Answer to the Question 07/00
ENERGETICS OF WALKThe question was:
Energy expenditure (power) for walking on a level for speed v smaller than 2m/s is approximately given by the expression: P=M+K*v. (At speeds higher than 2m/s the walk is naturally transformed into running. "Race-walking" has higher speeds but involves a very different kind of body motion.) In the expression for power, M represents basal energy spent even when we are standing, which is about 80W (see problem 08/98).
Why do we need to use energy in order to walk on a horizontal plane?
What is the coefficient K in the above expression?
(7/2000) The problem has been solved by Victor Ivanov from Faculty of Physics at Sofia University, Bulgaria (e-mail vgi@phys.uni-sofia.bg). His solution can be read here in postscript format, It also was solved by Travis Brooks from Stanford (e-mail travis@SLAC.Stanford.EDU), and by Lihi Goldsmith from Comverse Network Systems (e-mail Lihi_Goldsmith@icomverse.com).
The answer and solution:
This is what Philip and Phylis Morrison have to say about walking in the March 1999 issue of Scientific American:
Walking resembles the motions of a pendulum. Consider the walker at the moment when one leg is slanted behind, the other slanted ahead, making an inverted V. Soon the lagging leg rises to pass the other. When they pass more or less straight beneath the torso, both feet are in ground contact, and the walker's center of gravity is raised a little, given that the legs, nearly straight and close to vertical, hold the body's center of gravity higher off the floor than the slanted inverted V did. The inverted V forms again, only now with the legs exchanged. The total mechanical energy alternates in form, changing from the kinetic energy of swinging legs and rising body to gravitational potential energy near maximum body height, and back again. A pendulum handles energy similarly, exchanging kinetic energy near the midpoint of its arc for gravitational energy around the point of least height, then up again. The energy so stored is largely recovered and serves to reduce the energy cost of walking by about half of what would be needed were the swinging legs not also stores of gravitational energy. The walker rolls up and over the high midpoint of a step on near- straight legs, then smoothly down again under gravity.
The work that has to be done to raise the center of mass of the body is the most important energy expenditure. However, it is not the only energy required for walking: We also need to maintain some (varying amount) of kinetic energy of the center of mass, as well as some kinetic energy spent on relative (translational and rotational) motion of various limbs relative to the center of mass. The latter energy is relatively small (see, e.g., a very detailed discussion in the paper by P.A. Willems, G.A. Gavanga and N.C. Heglund in The Journal of Experimental Biology 198, 379-393 (1995)). The kinetic energy of the center of mass is rather significant and should be taken into account. On the other hand, the kinetic energy is partially used as to modify the potential energy of the center of mass. Thus, a very crude approximation would be by considering the work needed to raise center of mass. If the inverted V description is close to reality, then assuming the length of the legs of that V is about 1 m while the size of a step is l=0.5 m we find that the height of the center of mass oscillates between "position" /\ and "position" || by about 3 cm, which for 70 kg person would mean that the potential energy changes by A=20 J. If the walking speed is v, then the number of steps per unit time is v/l, and the power required for such walk is P'=A*v/l. Therefore, A/l=40 N is the coefficient K which we needed to find.
This is, of course, a very crude (under)estimate. Real measurements of energy expenditure show that the coefficient K=160 N (approximately).
P.S. We recommend the book Energies, by V. Smil (MIT Press, Cambridge, Mass., 1998). It has some interesting facts about energetics of walking, as well as many useful facts about energy in general.
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