Μηχανικές αρχές. Spiros Prassas National & Kapodistrian University of Athens

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1 Μηχανικές αρχές 1

2 Σηµαντικοί παράγοντες στην εκτέλεση από µηχανικής απόψεως ικανότητα απόκτησης ύψους ικανότητα περιστροφής ικανότητα αιώρησης ικανότητα προσγείωσης Δύναµη είναι η φυσική οντότητα η οποία προκαλεί η τείνει να προκαλεί αλλαγές στην ταχύτητα ενός σώµατος, δηλαδή προκαλεί επιτάχυνση. δύναµη Η σχέση µεταξύ δύναµης και κίνησης γίνεται κατανοητή µέσω των νόµων του Νεύτωνα. Πρώτος νόµος Νόµος της αδράνειας Δεύτερος νόµος Νόµος της επιτάχυνσης Τα υλικά σηµεία διατηρούν αµετάβλητη τη κινητική τους κατάσταση εκτός εάν αναγκαστούν από κάποια (εξωτερική) δύναµη να µεταπηδήσουν σε άλλη κατάσταση. Η επιτάχυνση του ΚΒ ενός σώµατος είναι ανάλογη της δύναµης που την προκαλεί, στην κατεύυνση αυτής της δύναµης και αντιέτως ανάλογη της µάζας: net 1 a a 1 a net a cm = ext m

3 Τρίτος Νόµος Νόµος Δράσης-Αντίδρασης Όταν ένα σώµα ασκεί µια δύναµη σε ένα άλλο τότε δέχεται µία αντίετη δύναµη δηλαδή µία δύναµη ίσου µέτρου αλλά αντίετης κατεύυνσης. Οι δυνάµεις στη φύση εµφανίζονται κατά ζεύγη. Είναι αδύνατον να εµφανισεί περιττός αριµός. Althugh the magnitude the actin/reactin rces is the same, their eect n the respective bjects are nt why? rictin rictin rictinal rces arise between bjects in cntact. They are parallel t the cntact surace and always ppse, r tend t ppse the relative mtin the bjects invlved rictin rictinal rces are equal t: W N N W = µn Where: µ is the ceicient rictin, and N is the perpendicular (Nrmal) rce between the tw bjects rictin Static rictin Kinetic rictin therere, rictinal rces can be altered by altering either µ: hw? Dierent lrs Dierent shes Dierent tires Dierent lubricants Arises between suraces at rest in relatin t each ther s µ s N N s Arises between suraces in relative mtin k = µ k N Or N: hw? Alter weight (mass) Alter psitin, is variable in magnitude w is independent velcity Will the skier speed all the way dwn? Why yes/nt? 3

4 Ροπή ροπή = * d Η ροπή αντιπροσωπεύει την επίδραση της δύναµης στην κυκλική κίνηση. Το µέγεος της ροπής εξαρτάται από το µέγεος της δύναµης, την κατεύυνση της δύναµης και την απόσταση από το σηµείο εξάσκησης της δύναµης µέχρι τον άξονα περιστροφής. d " = d sin τ = d (1) τ = d () τ d d W τ w W ροπή Ροπές που παράγουν η τείνουν να παράγουν cunterclckwise (CCW) κυκλικές κινήσεις είναι ετικές. clckwise (CW) κυκλικές κινήσεις είναι αρνητικές. Θετικές (CCW) η αρνητικές ροπές (CW) δεν πρέπει να συνδέονται αναγκαστικά µε σχετικές κινήσεις των αρρώσεων (δίπλωση, έκταση, κλπ.) Αρνητική ροπή/δίπλωση Θετική ροπή/έκταση The rtatinal equivalent t =ma (Newtn s nd Law) is: " = I 4

5 ροπή Κυκλική αδράνεια (I) αναφέρεται στην ιδιότητα των φυσικών όντων να αντιστέκονται σε αλλαγές της (υπάρχουσας) κυκλικής κίνησης m W=300N d w =35cm d m =3.5cm m =65 Παράγοντες που επηρεάζουν I: Μάζα Απόσταση της µάζας από το άξονα περιστροφής Practical Implicatins W Hw much rce will the muscle has t exert t hld the weight in that psitin? m I: "# = 0 W (.035)(sin 65 ) (300)(.35)(sin 90 ) = 0 m (300)(.35)(sin 90 ) m = = 3310N (.035)(sin 65 ) R mg/ m R =54N d R =60cm mg=500n d mg =1cm d m =5cm ind m R mg/ m " $ = I# = 0 mg ( m )( d ) ( R )( d ) ( )( d m R mg ) = 0 (54)(.60) + (50)(.1) m = = 3648N.05 Just in case yu are wndering 300N=67.4lb, just try t hld that much weight Nte: slightly less, because we did nt cnsider the weight the upper extremities still ~6Xbdy weight prjectile mtin Prjectile Mtin Gravity Air Resistance in sme activities, it is practical imprtance in sme, it is r it is cnsidered negligible when air resistance is negligible, trajectries are parablic the shape the parabla depends n the prjectin velcity a b c 5

6 prjectile mtin vy0 v0 Vertical Mtin vertical mtin The vertical mtin a prjectile is deined by the equatins r ree all, i.e mtin with cnstant acceleratin (gravity, g), where: H I the prjectin and landing pints are leveled, the height (H) a prjectile is given by the equatin : R vy0 v vy0 v vy(i) = v(i) sin, and vy() = v(i) sin - gt vx0 vyt H= (v i sin " ) g H Hrizntal Mtin vertical mtin I the prjectin and landing pints are NOT leveled, the height (H) a prjectile is given by the equatin : the vx is cnstant thrughut the light The hrizntal displacement r range (R) a prjectile is given by: H (v sin " ) H= i ±h g H Speed prjectin Angle prjectin Relative height v R = v x i.t air -h vx +h R= actrs Aecting Prjectile Mtin (v i sin" ) g R R= v0 sin cs + v0 cs (v0 sin ) + gh g actrs aecting Optimum Angle amng the (3) actrs aecting prjectile mtin, and range in particular, prjectin speed is the mst critical that is, an equal change in speed, eects mtin (R) mre than equivalent % changes in any the ther tw variables R= (v i sin" ) g In bimechanics, a requent bjective is t maximize hrizntal distance, r range This requires the selectin an ptimum angle. The magnitude this angle depends n the relative psitin the prjectin and landing pints 6

7 ptimum angle When the prjectin and landing pints are leveled, the ptimum angle is 45 degrees. ptimum angle When the landing pint is higher than the prjectin pint, the ptimum angle is mre than 45 degrees. When the landing pint is lwer than the prjectin pint, the ptimum angle is less than 45 degrees. the exact value this angle depends n bth, the relative height and speed prjectin. R= (v i sin" ) g Wrk Wrk Energy Pwer Mmentum Wrk Mechanical wrk is deined as : r rtary mtin, mechanical wrk is deined as : W = " d... W = " # W = d cs d Mechanical Energy is represented by the ability bjects t d Wrk because Their mtin (kinetic ) Psitin (gravitatinal ptential ) Cniguratin (elastic ) Wrk-Energy relatinship ME = KE + PE + EE ME = ( mv + I ) + (mgh) + ( kx ) Under special circumstances, the sum the kinetic and (gravitatinal) ptential energy a system is cnstant, i.e. it is cnserved. i PE is negligible, the Wrk-Energy relatinship is expressed as llws: KE = KEi + W Practical Implicatins 7

8 Wrk-Energy relatinship KE W = KE W = KE = KE + W i " KE i The amunt wrk that the H O des n the diver is set By diving deep int the pl, the rce ding the wrk is small W= d Again, the amunt wrk that the H O is ding n the diver is set the same as in the previus dive. I, hwever, the diver belly laps (r back laps as he/she did) int the pl, he/she pays the price W= d practical applicatins d d Pwer W d P = = = v t t W "# P = = = " t t # Practical signiicance: in bicycling What gear? in running What stride length/requency? r c e / P w e r V e l c i t y Pwer rce Maximum pwer is achieved in abut 30%- 50% maximum velcity cntractin Linear Mmentum B HB rm Hay, J 8

9 linear mmentum is the quantity (linear) mtin pssessed by an bject/bdy is prprtinal t the prduct the mass and the velcity pssessed by an bject/bdy M = mv in the absence external rces, the linear mmentum a system is cnstant (equatin) m1 vi + mvi = m1v + mv r m 1v1 + mv r 1 1 v = cnstant m v + m = 0 Impulse/impulse-mmentum relatinship The prduct rce and time let side the equatin abve is knwn as Impulse (J) and equatin (1) describes the Impulse-mmentum relatinship Since t = mv mv i (1) t The Impulse that the H O is ding n the diver is set By diving deep int the pl, the rce the Impulse will be small J= t t Again, the Impulse that the H O is ding n the diver is set the same as in the previus dive. I, hwever, the diver belly laps (r back laps) int the pl, he/she pays the price J= t Lcmtin Speeding up* r c e Negative Impulse Slwing dwn* Psitive Impulse T i m e Speed changes i there is a dierence between the psitive and negative impulses in the illustrated case, the subject will speed-up * (why?) *in this example, rward is the psitive directin 9

10 angular mmentum is the quantity (angular) mtin pssessed by an bject/bdy is prprtinal t the prduct the mment inertia and the angular velcity pssessed by an bject/bdy Cnservatin angular mmentum Angular mmentum is cnstant, i.e. it is cnserved in the absence external trques L = I Angular impulse/angular mmentum relatinship therere it changes nly when external trques act r a time perid ) * " *i & $$ + = I, = I '' ( t % + # t = I* " I*i = L Practical applicatins angular mmentum transer angular mmentum The ttal angular mmentum a multisegment system is made up the sum the angular mmenta its parts, i.e. The cnservatin L principle, plus the act that ttal L is made up the sum the angular mmenta its parts, is utilized in rder t transer mmentum Amng the parts, and Amng dierent axes rtatin L = l1 + l +... Practical implicatins Practical implicatins early release rm Kreighbaum, E (mdiied) 10

11 late release just right 11