless velocity.

SF, stride frequency; SL, stride length; int, internal power; ext, external power.

Fig. 2.

Residuals analysis of kinematics and mechanical power. (A) Residuals of internal power vs dimensionless velocity regressed on residuals of dimensionless stride frequency vs dimensionless velocity(r=0.48; P<0.001). Group 1 is represented by filled circles and Group 2 is represented by open circles. Note that Group 1 clusters together in the negative residuals quadrant. (B) Residuals of external power vs dimensionless velocity regressed on residuals of dimensionless stride length vs dimensionless velocity (r=0.21; P<0.001). Group 1 is represented by filled circles and Group 2 is represented by open circles. Note that Group 1 clusters in the positive residuals quadrant.

Fig. 2.

Residuals analysis of kinematics and mechanical power. (A) Residuals of internal power vs dimensionless velocity regressed on residuals of dimensionless stride frequency vs dimensionless velocity(r=0.48; P<0.001). Group 1 is represented by filled circles and Group 2 is represented by open circles. Note that Group 1 clusters together in the negative residuals quadrant. (B) Residuals of external power vs dimensionless velocity regressed on residuals of dimensionless stride length vs dimensionless velocity (r=0.21; P<0.001). Group 1 is represented by filled circles and Group 2 is represented by open circles. Note that Group 1 clusters in the positive residuals quadrant.

### Hypothesis 2: comparisons with other quadrupeds

To assess the likelihood that larger differences in inertial properties and kinematics lead to power output trade-offs, the infant baboons (combined Group 1 and Group 2) were compared with more cursorial non-primate quadrupeds. In the following comparative analyses, dogs and horses are used because data for inertial properties, kinematics and power outputs in these species are readily available in the literature.

Inertial properties for dogs were taken from Myers and Steudel(1997) and for horses from Buchner et al. (1997). Stride frequencies and stride lengths were calculated from equations in Minetti et al. (1999) for horses and from Williams et al. (2002) for dogs. Power outputs for dogs were taken from Heglund et al.(1982b) and Fedak et al. 1982), and power for horses was taken from Minetti et al.(1999). ANCOVAs cannot be used to compare non-primate kinematics and power outputs with those of the infant baboons because only least-squares regression equations relating kinematics and power to velocity in dogs and horses have been reported. Therefore, the 95% confidence intervals of the regression lines for the infant baboon sample were calculated and differences are suggested when the regression line for the non-primate data set fell outside of this interval. The limb length used to calculate dimensionless velocity for dogs was given in Williams et al.(2002) for kinematics and Fedak et al. (1982) for power. For the horse data set, limb length was estimated from allometric equations in Alexander et al. (1979) for mean horse body mass in Minetti et al.(1999) (limb length = 1.26 m).

Infant baboons have more distal forelimb and hindlimb centers of mass compared with dogs and horses (Table 5). Following the predicted relationships between inertial properties and kinematics, infant baboons use lower stride frequencies and longer strides at a given dimensionless velocity than either dogs or horses(Fig. 3A,B). The infant baboon sample has lower int(Fig. 3C) and higher ext compared with the non-primates (Fig. 3D). The lower int and higher ext in the infant baboons lead to similar tot in the infant baboon sample compared with the non-primate sample(Fig. 3E). These data are consistent with the presence of a trade-off mechanism where low stride frequencies are associated with relatively low internal power, long strides are associated with high external power, and total power does not differ significantly among the sampled taxa.

Table 5.

Center of mass positions as a fraction of limb length (95% C.I.) in the infant baboons, dogs and horses

VariablePapio (N=3)Canis (N=10)Equus (N=6)
Hindlimb D 0.410 (0.007) 0.304 (0.023) 0.327
Forelimb D 0.467 (0.005) 0.342 (0.068) 0.343
VariablePapio (N=3)Canis (N=10)Equus (N=6)
Hindlimb D 0.410 (0.007) 0.304 (0.023) 0.327
Forelimb D 0.467 (0.005) 0.342 (0.068) 0.343

Data for Canis (Myers and Steudel, 1997); data for Equus(Buchner et al., 1997).

N, number of animals; D, distance of the center of mass from the proximal joint.

Superficially, the results from the ontogenetic sample do not support the idea that differences in stride frequency and stride length among animals with different limb mass distributions will have opposite effects on int and ext. Although relatively low stride frequencies are associated with low int, the relationship between stride length and ext appears to be less clear.

The comparison of mechanical power in infant baboons with dogs and horses is, however, consistent with the presence of a mechanical power trade-off mechanism. The infant baboons have more distally heavy limbs and use longer strides and lower stride frequencies compared with dogs and horses. The infant baboons do less int and more ext compared with the dog and horse samples, and their values of tot do not differ significantly from those of the non-primate sample. It is possible that the trade-off mechanism is only apparent when stride frequencies and stride lengths differ greatly between taxa.

There are several possible reasons for the absence of trade-offs in the ontogenetic sample despite the presence of a trade-off mechanism in the inter-specific sample. First, the differences in inertial properties between infant groups were smaller than inter-specific differences. Perhaps slight differences in spatio-temporal kinematics simply equalize energy outputs,rather than having the trade-off effects. This possibility seems likely given the results of the manipulation of internal power calculations (see Fig. 1C). If young infants,with their relatively distally heavy limbs, used the kinematics of older aged infants, int would be approximately 20% higher.

Second, it is possible that the kinematic method of calculating center of mass displacement is not sensitive enough to make comparisons within small differences in stride length. Future examinations of the trade-off mechanism could use force-plate data to calculate center of mass displacements, although Gard et al. (2004) suggest that differences between the two techniques are minimal.

Finally, it is possible that ext, as calculated here,does not fully account for the power output to produce center of mass motion. Recently, Donelan et al.(2002) suggested that much of the work required to produce center of mass motion actually occurs as the center of mass is redirected at the end of the stance phase during walking(e.g. collisional costs; see also Bastien et al., 2003 for alternative method of calculating this cost). This work done during the double-contact phase of walking may represent a substantial portion of extand should scale with step length4(Donelan et al., 2002). If true, then ext as measured here does not fully account for the effects of longer strides on the power required to lift and accelerate the center if mass.

Fig. 3.

Kinematics in infant baboons compared with dogs and horses. In all panels,the least-squares regression (LSR) line for infant baboons in black, with 95%confidence limits for the regression line in gray, is compared with the LSR line for horses (dot-dashed line) and for dogs (dashed line). (A)Dimensionless stride frequencies over the range of dimensionless velocities for the combined infant baboon sample compared with dogs and horses. LSR line for infant baboons (95% C.I. in parentheses): slope=0.64 (0.02);intercept=-0.35 (0.01); r=0.96; P<0.001. (B)Dimensionless stride lengths for the infant baboons compared with dogs and horses. LSR line for infant baboons: slope=0.35 (0.02); intercept=0.35 (0.01); r=0.86; P<0.001. (C) Internal power in infant baboons compared with dogs and horses. LSR line for infant baboons: slope=1.82 (0.05);intercept=-0.19 (0.02); r=0.96; P<0.001. (D) External power in infant baboons compared with dogs and horses. LSR line for infant baboons: slope=1.68 (0.06); intercept=0.37 (0.03); r=0.92; P<0.001. (E) Total power in infant baboons compared with dogs and horses. LSR line for infant baboons: slope=1.68 (0.07); intercept=0.42 (0.03); r=0.92; P<0.001.

Fig. 3.

Kinematics in infant baboons compared with dogs and horses. In all panels,the least-squares regression (LSR) line for infant baboons in black, with 95%confidence limits for the regression line in gray, is compared with the LSR line for horses (dot-dashed line) and for dogs (dashed line). (A)Dimensionless stride frequencies over the range of dimensionless velocities for the combined infant baboon sample compared with dogs and horses. LSR line for infant baboons (95% C.I. in parentheses): slope=0.64 (0.02);intercept=-0.35 (0.01); r=0.96; P<0.001. (B)Dimensionless stride lengths for the infant baboons compared with dogs and horses. LSR line for infant baboons: slope=0.35 (0.02); intercept=0.35 (0.01); r=0.86; P<0.001. (C) Internal power in infant baboons compared with dogs and horses. LSR line for infant baboons: slope=1.82 (0.05);intercept=-0.19 (0.02); r=0.96; P<0.001. (D) External power in infant baboons compared with dogs and horses. LSR line for infant baboons: slope=1.68 (0.06); intercept=0.37 (0.03); r=0.92; P<0.001. (E) Total power in infant baboons compared with dogs and horses. LSR line for infant baboons: slope=1.68 (0.07); intercept=0.42 (0.03); r=0.92; P<0.001.

Infant baboons do show between-group differences in dimensionless step length (ANOVA results: Group 1 mean=0.94, Group 2 mean=0.88, P<0.001, F=3.86), which should lead to higher step-to-step costs in Group 1 infants. In fact, Group 1 step lengths raised to the fourth power (mean=0.78) are 23% higher than those of Group 2 (mean=0.59). This value is quite similar to the between-group difference in least-squares means for int (22%; see Table 2). Although differences in center of mass height changes associated with longer strides may not be large enough to impact ontogenetic changes in ext as calculated in this study, ontogenetic differences in step-to-step costs may in fact provide the trade-off.

Although the results presented above suggest a trade-off in mechanical costs, the relationship between mechanical and metabolic costs remains unclear. In a classic paper, Heglund et al. showed a disconnection between mechanical power and metabolic costs of locomotion in a wide range of taxa(Heglund et al., 1982a). This disconnection is due to the fact that muscles consume energy not only when they do work but also when they produce force isometrically(Kram and Taylor, 1990). Despite the large impact of isometric force production on metabolic costs of locomotion, increases in mechanical power at a given speed (driven by changes in stride frequency/length at a given speed) cause an increase in metabolic cost (Minetti et al., 1995). Thus, minimizing total power should reduce the metabolic costs of locomotion at a given speed. Minetti et al.(1999) suggest that a complete description of metabolic costs should therefore include a combination of mechanical work and power plus the costs associated with isometric muscle force production.

### Implications for mammalian limb design

These results may explain why mammals in general do not differ greatly in energy costs despite differences in limb mass distributions. Taylor et al.(1974) showed that the cheetah, the gazelle and the goat do not differ in their energetic costs despite having different limb mass distributions. Domestic cats use relatively longer strides than other mammalian cursors (see fig. 3 in Alexander and Jayes, 1983), and these long strides may be linked to their distally heavy limbs. If cheetahs use similar kinematics, then low stride frequencies and long strides may explain their similar energetic costs compared with the goat and gazelle. Cheetahs' low stride frequencies and long strides would reduce their internal power outputs while increasing their external power outputs, allowing them to maintain similar total power outputs compared with other mammals.

It is possible that all mammals follow a similar trade-off pattern that is dependent on their limb mass distributions. Alexander and Jayes(1983) showed that non-cursorial mammals use longer strides at a given dimensionless velocity compared with more cursorial mammals. Since non-cursors also have more distal limb mass concentrations than cursors(Grand, 1977; Myers and Steudel, 1997; Raichlen, 2004), non-cursors may be taking advantage of the trade-off mechanism to reduce total mechanical power.

The trade-off mechanism may have played an important role in allowing mammalian quadrupeds to evolve non-locomotor functions in their distal limb elements that would increase distal mass, without having a negative impact on their energy costs of locomotion. This type of mechanism would have been especially important for animals such as primates, who rely on their distally heavy limb muscles to control their grasping hands and feet. Grasping hands and feet are a hallmark of the primate order and were an essential element of the success of early primates (Cartmill,1972). The use of the trade-off mechanism would have allowed early primates to evolve grasping hands and feet without exacting an energetic price.

### Why concentrate limb mass proximally?

Although the results from this study suggest that mechanisms exist that reduce the impact of limb mass distributions on mechanical power outputs, the question of why cursorial mammals concentrate limb mass proximally remains unanswered. Based on the available evidence from studies of energy costs of transport in mammals, proximally concentrated limb mass does not lead to greatly reduced energy expenditures(Taylor et al., 1974; Taylor et al., 1982; Heglund, 1985). Other adaptive scenarios must therefore be examined.

Reducing distal limb mass leads to a reduction in the limb's mass moment of inertia (e.g. the limb's resistance to rotational acceleration). Limb mass moments of inertia affect limb acceleration relative to the body and therefore affect whole-body accelerations (Ropret et al., 1998; Pasi and Carrier,2003; Rahmani et al., 2003). Although it seems clear that the ability to sprint at high speeds is determined mainly by the ability to produce greater ground reaction forces(Weyand et al., 2000), studies of lower limb loading show significant reductions in limb velocity and consequently reductions in maximum sprint velocity with added distal loads(Ropret et al., 1998).

Accelerations are necessary to either catch prey or evade predators(Elliott et al., 1977), and thus structures that enhance an individual's ability to accelerate should be subjected to large selection pressures. Since the ability to rapidly accelerate is a strong predictor of successful predator evasion in many taxa(Elliott et al., 1977), the evolution of proximal limb mass may be the result of selection for acceleration capabilities. Although the results from the present study certainly cannot address why cursorial quadrupeds concentrate limb mass proximally, the context of predator-prey interactions may provide selection pressures for the evolution of proximally concentrated limb mass.

### Summary

Infant baboons show some ontogenetic evidence of a trade-off mechanism,although the relationship between external power and stride length appears to be weak. Compared with dogs and horses, the infant baboon sample has lower internal power outputs, higher external power outputs and more similar total power outputs. These results suggest that, on a broad scale, individuals may use a combination of stride frequency and stride length that is determined by their limb mass distributions and that minimizes total mechanical power outputs. These findings suggest that selection pressures for non-cursorial activities acting on the distal limb elements of primates, and perhaps other mammals in general, do not have to exact an energetic price. Quadrupeds may adjust their kinematics to accommodate limb mass distribution patterns that,superficially, seem detrimental to quadrupedal energetics. Additionally, the results from this study suggest that researchers should explore new explanations for the evolution of proximally concentrated limb mass in cursorial quadrupeds.

List of symbols

• D

distance of limb center of mass from the proximal end

•
• dNPP

dimensionless natural pendular period

•
• dNPP

min minimum dNPP during swing phase for a given stride

•
• Ecm

total energy of the body center of mass

•
• h

limb length

•
• I

mass moment of inertia of the limb about a transverse axis through its proximal joint

•
• B

body mass

•
• M

segment mass

•
• NPP

limb natural pendular period

•
• Wint

internal work

•
• int

internal power

•
• Wext

external work

•
• ext

external power

•
• Wtot

total work

•
• tot

total power

I thank Liza Shapiro for her help and guidance during this project and in the preparation of this manuscript. Thanks also to John Kappelman, Susan Larson, Deborah Overdorff, Marcus Pandy, Daniel Lieberman, Herman Pontzer,Adam Gordon, Robert Scott and three anonymous reviewers for their helpful discussions on this project and this manuscript. This research would not have been possible without the generosity of Jody Jensen, who provided access to her Motor Development Laboratory in the Kinesiology Department at University of Texas at Austin. Thanks to Jerry Finig and the staff of the University of Texas at Austin Animal Resource Center for their help in caring for the baboon sample. Thanks to Karen Rice and Dee Carey for their help in facilitating the use of infant baboons for this study. Magdalena Muchlinski, Katherine Whitcome and Amber Copeland provided help with animal care that was much appreciated. Thanks also to John Allison and Larry Isgur for design and construction of the Lexan tunnel. This project was supported by the L.S.B. Leakey Foundation and by NSF BCS-0313821.

Alexander, R. McN. and Jayes, A. S. (
1983
). A dynamic similarity hypothesis for the gaits of quadrupedal mammals.
J. Zool. (Lond.)
201
,
135
-152.
Alexander, R. McN., Jayes, A. S., Maloiy, G. M. O. and Wathuta,E. M. (
1979
). Allometry of the limb bones of mammals from shrews (Sorex) to elephant (Loxodonta).
J. Zool.(Lond.)
194
,
539
-552.
Bastien, G. J., Heglund, N. C. and Schepens, B.(
2003
). The double contact phase in walking children.
J. Exp. Biol.
206
,
2967
-2978.
Buchner, H., Savelberg, H., Schamhardt, H. and Barneveld, A.(
1997
). Inertial properties of Dutch Warmblood horses.
J. Biomech.
30
,
653
-658.
Cartmill, M. (
1972
). Arboreal adaptations and the origin of the order Primates. In
The Functional and Evolutionary Biology of Primates
(ed. R. H. Tuttle), pp.
3
-35. Chicago: Aldine-Atherton.
Cavagna, G. A. and Franzetti, P. (
1986
). The determinants of the step frequency in walking in humans.
J. Physiol.
373
,
235
-242.
Cavagna, G. A., Heglund, N. C. and Taylor, C. R.(
1977
). Mechanical work in terrestrial locomotion: two basic mechanisms for minimizing energy expenditure.
Am. J. Physiol.
233
,
R243
-R261.
Cavagna, G. A., Willems, P. A., Franzetti, P. and Detrembleur,C. (
1991
). The two power limits conditioning step frequency in human running.
J. Physiol.
437
,
95
-108.
Dellanini, L., Hawkins, D., Martin, R. B. and Stover, S.(
2003
). An investigation of the interactions between lower-limb bone morphology, limb inertial properties and limb dynamics.
J. Biomech.
36
,
913
-919.
Donelan, J., Kram, R. and Kuo, A. (
2002
). Mechanical work for step-to-step transitions is a major determinant of the metabolic cost of human walking.
J. Exp. Biol.
205
,
3717
-3727.
Elliott, J., McTaggart, C. and Holling, C.(
1977
). Prey capture by the African lion.
Can. J. Zool.
55
,
1811
-1828.
Fedak, M. A., Heglund, N. C. and Taylor, C. R.(
1982
). Energetics and mechanics of terrestrial locomotion. II. Kinetic energy changes of the limbs and body as a function of speed and body size in birds and mammals.
J. Exp. Biol.
79
,
23
-40.
Fenn, W. O. (
1930
). Work against gravity and work due to velocity changes in running.
Am. J. Physiol.
93
,
433
-462.
Gard, S., Miff, S. and Kuo, A. (
2004
). Comparison of kinematic and kinetic methods for computing the vertical motion of the body center of mass during walking.
Hum. Mov. Sci.
22
,
597
-610.
Grand, T. I. (
1977
). Body weight: its relation to tissue composition, segment distribution, and motor function. I. Interspecific comparisons.
Am. J. Phys. Anthropol.
47
,
211
-240.
Heglund, N. C. (
1985
). Comparative energetics and mechanics of locomotion: how do primates fit in? In
Size and Scaling in Primate Biology
(ed. W. L. Jungers), pp.
319
-335. New York: Plenum Press.
Heglund, N. C. and Schepens, B. (
2003
). Ontogeny recapitulates phylogeny? Locomotion in children and other primitive hominids. In
Vertebrate Biomechanics and Evolution
(ed. V. L. Bels, J. P. Gasc and A. Casinos), pp.
283
-295. Oxford: BIOS Scientific Publishers.
Heglund, N. C., Fedak, M. A., Taylor, C. R. and Cavagna, G. A. (
1982a
). Energetics and mechanics of terrestrial locomotion. IV. Total mechanical energy changes as a function of speed and body size in birds and mammals.
J. Exp. Biol.
97
,
57
-66.
Heglund, N. C., Cavagna, G. A. and Taylor, C. R.(
1982b
). Energetics and mechanics of terrestrial locomotion. III. Energy changes of the centre of mass as a function of speed and body size in birds and mammals.
J. Exp. Biol.
79
,
41
-56.
Hildebrand, M. (
1985
). Walking and running. In
Functional Vertebrate Morphology
(ed. M. Hildebrand,D. M. Bramble, K. F. Liem and D. B. Wake), pp.
38
-57. Cambridge: Harvard University Press.
Hildebrand, M. and Hurley, J. (
1985
). Energy of the oscillating legs of a fast-moving cheetah, pronghorn, jackrabbit, and elephant.
J. Morphol.
184
,
23
-31.
Hof, A. L. (
1996
). Scaling gait data to body size.
Gait Posture
4
,
222
-223.
Holt, K. G., Hammil, J. and Andres, R. O.(
1990
). The force-driven harmonic oscillator as a model for human locomotion.
Hum. Mov. Sci.
9
,
55
-68.
Holt, K. G., Hamill, J. and Andres, R. O.(
1991
). Predicting the minimal energy costs of human walking.
Med. Sci. Sports Exerc.
23
,
491
-498.
Kram, R. and Taylor, C. R. (
1990
). Energetics of running: a new perspective.
Nature
346
,
265
-267.
Lieberman, D. E., Pearson, O. M., Polk, J. D., Demes, B. and Crompton, A. W. (
2003
). Optimization of bone growth and remodeling in response to loading in tapered mammalian limbs.
J. Exp. Biol.
206
,
3125
-3138.
Marsh, R. L., Ellerby, D. J., Carr, J. A., Havalee, T. H. and Buchanan, C. I. (
2004
). Partitioning the energetics of walking and running: Swinging the limbs is expensive.
Science
303
,
80
-83.
Minetti, A. E., Capelli, C., Zamparo, P., Prampero, P. E. and Saibene, F. (
1995
). Effects of stride frequency on mechanical power and energy expenditure of walking.
Med. Sci. Sports Exerc.
27
,
1194
-1202.
Minetti, A. E., Ardigo, L. P., Reinach, E. and Saibene, F.(
1999
). The relationship between mechanical work and energy expenditure of locomotion in horses.
J. Exp. Biol.
202
,
2329
-2338.
Mochon, S. and McMahon, T. A. (
1980
). Ballistic walking.
J. Biomech.
13
,
49
-57.
Mochon, S. and McMahon, T. A. (
1981
). Ballistic walking: An improved model.
Math. Biosci.
52
,
241
-260.
Modica, J. R. and Kram, R. (
2005
). Metabolic energy and muscular activity required for leg swing in running.
J. Appl. Physiol.
98
,
2126
-2131.
Myers, M. and Steudal, K. (
1985
). Effect of limb mass and its distribution on the energetic cost of running.
J. Exp. Biol.
116
,
363
-373.
Myers, M. J. and Steudel, K. (
1997
). Morphological conservation of limb natural pendular period in the domestic dog(Canis familiaris): implications for locomotor energetics.
J. Morphol.
234
,
183
-196.
Pasi, B. M. and Carrier, D. R. (
2003
). Functional trade-offs in the limb muscles of dogs selected for running vs. fighting.
J. Evol. Biol.
16
,
324
-332.
Pontzer, H. (
2005
). A new model predicting locomotor cost from limb length via force production.
J. Exp. Biol.
208
,
1513
-1524.
Preuschoft, H. and Gunther, M. M. (
1994
). Biomechanics and body shape in primates compared with horses.
Z. Morphol. Anthropol.
80
,
149
-165.
Rahmani, A., Locatelli, E. and Lacour, J.(
2004
). Differences in morphology and force/velocity relationship between Senegalese and Italian sprinters.
Eur. J. Appl. Physiol.
91
,
399
-405.
Raichlen, D. A. (
2004
). Convergence of forelimb and hindlimb natural pendular periods in baboons (Papio cynocephalus)and its implication for the evolution of primate quadrupedalism.
J. Hum. Evol.
46
,
719
-738.
Raichlen, D. A. (
2005a
). Effects of limb mass distribution on the ontogeny of quadrupedalism in infant baboons (Papio cynocephalus) and implications for the evolution of primate quadrupedalism.
J. Hum. Evol.
49
,
415
-431.
Raichlen, D. A. (
2005b
). Ontogeny of limb mass distribution in infant baboons (Papio cynocephalus).
J. Hum. Evol.
49
,
452
-467.
Ropret, R., Kukolj, M., Ugarkovic, D., Matavulj, D. and Jaric,S. (
1998
). Effects of arm and leg loading on sprint performance.
Eur. J. Appl. Physiol
77
,
547
-550.
Schepens, B., Willems, P. A., Cavagna, G. A. and Heglund, N. C. (
2001
). Mechanical power and efficiency in running children.
Pflügers Arch.
442
,
107
-116.
Schepens, B., Bastien, G. J., Heglund, N. C. and Willems, P. A. (
2004
). Mechanical work and muscular efficiency in walking children.
J. Exp. Biol.
207
,
587
-596.
Steudel, K. (
1990
). The work and energetic cost of locomotion. I. The effects of limb mass distribution in quadrupeds.
J. Exp. Biol.
154
,
273
-285.
Steudel-Numbers, K. L. (
2003
). The energetic cost of locomotion: humans and primates compared to generalized endotherms.
J. Hum. Evol.
44
,
255
-262.
Taylor, C. R., Shkolnik, A., Dmpel, R., Baharav, D. and Borut,A. (
1974
). Running in cheetahs, gazelles, and goats: energy cost and limb configuration.
Am. J. Physiol.
227
,
848
-850.
Taylor, C. R., Heglund, N. C. and Maloiy, G. M. O.(
1982
). Energetics and mechanics of terrestrial locomotion. I. Metabolic energy consumption as a function of body size in birds and mammals.
J. Exp. Biol.
97
,
1
-21.
Turnquist, J. E. and Wells, J. P. (
1994
). Ontogeny of locomotion in rhesus macaques (Macaca mulatta). I. Early postnatal ontogeny of the musculoskeletal system.
J. Hum. Evol.
26
,
487
-499.
Weyand, P., Sternlight, D., Bellizzi, M. and Wright, S.(
2000
). Faster top running speeds are achieved with greater ground reaction forces not more rapid leg movements.
J. Appl. Physiol.
89
,
1991
-1999.
Willems, P. A., Cavagna, G. A. and Heglund, N. C.(
1995
). External, internal and total work in human locomotion.
J. Exp. Biol.
198
,
379
-393.
Williams, T. M., Ben-David, M., Noren, S., Rutishauser, M.,McDonald, K. and Heyward, W. (
2002
). Running energetics of the North American river otter: Do short legs necessarily reduce efficiency on land?
Comp. Biochem. Physiol.
133A
,
203
-212.
Zarrugh, M. and Radcliffe, C. (
1978
). Predicting metabolic cost of level walking.
Eur. J. Appl. Physiol.
38
,
215
-223.


text:
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## Example Numerical Question Requiring a Computed Response

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format: plain


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# Solving the functional equation $f(x+1) - f(x-1) = g(x)$

Given a function $g(x)$, is it possible to find a function $f(x)$ that satisfies

$$f(x+1) - f(x-1) = g(x)$$

-
What kind of functions are we talking about here? – Qiaochu Yuan Jan 13 '11 at 2:00

Pick any choice of $f$ on the interval $[-1,1)$. By using the property $f(x) = g(x-1) + f(x-2)$, you can extend this definition to $[1,3)$, and so on to the rest of the positive real line. Similarly, you can extend $f$ to the negative real line using $f(x) = f(x+2) - g(x+1)$. This is one solution to your equation.

Of course, a different choice of $f$ on $[-1,1)$ will give you a different result. This corresponds to the fact that adding any periodic function $h$ with period 2 to $f$ still satisfies the original equation. In other words, there are infinitely many solutions.

For particular restricted classes of functions, as Qiaochu alludes to, there may be a single canonical solution. For example, if $g$ is a polynomial of degree $n$, there is a unique polynomial solution for $f$. This is a polynomial of degree $n+1$, and you can find it by expanding out the original equation and comparing powers starting with the highest. A similar fact holds for exponentials and trigonometric functions (except of course when the trigonometric functions have period 2, in which case you would have to include multiples of $x \sin \pi x$ and $x \cos \pi x$ in $f$ to get a solution).

-

Let $f(x)=\frac{1}{2}x g(0)$ for $-1\le x \le 1$, $f(x)=g(x-1) + f(x-2)$ for $x>1$, and $f(x)=f(x+2)-g(x+1)$ for $x<-1$.

-

There might be something to gain by interpreting $g(x)$ as being an approximation to the derivative of $f(x)$, (barring a factor of 2). It's not really the derivative, but the centered finite-difference.

-

I don't know whether this procedure is correct or not:

$f(x+1)-f(x-1)=g(x)$

$x\to x+1$:

$f(x+2)-f(x)=g(x+1)$

$x\to2x$:

$f(2x+2)-f(2x)=g(2x+1)$

$f(2(x+1))-f(2x)=g(2x+1)$

$f(2x)=\sum_x g(2x+1)+\Theta_1(x)$, where $\Theta_1(x)$ is an arbitrary periodic function with unit period

$f(x)=h\left(\dfrac{x}{2}\right)+\Theta(x)$, where $h(x)=\sum_x g(2x+1)$ and $\Theta(x)$ is an arbitrary periodic function with period $2$

If $\lim_{x\to+\infty}g(x)=0$, according to http://en.wikipedia.org/wiki/Indefinite_sum#Mueller.27s_formula, the result can be further simplified to $f(x)=\sum_{n=0}^\infty(g(2n+1)-g(x+2n+1))+\Theta(x)$, where $\Theta(x)$ is an arbitrary periodic function with period $2$, $\sum_{n=0}^\infty(g(2n+1)-g(x+2n+1))$ i.e. kernelisation of $\sum_{n=0}^{\frac{x}{2}-1}g(2n+1)$

-
In other words: solutions to this problem are equivalent to solutions of the "indefinite summation" problem. – GEdgar Jul 1 '12 at 22:24


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# Trying to figure out integral

Hi,

I have a couple of problems I'm trying to figure out:

1) 1/x^3-x^2 dx

Is this a partial fraction problem and if so am I doing it correctly?

I think I should break down the denominator into x^2(x-1) then I have A/x^2 + B/(x-1)

Then I get A= -3 and B = 1 so I figure it to be -3ln[x^2] + ln[x-1]?

I may be way off and if I am could you please point me in the right direction.

2) [5x^3 + x^2 - square root (x)] / 2x

I know this is another partial fractions problem but I have no idea what to do when I only have 2x in the denominator. Could you give me a way to start this problem so I can try to figure it out too?

Thanks for the help

Char. Limit
Gold Member

Hi,

I have a couple of problems I'm trying to figure out:

1) 1/x^3-x^2 dx

Is this a partial fraction problem and if so am I doing it correctly?

I think I should break down the denominator into x^2(x-1) then I have A/x^2 + B/(x-1)

Then I get A= -3 and B = 1 so I figure it to be -3ln[x^2] + ln[x-1]?

I may be way off and if I am could you please point me in the right direction.

Okay, here we go. First, you're right that it's a partial fractions problem. However, when you have x*x*(x-1), you have to have a term for 1/x^2 and 1/x. So in reality, you have A/x^2 + B/x + C/(x-1). That will help.

Also, something that will help is that the integral of 1/x^2 is not ln(x^2). You have to use the power rule for that one.

2) [5x^3 + x^2 - square root (x)] / 2x

I know this is another partial fractions problem but I have no idea what to do when I only have 2x in the denominator. Could you give me a way to start this problem so I can try to figure it out too?

Thanks for the help

This one is actually really easy. Just split the numerator and then you'll get a formula in terms of x^2, x, and 1/sqrt(x). Each of these can be integrated using the power rule.

HallsofIvy
Homework Helper

Actually, it's impossible to tell if it is a "partial fractions" problem without knowing whether you meant
$$\frac{1}{x^3}- x^2$$
(what you actually wrote)
or
1/(x^3- x^2) which would be
$$\frac{1}{x^3- x^2}$$

If it is the latter, then, yes, factor and write as partial fractions.


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# Team:Toulouse

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# Stoichiometry - A Review

Skills Tested by This Quiz

• evaluate molecular weight for a given formula
• evaluate weight (mass) percentages of elements for a given formula
• evaluate amounts (in mass and mole units) produced in a chemical reaction from given conditions
• classify reactions by types: combination, combustion, displacement, formation, etc
• determine the chemical formula when weight percentages are given and then evaluate the mole percentages of elements in the formula
• determine the chemical formula when weight percentages are given and molecular weight is known
• determine the amount produced, the actual yield, and other stoichiometry quantities for a given reaction

Review Purposes

• To get an overall view of stoichiometry.
• Apply skills learned to perform quantitative chemical analysis.
• Apply theories and rules of chemistry to solve problems.
• Assess areas of strength and weakness for review purposes.
• Improve problem solving strategy and learning efficiency.

## Stoichiometry

STOICHIOMETRY is the quantitative relationship of reactants and products. This unit has been divided into the following objects. A brief review is given here so that you can get a birds'-eye or overall view of stoichiometry.

1. Amounts of substances
Express amounts of substance in mass units of g, kg, tons, and convert them to moles, kilomoles, or millimoles.
2. Chemical formulas
Represent a substance with a formula that reflects its chemical composition, structure, and bonding; evaluate weight and mole percentages of elements in a substance; and determine chemical formula by elemental analysis.
3. Reaction features
Define some common features of chemical reactions; classify chemical reactions by common features such as combination, combustion, decomposition, displacement, and redox reactions.
4. Reaction equations
Express quantitative relationships using chemical reaction equations; evaluate quantities of reactants and products in a chemical reaction; and solve reaction stoichiometry problems.
5. Excess and limiting reagents
Define excess and limiting reagents; determine excess and limiting reagents in a reaction mixture; and determine quantities produced in a chemical reaction.
6. Yields
Define theoretical and actual yields due to limiting reagent; apply the concept of limiting reagent to evaluate theoretical yield; convert actual yield to percentage yields.

## A Typical Quiz

In these quizzes, we give a set of question that are interrelated.

Problem 1

You have weighed x g of $$\ce{CaCl2}$$ (molar mass 111) for an experiment dealing with chloride ions and you need to find the number of moles of $$\ce{Cl-}$$ ions. Show the conversion from x g $$\ce{CaCl2}$$ to moles of $$\ce{Cl-}$$ ions.

Hint...
The conversion logic in given below:

$$\ce{x\: g\: CaCl2 \times \dfrac{1 mol\: CaCl2}{111 g\: CaCl2}\times\dfrac{2 mol\: Cl- }{1 mol\: CaCl2}=\dfrac{2 x}{111}mol\: Cl-}$$

Skill...

Convert amounts of substances in various units.

Problem 2

You have a sample weighing 2.345 g containing equal weight of $$\ce{CaCl2}$$ (f.wt. 111) and $$\ce{AlCl3}$$ (f.wt. 133). How many mole of $$\ce{Cl-}$$ ions do you have?

Hint...
Assuming you understand the previous example, then you can give

$$\mathrm{\left(\dfrac{2\: mol}{111\: g}+\dfrac{3\: mol}{133\: g} \right )\times\dfrac{2.345\: g}{2}= 0.04757\: mol\: Cl^-}$$

Skill...

Develop skills to calculate moles of chloride ions in problems containing unequal weights of $$\ce{CaCl2}$$ and $$\ce{AlCl3}$$.

Problem 3

When a 2.00-g mixture of $$\ce{CaCl2}$$ (111) and $$\ce{RbCl}$$ (121) was analyzed, the result reports that the mixture contains a total of 0.0241 mol $$\ce{Cl-}$$ ions. Find weight percentage of $$\ce{RbCl}$$ in the sample.

Hint...
Using the method in the previous examples, and assuming x g $$\ce{RbCl}$$ are present in the mixture, you have

$$\dfrac{x}{121}+\dfrac{2 (2.00-x)}{111}= 0.0241$$

Solving for x, x = 1.23 g (Practice your solution)

$$\mathrm{Percentage = \dfrac{1.23}{2.00} = 61.4 \%\: RbCl}$$

Skill...
Making an assumption to make the problem solvable.

Problem 4

When a 2.00-g mixture of $$\ce{CaCl2}$$ (111) and $$\ce{RbCl}$$ (121) was received by an analytical laboratory, the technician dissolved it in a solution and then he used an excess amount of $$\ce{AgNO3}$$ to precipitate the chloride as $$\ce{AgCl}$$. He collected 3.456 g $$\ce{AgCl}$$. What is the percentage of $$\ce{RbCl}$$ in the sample?

Hint...
We should convert 3.456 g $$\ce{AgCl}$$ into moles of $$\ce{AgCl}$$ or $$\ce{Cl-}$$ ions.

$$\mathrm{3.456\: g\: AgCl\times\dfrac{1\: mol\: AgCl}{143.4\: g\: AgCl}= 0.0241\: mol\: AgCl\: or\: Cl^-}$$

The rest of the solution has already been given in Problem 3.

Skill...
Building skills to do useful work.

Problem 5

You are interviewed for a manager's job, and you were asked what you should instruct your technician to do for the following situation. Company X has received a sample containing a mixture of $$\ce{CaCl2}$$ and $$\ce{BaCl2}$$. The client of Company X needs to know the percentages of the two components.

Hint...
Instruct your technician to find out if the sample contains any other substance than $$\ce{CaCl2}$$ and $$\ce{BaCl2}$$.

If it does not contain other substances than $$\ce{Ca^2+}$$, $$\ce{Ba^2+}$$, and $$\ce{Cl-}$$ ions, instruct him to precipitate the chloride ions with an excess amount of silver ion.

Instruct the technician how to calculate the moles of chloride present, and how to calculate the percentages.

Skill...
Write instructions for laboratory process.

## Example Problems

1. When properly used, urea, $$\ce{CH4N2O}$$, is a good candidate for fertilizer. Unocal is one of the companies that produces urea in Kenai, Alaska. It produces 1.1 million tons of urea each year. How many moles are this amount?

Hint: 18 trillion moles

Urea is also called carbamide or carbonyldiamide. Since this is a US factory, 1 ton is taken as 983 kg that is equivalent to 18 thousand moles.

Skill -
Converting from mass units to moles and vice versa.

2. Urea is made by reacting carbon dioxide ($$\ce{CO2}$$) with anhydrous ammonia ($$\ce{NH3}$$) under 3,000 psi (pounds per square inch) pressure and at 350°F (177°C or 450 K).

\begin{alignat}{3} \ce{ &CO2 + &&2 NH3 &&=\: &&CO(NH2)2 + &&H2O\\ &\:44 &&\:\:34 &&= &&\:\:\:\:\:\:60 &&\:\:18 \:(molar\: mass)} \end{alignat}

The removal of water that occurs during the reaction is referred to as "dehydration." The resulting molten mixture is further processed into either prills or granules. For each ton of urea made, how much water is removed?

Hint: 0.3 tons

For 1.3 tons of reactants, 1.0 ton of urea is produced. By the way, 3,000 psi is equivalent to 200 atmosphere pressure.

Skill -
Interpret stoichiometry among products and reactants.

3. The NPK numbers (for example, 25, 3, 7) used in the fertilizer industry refer to the percentage of nitrogen, $$\ce{N}$$, phosphorus, $$\ce{P}$$, and potassium, $$\ce{K}$$. What are the $$\ce{NPK}$$ numbers for pure urea?

Hint: 46 0 0

Based on the formula, urea contains 46.7% $$\ce{N}$$. However, the factory usually gives 46 because most products contain water.

Skill -
Calculate weight and mole percentage from chemical formula.

4. Conversion of nitrogen into protein by plants is a complicated process. First, urea is hydrolyzed and converted to ammonium carbonate. The hydrolysis of urea is brought about by the urease enzyme, but the reaction may be written as follows:

$$\ce{CO(NH2)2 + 2 H2O \rightarrow (NH4)2CO3}$$

The product is a salt. The solubility of urea is reported to be 1 g in 1 ml of water. Assuming that this solution is allowed to hydrolyze by the urease enzyme, which is the limiting reagent, urea or water?

Hint: urea

One g of urea is 17 millimoles, which requires 34 millimoles of water to react from the stoichiometry of the reaction equation. One g of water is 56 millimoles; that is more than 34 millimoles.

Skill -
Determine excess and limiting reagent.

5. Alanine $$\ce{CH3CH(NH2)(COOH)}$$ is one of the amino acids that are the building blocks of proteins. What is the percentage of nitrogen in this amino acid?

Hint: 16% $$\ce{N}$$ in alanine

Skill -
Determine an element percentage from chemical formula.

6. An old method for protein analysis was to measure the nitrogen content. The amount of protein present was assumed to be 16 times the amount of nitrogen.

Assume that 50 % of the nitrogen in urea is converted to protein by plants. How much protein is produced for one gram of urea?

Hint: 3.68 g protein

Percentages of protein in plant material are usually very low.

Skill -
Calculate theoretical yield, actual yield, and percentage yield.

7. A solid mixture contains $$\ce{Na2SO4}$$ and $$\ce{NaCl}$$ . What compound would you use to precipitate the $$\ce{Cl-}$$ ion?

Hint: silver ion, $$\ce{Ag+}$$

Skill -
Apply what you know of substances. $$\ce{AgCl}$$ is one of the few insoluble chlorides.

8. A solid mixture contains $$\ce{Na2SO4}$$ and $$\ce{NaCl}$$ . What compound would you use to precipitate the $$\ce{SO4^2-}$$ ion?

Hint: $$\ce{BaCl2}$$

$$\ce{BaCl2}$$ is a soluble salt, but $$\ce{BaSO4}$$ is not.

Skill -
Apply what you know of substances.

9. A 2.21 g mixture containing $$\ce{Na2SO4}$$ and $$\ce{NaCl}$$ dissolved in water is treated with $$\ce{BaCl2}$$. The resulting dry $$\ce{BaSO4}$$ weighs 2.57 g. How many mole(s) of $$\ce{BaSO4}$$ were collected? (formula wt. $$\ce{BaSO4}$$, 233.39)

Hint: 0.0110 mol

$$\mathrm{\dfrac{2.57\: g}{233.39\: g/mol} =\: ?\: mol}$$

How many mole(s) of $$\ce{Na2SO4}$$ were present in the mixture? (Also 0.0110 mol)

Skill -

10. A 2.21 g mixture containing $$\ce{Na2SO4}$$ and $$\ce{NaCl}$$ dissolved in water is treated with $$\ce{BaCl2}$$, and analysis shows 0.01101 mol $$\ce{Na2SO4}$$ was present. What is the weight of $$\ce{Na2SO4}$$? (formula wt. $$\ce{Na2SO4}$$, 142.04)

Hint: 1.56 g

Skill -

11. A 2.21 g mixture containing 1.56 g $$\ce{Na2SO4}$$ and some unknown impurities is treated with $$\ce{BaCl2}$$. How many grams of $$\ce{BaSO4}$$ should form? (formula wt. $$\ce{Na2SO4}$$, 142.04, $$\ce{BaSO4}$$, 233.39)

Hint: 2.57 g

Conversion calculation:

$$\mathrm{1.56\: g\: NaSO_4\times\dfrac{233.39\: g/mol\: BaSO_4}{142.04\: g/mol\: Na_2SO_4}=\: ?\: g\: BaSO_4}$$

Skill -

You have acquired all the skills to perform chemical analysis to find out the percentage of certain components in a mixture. For example, consider this problem.

A 2.21 g mixture containing $$\ce{Na2SO4}$$ and $$\ce{NaCl}$$ dissolved in water is treated with $$\ce{BaCl2}$$, and 2.57 g of $$\ce{BaSO4}$$ was obtained. What is the percentage of $$\ce{NaCl}$$ in the sample?

12. Which one of the following samples has the highest percentage of chloride?
1. pure $$\ce{CaCl2}$$
2. pure $$\ce{AlCl3}$$
3. equal weight mixture of $$\ce{CaCl2}$$ and $$\ce{AlCl3}$$
4. pure $$\ce{NaCl}$$
5. pure $$\ce{BaCl2}$$

Hint: $$\ce{AlCl3}$$

Skill -
Interpret chemical formula

13. The reaction

$$\ce{C3H8 + 5 O2 \rightarrow 3 CO2 + 4 H2O}$$

is best classified as a

1. formation reaction
2. decomposition reaction
3. combustion reaction
4. exchange reaction
5. displacement reaction

Hint: c.

Skill -
Describe the major features of chemical reactions.

14. What chemical (give the formula) would you use in order to determine the percentage of $$\ce{CaCl2}$$ in a solid mixture of $$\ce{CaCl2}$$ and $$\ce{Ca(NO3)2}$$. Assume you can first dissolve the solid in water.

Hint: silver nitrate

Skill -
Apply the rule: salts of nitrates are usually soluble.

15. A 1.140 g mixture of $$\ce{NaCl}$$ and $$\ce{CaCl2}$$ dissolved in water is mixed with sufficient solution of $$\ce{AgNO3}$$ to give 2.868 g of dry $$\ce{AgCl}$$. Calculate the WEIGHT percentage of $$\ce{NaCl}$$. ($$\ce{Na}$$, 23.0; $$\ce{Cl}$$, 35.5; $$\ce{Ca}$$, 40; $$\ce{Ag}$$, 107.9)

Hint: 51.3% $$\ce{NaCl}$$

Skill -
Technique suggestion:

$$\mathrm{2.868\: g\: AgCl = 0.0200\: mol\: (Cl^-\: or\: AgCl)}$$

Assume the mixture contains x g $$\ce{NaCl}$$, then (1.140-x) g is $$\ce{CaCl2}$$. Thus, we have:

$$\mathrm{\#\: mol\: NaCl + 2\, \#\: mol\: CaCl_2 = \#\: mol\: Cl^-}$$ lead to

$$\mathrm{\dfrac{x\: g\: NaCl}{58.5\: g /mol}+\dfrac{2 (1.140 - x)\: CaCl_2}{111.1\: g/mol}= 0.0200\: mol}$$

Solve the equation for x (x = 0.585 g) and calculate the percentage.

16. A 1.140 g mixture of $$\ce{NaCl}$$ and $$\ce{CaCl2}$$ dissolved in water is mixed with sufficient solution of $$\ce{AgNO3}$$ to give 2.868 g of dry $$\ce{AgCl}$$. Calculate the MOLE percentage of $$\ce{NaCl}$$. ($$\ce{Na}$$, 23.0; $$\ce{Cl}$$, 35.5; $$\ce{Ca}$$, 40; $$\ce{Ag}$$, 107.9)

Hint: 66% by mole

Skill -
Determine the weight and mole percentages of mixtures.


text:
# Measurement of prompt and nonprompt $$\mathrm{J}/{\psi }$$ J / ψ production in $$\mathrm {p}\mathrm {p}$$ p p and $$\mathrm {p}\mathrm {Pb}$$ p Pb collisions at \sqrt{s_{\mathrm {NN}}} =5.02\,\text …

Overview of attention for article published in European Physical Journal C -- Particles & Fields, April 2017


text:
# LyricsNeed You

## Daniel Vauti

Last update on: April 20, 2020
No translations available

I saw you right there sitting at the table Can I get your attention maybe? Looking right in your eyes

Feel my heart beating, I start to smile Put my hand around your hip I'm reaching out for your first kiss I need you now, I need you I need you now, I need you I need you now, I need you I need you now, I need you I need you now, I need you I need you now, I need you Always The time it runs you have to go I say I'll text you tomorrow Lookin' right in your eyes I see a universe inside Feeling strong holding your hand Wish you'd go with me, but you can't I need you now, I need you I need you now, I need you I need you now, I need you I need you now, I need you I need you now, I need you I need you now, I need you Always

No translations available


text:
# What are the physical optical limitations photonic metamaterials can offer?

Modern technology has introduced new photonic metamaterials that can extend optical properties beyond what standard materials can offer. For example a negative index of refraction is now possible which can lead to extraordinary abilities such as invisibility (optical cloaking) of massive objects, etc.

So we once believed that our ability to resolve an image was limited to, for example the diffraction limit derived by Abbe which is a function of the wavelength of the light, and index of refraction of the material.

Given these new metamaterials, do these laws still apply? Can we, again for example, just plug-in the negative index of refraction to Abbe's expression for diffraction limit? If so, how does one interpret a negative diffraction limit?

It also leads me to the question of what does a zero index of refraction mean? What physically amazing things might a zero index entail, and is a zero index of refraction even possible to engineer with metamaterials?

• Suggest you read the introductory explanations before asking a broad question like this. – Carl Witthoft Jun 8 '15 at 15:14
• Sub-wavelength (superresolution) imaging can be achieved without meta-materials. The Abbe-criterion only applies to passively illuminated objects in the far field without the use of deconvolution of high-SNR images. One can do quite a bit better with laser excited dye stained samples and high quality large NA optics. Negative refractive indices won't change much about that, they do allow to build near ideal optical elements, though, which is not possible with positive index of refraction optics, alone. – CuriousOne Jun 8 '15 at 15:33

The reason that the diffraction limit actually occurs is due to the decay of evanescent EM fields which decay exponentially from the surface of the material, and thus are lost in the far field. However, a material with a refractive index of $-1$ will (as proven by John Pendry) actually cause these evanescent fields to increase in strength (they carry no energy, and therefore conservation of energy is not broken). This is the basis of the so-called super lens (for more information on how this field has developed, take a look at hyper lenses, or meta lenses). It should be pointed out, however, that no-one has yet made a metamaterial with sufficiently exact properties to realise the super lens fully. We have achieved sub-wavelength imagery, but we have not yet made the perfect lens.
I haven't looked into work which aims to have a refractive index of zero, however we certainly can get refractive indices of less than $1$ (see, for instance, $-0.6$, I believe, in a paper by Gunnar Dolling, achieved at $780nm$). In this instance, the wave moves faster than the speed of light. However, causality is not compromised because it is only the phase velocity of the light which is moving faster than the speed of light, and not the group velocity, meaning that no information would be transferred faster (you can only transfer information along a wave by modulating that wave somehow, say by blocking it and unblocking in Morse code). In another paper by the same author, they also note a negative group velocity, but I don't remember if that has a magnitude of more or less than $1$.


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# From cartesian to cylindrical

1. Mar 1, 2005

### ViktigLemma

I find this passage $$\frac{\partial}{\partial x} = \cos(\phi)\frac{\partial}{\partial \rho } - \frac{\sin(\phi)}{\rho}\frac{\partial}{\partial \phi}$$ difficult to understand.

My teacher wrote this as an explanation:

$$\frac{\partial V}{\partial x} = \frac{\partial\rho}{\partial x}\frac{\partial V}{\partial \rho} + \frac{\partial\phi}{\partial x}\frac{\partial V}{\partial \phi} + \frac{\partial z}{\partial x}\frac{\partial V}{\partial z}$$ *

And then inserting for $$\rho$$ and $$\phi$$, which will yield a correct result.

What I don't understand is how * can be correct? To me it seems that the right side of the equation is equal to $$3\frac{\partial V}{\partial x}$$

Last edited: Mar 1, 2005
2. Mar 1, 2005

### ViktigLemma

There, I fixed it :)

3. Mar 1, 2005

### HallsofIvy

Staff Emeritus
Could you tell us WHY you think there should be a "3" in that?

$$\frac{\partial V}{\partial x} = \frac{\partial\rho}{\partial x}\frac{\partial V}{\partial \rho} + \frac{\partial\phi}{\partial x}\frac{\partial V}{\partial \phi} + \frac{\partial z}{\partial x}\frac{\partial V}{\partial z}$$
is simply the chain rule!
That would be 3 times dV/dx only if all those were equal to dV/dx separately and they are not!

Take a simple example. Suppose V(u,w)= u2+ 2w2 and
that u= 3x-1, v= 2x+3.
One way of finding dV/dx would be to substitute. V(x)= (3x-1)2+ 2(2x+3)2= 9x2- 6x+ 1+ 2(4x2+ 12x+ 9)= 9x2- 6x+ 1+ 8x2+ 24x+ 18= 17x2+ 18x+ 19.

dV/dx= 34x- 30.

But a simpler way is to use the chain rule: $\frac{dV}{dx}= \frac{dV}{du}\frac{du}{dx}+ \frac{dV}{dw}\frac{dw}{dx}$= 2u(3)+ 4w(2)= 6(3x-1)+ 8(2x-3)= 18x- 6+ 16x- 24= 34x- 30 just as before. dV/dx is that sum, not the individual parts.

4. Mar 1, 2005

### Gamma

$$\frac{\partial}{\partial x} = \cos(\phi)\frac{\partial}{\partial \rho } - \frac{\sin(\phi)}{\rho}\frac{\partial}{\partial \phi}$$

If you understand how chain rule works just as explained by HallsofIvy,

this is not so difficult.
Lets say $$V=V(\rho, \phi, z)$$

$$\frac{\partial V}{\partial x} = \frac{\partial V}{\partial \rho}\frac{\partial\rho}{\partial x} +\frac{\partial V}{\partial \phi} \frac{\partial\phi}{\partial x}+ \frac{\partial V}{\partial z}\frac{\partial z}{\partial x}$$

Transformation from cylindrical to cartisian coordinates,
$$x= \rho cos \phi ,\\ y= \rho sin \phi ,\\ z= z$$

$$\rho = \sqrt(x^2 +y^2)$$

$$tan \phi = \frac{y}{x}$$

From here,

$$\frac{\partial\rho}{\partial x}= \frac{x}{\rho}$$

$$\frac{\partial\phi}{\partial x}= \frac{-sin\phi}{\rho}$$

$$\frac{\partial z}{\partial x}= 0$$

5. Mar 1, 2005

### ViktigLemma

Thank you both very much! I was thinking in the wrong paths entirely. To explain:

What I figured, not being well versed in this kind of manipulation, was this:

$$\frac{\partial V}{\partial \phi}\frac{\partial \phi}{\partial x} = \frac{\partial V}{\partial x}\frac{\partial \phi}{\partial \phi} = \frac{\partial V}{\partial x}$$

Because I remember doing something like that once.

6. Mar 1, 2005

### vincentchan

this is not valid in general, you can only do this in a very special case, i.e.

$$V = V(\phi(x))$$

7. Mar 1, 2005

### ViktigLemma

Yes that's the chain rule doing its magic I guess :D