QUALITATIVE AND QUANTITATIVE ANALYSIS OF MUSCLE POWER Jey N. Baham Anand B. Shetty Mechanical Kinesiology Laboatoy Depatment of Kinesiology Univesity of Nothen Coloado Geeley, Coloado Muscle powe is one mechanical quantity that is not clealy undestood by many coaches and teaches of human movement. This is pobably due to an inadequate diffeentiation between the concepts of wok/enegy and of powe. Thus we will begin this pape with a discussion of these concepts. The analysis of the vetical jump a8 a measue of wok, enegy, and powe will also be discussed. THE CONCEPTS OF MECHANICAL WORK AND ENERGY A moto task commonly used as a demonstation of muscle poduced wok and powe is the vetical jump. Thee is a majo poblem, howeve, with this demonstation in that the height of a jump is poduced by the instantaneous velocity of the jumpe at the moment of elease fom the suppoting suface and neithe the height of the jump no the elease velocity ae measues of powe. These quantities ae the poducts of the wok that has been done duing the act of jumping and not of powe. In othe wods, the task of vetical jumping equies that a muscle poduced foce act upon the segments of the lowe extemity though a distance which esults in the poduction of mechanical enegy. The amount of wok done is equal to the amount of enegy poduced. The analysis of a vetical jump equies that we measue thee positions of the cente of mass and two displacements. The thee positions ae Y l, the lowest couched position; Y 2 the extended position just as the body eleases fom the floo; and Y 3, the highest point of the jump. These positions ae shown in Figue 1. The two displacements ae defined fom the position data as follows: --The fist displacement. which we will call height one (h l ) is the diffeence between positions one and two 1 i.e. This is the distance that the cente of mass is aised duing the suppot phase of the jump. 383
VJ V, VI Figue 1. Thee positions involved in the analysis of a vetical jump. These positions ae Y1' the lowest couched position; Y 2 the extended position just as the body eleases fom the floo; and Y 3, the highest point of the jump. E.I / lb I.' I.....' I~ ~ ~ ~l / ;... :... : I t. t. Figue 2. Two unnes A and B obtain the same kinetic enegy E k, but unne A does it in the shote time ta' while unne B does it in longe time t b Runne A, theefoe, is by definition the moe poweful of the two. 384
--The second displacement, which we will call height two (h 2 ) is the diffeence between positions one and thee, i.e., (Equation 2). This is the total distance that the cente of mass is aised duing the jump. --A thid displacement, which is the height to which the cente of mass is pojected duing the aibone phase of the jump, is defined as the diffeence between heights one and two, i.e., (Equation 3). The Measuement of Wok and Enegy The thee displacements can be used in the measuement of wok and the amount of enegy poduced duing a vetical jump as follows: --Wok is defined as the poduct of foce (F) and the distance (d) though which the foce acts. Thus duing a vetical jump the muscles of the lowe extemities poduce foce though the fist distance (hi) and theeby do wok upon the jumpe. In fomula fom this would be: Wok e Fd e Fh l (Equation 4). --Enegy (E) is defined as the ability to do wok. The two types of mechanical enegy involved in a vetical jump ae identified as gavitational potential enegy (E ) and kinetic enegy (E k ). These two foms of enegy ag defined as follows: --Gavitational Potential Enegy (Equation 5). When the cente of gavity of the jumpe is aised to h 2 the foce of gavity (mg) then does wok though that distance in binging the body of the jumpe back to the stating point (Y I ). Thus the muscles do wok in aising the body to hi' and gavity doea wok in loweing the body back though the distance h 2 385
--Kinetic Enegy E k : 1/2mV 2 (Equation 6). When the body of the jumpe eleases fom the suface it has the the velocity of elease V, which gives the body its kinetic enegy 1/2~V2 and it is this enegy that then does the wok of pojecting the body into the ai to the height (A h). The kinetic enegy finally ends up as potential enegy (mga h) at the top of the jump. It should be emembeed that the potential enegy at the top of the jump (mg A h) is equal in ma~nitude to the kinetic enegy at elease (1/2 mv ). The total amount of wok that the muscles poduce can be measued though Newton's Second Law expessed In tems of wok/enegy elationships as: Wok I'd : Fh l : mgh l + 1/2 mv 2 u This equation shows that the wok done duing a vetical jump when muscle foces act though a distance (Fh l ) is equal to the gavitational potential enegy plus the kinetic enegy that it poduces. (equation 7). THE CONCEPT OF POWER AND ITS SIGNIFICANCE The measuement of powe equies us to divide both sides of equation 7 by tlte time (t) to obtain Powe : Fh l mgh l + 1/2mv 2 (equation 8). Th~ ight side of equation 8 is especially inteesting hecause it gives the ate of enegy poduction. In many human pefomance situations it is the ate of kinetic enegy poduction that is the most impotant. A given amount of kinetic enegy can be poduced by eithe a lage foce acting though a shot distance o a smalle foce acting though a longe distance. Applying the foce, howeve, though the longe distance will usually equie moe time so that the ate at which the kinetic enegy is poduced is less and by definition thee is less muscle powe. We can use spinting as an example of a powe event. Let us assume hat we have two spintes (A and B) and that each has the same mass and each is capable of poducing the same kinetic enegy. If we constuct a kinetic enegy/time gaph it might look like that shown in Figue 2. Both unnes A and B develop the same kinetic enegy, but unne A does it moe apidly and is, theefoe, the moe poweful of the two. When execution time is a limiting facto in pefomance as in baseball batting and in spinting, how fast one can develop kinetic enegy of eithe the body as a whole o of its pats becomes a cucial facto. Thus because of the impotance of powe in vaious types of pefomance it is essential that we have a valid method of measuing it. 386
ANALYSIS OF THE VERTICAL JUMP Two papes wee pesented at the thid ISBS Symposium held in Geeley, Coloado last yea (1985) that descibe some of ou wok at the Univesity of Nothen Coloado in the analysis of muscle powe. The fist was a pape by Paul A. Lightsey, Depatment of Physics, Univesity of Nothen Coloado in which he pesented an oiginal fomula fo the measuement of powe fom displacement data obtained fom a filmed vetical jump. The second pape was by Shetty, et. al., (1985) which descibed a pilot study designed to validate the Lightsey fomula. A thid pape in this seies, which summaizes the fist two papes and epots the esults of a followup study is pesented by Shetty and Baham (1986) elsewhee in these poceedings. In this thid pape it is concluded that the displacement based algoithm developed by Lightsey is valid and that it can be used as a eliable and inexpensive method of measuing muscle powe. The Lightsey algoithms can also be used to measue aveage foce, aveage velocity, duation of the suppot phase of the jump, as well as the aveage wok/enegy and powe involved in the jump. The Analysis ~ Aveage Foce --Poposition: wok done by the muscles of the lowe extemity duing the suppot phase of the jump is equal to the potential enegy of the cente of mass at its geatest height in the ai, Le., so aveage foce (i) is (Equation 9). --The atio h 2 /h 1 can be put in a diffeent fom as follows: --Accoding to the pinciples of paabolic motion, the vetical displacement ( t. h) of the jumpe afte elease fom the suface is given by: (Equation 10). --The height (hi) that the cente of mass is aised duing suface contact is given by: (Equation 11). --So: h ; - 2 h +Ah ; l+ah ; 1 l+v ; l+a (Equation 12). ---- -- hi hi hi gt g 387
--When Equation 12 is substituted into Equation 9 we obtain F mg(h 2 /h l ) ; mg(l+a/g) ; mg + ma (Equation 13). --This equation (#13) shows that the total aveage foce poduced by the muscles of the lowe extemity duing the suppot phase of the jump is equal to the foce equied to move the body weight (mg) plus the net foce that acceleates the body mass (ma) to the velocity at which the jumpe eleases fom the suface (V). It is the elease velocity (V ), of couse, t~at poduces the height of the jump (e. Equation 10). Analysis ~ Aveage Velocity --Aveage velocity (V), when the oiginal velocity is zeo, is given by v V + V 1 /2V (Equa tion 14). o 2 --Reaanging Equation 10 we obtain the elease velocity (V ) as being equal to V ;~ ( Equa t ion 15). --The velocity of elease is also given by V ;~ (Equation 16). whee a is net acceleation, Le., a ; F/m - g (Equation 17). --Substituting Equation 17 into Equation 16 we obtain V ;"y2( F/m - g)h l (Equation 18). --Substituting Equation 18 into Equation 14 we obtain V; l/212(f/m - g)h l ; 'Vl/2(F/m - g)h l (Equation 19). --Substituting Equation 9, fom the section on foce analysis, into Equation 19 we obtain vet 1/2 [mg( h 2 /h l ) - g] hi m ;1 1/ 2 [g( h 2 /h l ) - g] hi =11/2g[( h 2 /h l ) - 1 J hi (Equation 20). Thus the two vaiables that detemine the aveage velocity of the jumpe duing the suppot phase of the jump ae the magnitudes of hi and h 2 388
Analysis of the Duation ~ ~ Suppot Phase --The duation (t) of the suppot phase of the jump i~ equal to hi divided by the aveage velocity (V), Le. (Equation 21). --Substituting Equation 20 into Equation 21 we obtain t ~~ V (Equation 22). Analysis ~ Wok and Powe --Powe has been defined as the ate of doing wok and accoding to Equation 8 it is given by --When the aveage foce equation (#9) and the time equation (#22) ae substituted into Equation 8 we obtain: mg h 2 "'V 1/2pj h2/hc1 )h 1 hi mg( h 2 /h l )11/2pj h 2 /h l -l)h 1 mpj h 2 /h l )11/2pj h 2 - hi) mg( h 2 /h 1 )"'( 1/2g(.0. h) (Equation 23). Equation 23 is Lightsey's displacement based algoithm fo the ~alculation of muscle powe fom the two height measues obtsined duing a vetical jump. Lightsey, P. A. A Fomula fo Measuement of Leg Powe in the Vetical Jump. Biomechanics in Spots 11, edited by Juis Teauds and Jey N. Baham. Reseaach Cent e fo Spots, Del Ma, Califonia, 1985. Shetty, A., K. Spoone, J. N. Baham, and P. A. Lightsey. Validation of Lightsey Leg Powe Fomula. Biomech;anics in Spots 11, edited by Juis Teauds and Jey N. Bahan. ReseachCente-{o-Spot"g;- Del :-ta, Ca11fon1a-:-1985. - - ---- ---- 389