Spiros Prassas National & Kapodistrian University of Athens Μηχανικές αρχές …

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1 Spiros Prassas National & Kapodistrian University of Athens Μηχανικές αρχές …

2 Spiros Prassas National & Kapodistrian University of Athens

3 Spiros Prassas National & Kapodistrian University of Athens

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10 Spiros Prassas National & Kapodistrian University of Athens

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12 Spiros Prassas National & Kapodistrian University of Athens

13 Spiros Prassas National & Kapodistrian University of Athens Σημαντικοί παράγοντες στην εκτέλεση από μηχανικής απόψεως …ικανότητα απόκτησης ύψους …ικανότητα περιστροφής …ικανότητα αιώρησης …ικανότητα προσγείωσης

14 Spiros Prassas National & Kapodistrian University of Athens Δύναμη …είναι η φυσική οντότητα η οποία προκαλεί η τείνει να προκαλεί αλλαγές στην ταχύτητα ενός σώματος, δηλαδή προκαλεί επιτάχυνση.

15 Spiros Prassas National & Kapodistrian University of Athens … δύναμη Η σχέση μεταξύ δύναμης και κίνησης γίνεται κατανοητή μέσω των νόμων του Νεύτωνα.

16 Spiros Prassas National & Kapodistrian University of Athens Πρώτος νόμος — Νόμος της αδράνειας Τα υλικά σημεία διατηρούν αμετάβλητη τη κινητική τους κατάσταση εκτός εάν αναγκαστούν από κάποια (εξωτερική) δύναμη να μεταπηδήσουν σε άλλη κατάσταση.

17 Spiros Prassas National & Kapodistrian University of Athens Δεύτερος νόμος – Νόμος της επιτάχυνσης Η επιτάχυνση του ΚΒ ενός σώματος είναι ανάλογη της δύναμης που την προκαλεί, στην κατεύθυνση αυτής της δύναμης και αντιθέτως ανάλογη της μάζας:

18 Spiros Prassas National & Kapodistrian University of Athens F1F1 a1a1 F2F2 a2a2 F net a net

19 Spiros Prassas National & Kapodistrian University of Athens Τρίτος Νόμος — Νόμος Δράσης-Αντίδρασης Όταν ένα σώμα ασκεί μια δύναμη σε ένα άλλο τότε δέχεται μία αντίθετη δύναμη δηλαδή μία δύναμη ίσου μέτρου αλλά αντίθετης κατεύθυνσης. Οι δυνάμεις στη φύση εμφανίζονται κατά ζεύγη. Είναι αδύνατον να εμφανισθεί περιττός αριθμός. Although the magnitude of the action/reaction forces is the same, their effect on the respective objects are not…why?

20 Spiros Prassas National & Kapodistrian University of Athens Friction

21 Spiros Prassas National & Kapodistrian University of Athens …friction Frictional forces arise between objects in contact. They are parallel to the contact surface and always oppose, or tend to oppose the relative motion of the objects involved

22 Spiros Prassas National & Kapodistrian University of Athens W N

23 Spiros Prassas National & Kapodistrian University of Athens W N f

24 Spiros Prassas National & Kapodistrian University of Athens …friction Frictional forces are equal to: Where:  is the coefficient of friction, and N is the perpendicular (Normal) force between the two objects

25 Spiros Prassas National & Kapodistrian University of Athens …friction …therefore, frictional forces can be altered by altering either  : how?  Different floors  Different shoes  Different tires  Different lubricants Or N: how?  Alter weight (mass)  Alter position,

26 Spiros Prassas National & Kapodistrian University of Athens Static friction Arises between surfaces at rest in relation to each other… w N F fsfs …is variable in magnitude…

27 Spiros Prassas National & Kapodistrian University of Athens Kinetic friction Arises between surfaces in relative motion… …is independent of velocity… Will the skier speed all the way down? Why yes/not?

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29 Spiros Prassas National & Kapodistrian University of Athens Ροπή

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31 Spiros Prassas National & Kapodistrian University of Athens …ροπή Η ροπή αντιπροσωπεύει την επίδραση της δύναμης στην κυκλική κίνηση. Το μέγεθος της ροπής εξαρτάται από το μέγεθος της δύναμης, την κατεύθυνση της δύναμης και την απόσταση από το σημείο εξάσκησης της δύναμης μέχρι τον άξονα περιστροφής.

32 Spiros Prassas National & Kapodistrian University of Athens  d F  = F d (1)  =F d (2)

33 Spiros Prassas National & Kapodistrian University of Athens dFdF F   W dWdW ff ww

34 Spiros Prassas National & Kapodistrian University of Athens Ροπές που παράγουν η τείνουν να παράγουν counterclockwise (CCW) κυκλικές κινήσεις είναι θετικές. … clockwise (CW) κυκλικές κινήσεις είναι αρνητικές. Θετικές (CCW) η αρνητικές ροπές (CW) δεν πρέπει να συνδέονται αναγκαστικά με σχετικές κινήσεις των αρθρώσεων (δίπλωση, έκταση, κλπ.)  

35 Spiros Prassas National & Kapodistrian University of Athens Αρνητική ροπή/δίπλωση Θετική ροπή/έκταση

36 Spiros Prassas National & Kapodistrian University of Athens …ροπή The rotational equivalent to F=ma (Newton’s 2 nd Law) is:

37 Spiros Prassas National & Kapodistrian University of Athens …ροπή Κυκλική αδράνεια (I) αναφέρεται στην ιδιότητα των φυσικών όντων να αντιστέκονται σε αλλαγές της (υπάρχουσας) κυκλικής κίνησης… Παράγοντες που επηρεάζουν I: Μάζα Απόσταση της μάζας από το άξονα περιστροφής

38 Spiros Prassas National & Kapodistrian University of Athens Practical Implications

39 Spiros Prassas National & Kapodistrian University of Athens W FmFm W=300N d w =35cm d F m =3.5cm  F m =65 o How much force will the muscle has to exert to hold the weight in that position?

40 Spiros Prassas National & Kapodistrian University of Athens W FmFm Just in case you are wondering…300N=67.4lb, just try to hold that much weight…

41 Spiros Prassas National & Kapodistrian University of Athens If: F R =254N d F R =60cm mg=500N d mg =12cm d F m =5cm Find F m FmFm FRFR mg/2

42 Spiros Prassas National & Kapodistrian University of Athens FmFm FRFR mg/2 Note: slightly less, because we did not consider the weight of the upper extremities—still ~6Xbody weight!

43 Spiros Prassas National & Kapodistrian University of Athens Projectile Motion

44 Spiros Prassas National & Kapodistrian University of Athens Gravity Air Resistance  …in some activities, it is of practical importance  …in some, it is—or it is considered—negligible

45 Spiros Prassas National & Kapodistrian University of Athens …projectile motion… …when air resistance is negligible, trajectories are parabolic a b c …the shape of the parabola depends on the projection velocity

46 Spiros Prassas National & Kapodistrian University of Athens v0v0 vx0vx0 vy0vy0 R H …projectile motion…

47 Spiros Prassas National & Kapodistrian University of Athens Vertical Motion The vertical motion of a projectile is defined by the equations for free fall, i.e motion with constant acceleration (gravity, g), where: v y (i) = v (i) sin , and v y(f) = v (i) sin  - gt vy0vy0 vovo vytvyt 

48 Spiros Prassas National & Kapodistrian University of Athens …vertical motion… If the projection and landing points are leveled, the height (H) of a projectile is given by the equation : vy0vy0 vovo H 

49 Spiros Prassas National & Kapodistrian University of Athens …vertical motion… If the projection and landing points are NOT leveled, the height (H) of a projectile is given by the equation : H -h +h H

50 Spiros Prassas National & Kapodistrian University of Athens Horizontal Motion R  vovo vxovxo …the v x o is constant throughout the flight… The horizontal displacement or range (R) of a projectile is given by:

51 Spiros Prassas National & Kapodistrian University of Athens Factors Affecting Projectile Motion Speed of projection Angle of projection Relative height

52 Spiros Prassas National & Kapodistrian University of Athens

53 Spiros Prassas National & Kapodistrian University of Athens …factors affecting … …among the (3) factors affecting projectile motion, and range in particular, projection speed is the most critical…  …that is, an equal change in speed, effects motion (R) more than equivalent % changes in any of the other two variables…

54 Spiros Prassas National & Kapodistrian University of Athens Optimum Angle In biomechanics, a frequent objective is to maximize horizontal distance, or range… This requires the selection of an “optimum” angle. The magnitude of this angle depends on the relative position of the projection and landing points…

55 Spiros Prassas National & Kapodistrian University of Athens …optimum angle… When the projection and landing points are leveled, the “optimum” angle is 45 degrees.

56 Spiros Prassas National & Kapodistrian University of Athens …optimum angle… When the landing point is higher than the projection point, the “optimum” angle is more than 45 degrees. When the landing point is lower than the projection point, the “optimum” angle is less than 45 degrees.  …the exact value of this angle depends on both, the relative height and speed of projection.

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58 Spiros Prassas National & Kapodistrian University of Athens Work Energy Power Momentum

59 Spiros Prassas National & Kapodistrian University of Athens Work Mechanical work is defined as : d F

60 Spiros Prassas National & Kapodistrian University of Athens Work …for rotary motion, mechanical work is defined as :

61 Spiros Prassas National & Kapodistrian University of Athens …Mechanical Energy… …is represented by the ability of objects to do Work because of…  Their motion (kinetic…)  Position (gravitational potential…)  Configuration (elastic…)

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63 Spiros Prassas National & Kapodistrian University of Athens Work-Energy relationship Under special circumstances, the sum of the kinetic and (gravitational) potential energy of a system is constant, i.e. it is “conserved”. …if PE is negligible, the Work-Energy relationship is expressed as follows: Practical Implications

64 Spiros Prassas National & Kapodistrian University of Athens Work-Energy relationship …practical applications

65 Spiros Prassas National & Kapodistrian University of Athens W= F d F d The amount of work that the H 2 O does on the diver is set… By diving deep into the pool, the force doing the work is small…

66 Spiros Prassas National & Kapodistrian University of Athens d F Again, the amount of work that the H 2 O is doing on the diver is set—the same as in the previous dive. If, however, the diver belly flaps (or back flaps—as he/she did) into the pool, he/she pays the price… W= F d

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68 Spiros Prassas National & Kapodistrian University of Athens Power #2 Practical significance:  …in bicycling What gear?  …in running What stride length/frequency?

69 Spiros Prassas National & Kapodistrian University of Athens Force/PowerForce/Power V e l o c i t y Power Force Maximum power is achieved in about 30%- 50% of maximum velocity of contraction

70 Spiros Prassas National & Kapodistrian University of Athens From Hay, J…

71 Spiros Prassas National & Kapodistrian University of Athens FB HB

72 Spiros Prassas National & Kapodistrian University of Athens Linear Momentum

73 Spiros Prassas National & Kapodistrian University of Athens …linear momentum… …is the “quantity of (linear) motion” possessed by an object/body …is proportional to the product of the mass and the velocity possessed by an object/body…

74 Spiros Prassas National & Kapodistrian University of Athens …in the absence of external forces, the linear momentum of a system is constant… (equation)

75 Spiros Prassas National & Kapodistrian University of Athens or constant or 0

76 Spiros Prassas National & Kapodistrian University of Athens Impulse/impulse-momentum relationship The product of Force and time—left side of the equation above—is known as Impulse (J) and equation (1) describes the Impulse-momentum relationship (1) Since…

77 Spiros Prassas National & Kapodistrian University of Athens J= F t F t The Impulse that the H 2 O is doing on the diver is set… By diving deep into the pool, the force of the Impulse will be small…

78 Spiros Prassas National & Kapodistrian University of Athens t F Again, the Impulse that the H 2 O is doing on the diver is set—the same as in the previous dive. If, however, the diver belly flaps (or back flaps!) into the pool, he/she pays the price… J= F t

79 Spiros Prassas National & Kapodistrian University of Athens ForceForce Negative Impulse Positive Impulse T i m e Slowing down * Speeding up * Speed changes if there is a difference between the positive and negative impulses—in the illustrated case, the subject will “ speed-up ” * (why?) *in this example, forward is the positive direction Locomotion

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82 Spiros Prassas National & Kapodistrian University of Athens …angular momentum… …is the “quantity of (angular) motion” possessed by an object/body …is proportional to the product of the moment of inertia and the angular velocity possessed by an object/body…

83 Spiros Prassas National & Kapodistrian University of Athens Conservation of angular momentum …Angular momentum is constant, i.e. it is “conserved” in the absence of external torques…

84 Spiros Prassas National & Kapodistrian University of Athens Angular impulse/angular momentum relationship …therefore it changes only when external torques act for a time period Practical applications

85 Spiros Prassas National & Kapodistrian University of Athens …angular momentum… …The total angular momentum of a multi- segment system is made up of the sum of the angular momenta of its parts, i.e. Practical implications

86 Spiros Prassas National & Kapodistrian University of Athens …”transfer” of angular momentum… The conservation of L principle, plus the fact that total L is made up of the sum of the angular momenta of its parts, is utilized in order to “transfer” momentum…  Among the parts, and  Among different axes of rotation Practical implications

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88 Spiros Prassas National & Kapodistrian University of Athens From Kreighbaum, E…(modified)

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90 Spiros Prassas National & Kapodistrian University of Athens …early release

91 Spiros Prassas National & Kapodistrian University of Athens …late release

92 Spiros Prassas National & Kapodistrian University of Athens …just right