Angular Kinetics. Learning Objectives: Learning Objectives: Properties of Torques (review from Models and Anthropometry) T = F d

Transcription

1 Angular Kinetics Readings: Chapter 11 [course text] Hay, Chapter 6 [on reserve] Hall, Chapter 13 & 14 [on reserve] Kreighbaum & Barthels, Modules I & J [on reserve] 1 Learning Objectives: By the end of this lecture, you should be able to: For a body or system, calculate and define center of mass, moment of inertia, and radius of gyration (given appropriate anthropometric information) Use the parallel axis theorem to calculate moment of inertia of a body or system about various axes For a given movement (e.g. running, throwing, diving), describe how these parameters change 2 Learning Objectives: Use the concepts of moment of inertia, angular momentum, angular impulse, conservation of angular momentum, work, and conservation of energy to calculate the effects of torques (applied over different time periods and distances) on motion Use the concepts of reaction torques, moment of inertia, conservation of momentum, and angular momentum about multiple axes to explain changes in body position during projectile motion Properties of Torques (review from Models and Anthropometry) 1. Axis of rotation 2. Force 3. Moment arm perpendicular distance between force line of action and axis of rotation T = F d 3 moment arm 4 1

2 Centre of Mass (review from Models and Anthropometry) Sum of moments (torques) on all sides of a segment s centre of mass equals zero i.e. the mass is equally distributed about the centre of mass This is not the same as having equal mass on each side! 60 N 20 N Measuring Center of Mass Reaction board method: In static equilibrium, net F = 0 and net M = 0 i.e. F=0 and M=0 Goal: to measure the vertical height of the center of mass of a person. 0.5 m 1.5 m 5 6 R 1 Measuring Center of Mass Reaction board method: Step 1 weigh subject (mg s ). Step 2 empty board positioned Measuring Center of Mass Reaction board method: R 2 Step 3 person + board Scale Reading = R 1 Length of Board = 2 m Scale Reading = R 2 M = 0: -(R 1 )(2) + (mg b )(b) = 0 M = 0: -(R 2 )(2) + (mg b )(b) + (mg s )(s)= 0 R 1-2R 1 + bmg b = 0 2R 1 = bmg b R 2 2 m mg board b m 7 2 m s m mg mg subject 8 board b m 2

3 Measuring Center of Mass Reaction board method: Step 2 R 2 But we know 2R 1 = bmg b M = 0: -2R 2 + bmg bb + smg s = 0-2R 2 + 2R 1 = -smg s Calculating Center of Mass We can calculate the locations of the segment center of masses from segment end points and anthropometric information. If we know the locations of the segment center of masses, we can calculate the total body center of mass. R 2 2 m s m mg mg subject 9 board b m 10 Calculating Center of Mass Calculating Center of Mass 1 (5, 30) 0.5 kg 2 (20, 20) 1 kg Find x-coordinate of system center of mass (x T ): We know that M = F d and M = 0: (m T g)(x T ) = (m 1 g)(x 1 ) + (m 2 g)(x 2 ) + (m 3 g)(x 3 ) y (cm) 0 Question: What are the coordinates of the system s center of mass? x (cm) 3 (30, 10) 2 kg 12 All the g s cancel: m T x T = m 1 x 1 + m 2 x 2 + m 3 x 3 ( )(x T ) = (0.5)(5) + (1)(20) + (2)(30) (3.5)x T = (3.5)x T = 82.5 x T = 82.5/3.5 = 23.6 cm 13 3

4 Stability Stability Pyramid Quite Stable Inverted Pyramid Very Unstable Pendulum Highly Stable Ball Neutral Stability Inverted Pendulum Erector Spinae Anti-Gravity Musculature Abdominals Gluteus Maximus Quadriceps Gastrocnemius and Soleus 4

5 Increased Stability Stability Increased base of support Lower centre of gravity Higher mass Position of centre of gravity with respect to base of support (e.g. leaning into a tackle) Which shape is more stable? To be confident in how to answer that question you would need to ask stable in which direction? Ergonomic Chairs The standard is for five legs to increase base of support. Chairs? Is this a good idea? 5

6 Buckling If the work done on the spine (energy applied) is greater than the work the muscles can do to stiffen the spine, then the spine will buckle. Moments Created by Mass The moments created by segment masses can be used to provide resistance during exercise or rehabilitation. 27 Angular Kinetic Variables Angular kinetics is similar to linear kinetics, but uses angular equivalents to many variables. Torque or moment can be thought of as the angular equivalent of force. In static situations, segment mass, length, and center of mass location is sufficient for analysis. In dynamic activities, we also need to know segment moment of inertia because the segments rotate. 28 Moment of Inertia A segment s moment of inertia (I) represents its resistance to rotation about a given axis. Moment of inertia is the angular equivalent of mass. Moment of inertia depends on the distribution of mass about the axis of rotation: n 2 " i i=1 I= m ir The units of I are kg m

7 Changes to moment of inertia The moment of inertia of a body or system often changes during the course of a movement. Moment of inertia of leg about hip (kg m 2 ) Leg Position Radius of Gyration A segment s radius of gyration (k) represents the distance at which all of its mass can be said to act. Radius of gyration is the angular equivalent of center of mass. I = mk 2 k Questions: k L I = mk 2 Radii of Gyration / segment length (transverse axis) Segment Centre of Mass Proximal Distal head, neck, trunk upper arm arm hand thigh leg foot If the leg (shank) of a subject has a mass of 3.6 kg and is 0.4 m long, what is I g? 2. Why is the ratio in the table usually higher for a transverse axis at the distal end than a transverse axis at the proximal end? 32 Radii of Gyration / segment length (transverse axis) Segment Centre of Mass Proximal Distal head, neck, trunk upper arm arm hand thigh leg foot Answers: I = mk 2 1. Value from table (a) = k/l Therefore k = a L. I g = mk 2 = m(al) 2 = (3.6kg)(0.302 x 0.4m) 2 = kg m 2 2. Muscle mass is usually distributed closer to proximal joints. This favours speed of movement. 33 7

8 Parallel axis theorem If you know the moment of inertia of a segment or body about an axis through its center of mass, you can calculate its moment of inertia about any parallel axis using the parallel axis theorem: I // = I g + mr 2 In this equation: I // I g r Moment of Inertia Examples I // is an axis of rotation parallel to I g I g is the moment of inertia about the center of mass m is the mass of the segment or body r is the distance between I // and I g [Hay, 1993 (gymnasts); Hall, 1999] Angular Angular momentum (H) is the quantity of angular motion of an object. Angular momentum is the angular equivalent of linear momentum, and is also a vector: Conservation of Angular Conservation of Angular : When there is no net external torque acting on a system, then H is constant. H = I ω angular momentum p = m v linear momentum The units of angular momentum are: (kg m 2 )(rad/s) = kg m 2 s 38 e.g. when gravity is the only external force acting on an object, such as a projectile with no air resistance 39 8

9 Conservation of Angular H=Iω I = (mi ri2) If momentum remains constant, changing moment of inertia will change the angular velocity. Tuck position Moment of inertia Angular velocity Time (sec) Diving: watch?v=wtcatdancw&feature=related Trampolining: watch? v=wbv5ajnnl1w&featur e=related Figure skating: (spot the fake physics terminology!) 40 Conservation of angular 41 Maximum Linear Velocity Transfer of momentum via the kinetic chain can maximize linear velocity of distal segments. Question: Why don t bikes fall over? Answer: Partly due to gyroscopic effects created by conservation of angular momentum. Conservation of Angular Relevant equations for this concept: vt = r ω τ=iα I = (mi ri2) 42 H=Iω 46 9

10 Kinetic Chain Kinetic Chain Some Linear and Angular Analogues Some Linear and Angular Analogues Linear variable Linear displacement Angular variable Δx Δy Angular displacement Δθ Linear velocity v Angular velocity ω Linear acceleration a Angular acceleration α Mass m Center of mass Moment of inertia I Radius of gyration k Force F Torque τ M Linear momentum ρ Angular momentum H Linear equation F = m a ρ = m v Δρ = F t F t = m Δv KE linear = ½ m v 2 W = F d Angular equation τ = I α H = I ω ΔH = τ t τ t = ΔI Δω KE rotational = ½ I ω 2 W = τ θ

11 Some Linear and Angular Analogues Some Linear and Angular Analogues Newton s First Law An object at rest stays at rest, and an object in motion moves with constant velocity, unless acted on by an external force. Newton s Second Law A change in motion of an object is proportional to the net force acting on the object, and occurs in the same direction as the force. An object at rest stays at rest, and a rotating object moves with constant angular momentum, unless acted on by an external moment/torque. A change in angular momentum of an object is proportional to the net moment/torque acting on the object, and occurs in the same direction as the moment/torque acts KErotational = ½ I ω2 Angular Impulse Work and Energy The longer you can apply a torque for, the greater the change in angular velocity. Similar to linear kinetics, when angular work is performed on a system, it changes the rotational energy of the system. The more torque you can apply, the greater the change in angular velocity. The angular work done on a system is equal to the change in rotational energy of the system for our purposes, this usually means a change in rotational kinetic energy. ΔH = τ t = ΔI Δω W = ΔE 54 τ θ = ΔPE + ΔKE = ½ Δ(I ω2) 55 11

12 angular rotational Work and Energy Work and energy If there is no change in rotational potential energy and moment of inertia: The total work (linear + angular) performed on a system is equal to the total change in energy (linear + angular) of the system. The larger the angle that you can apply a torque over, the greater the change in angular velocity. W = ΔE The more torque you can apply, the greater the change in angular velocity. Wlinear + Wrotational = ΔPE + ΔKE W = τ θ = ½ Δ(I ω2) ΔE Fd +τθ = ΔPElinear + ΔKElinear + ΔPErotational + ΔKErotational ΔKE this is called an eccentric force Work and Energy Work and Energy A force whose line of action passes through an object s center of mass causes translation but no rotation: A force whose line of action does not pass through an object s center of mass causes translation and rotation: W = ΔE If we assume ΔPE=0: F d F d θ F d = ΔPE + ΔKE F d = ½ Δ(m v2) 58 Note that the center of mass of the object does not move as far as in the last slide. W = ΔE = ΔElinear + ΔErotational If we assume ΔPE=0: F d + τ θ = ΔKElinear + ΔKErotational F d + τ θ = ½ Δ(m v2) + ½ Δ(I ω2) 59 12

13 Some Linear and Angular Analogues Example: To generate a topspin on a tennis ball, the racket applies an eccentric force to the ball. net force of racket on ball Newton s Third Law If one object applies a force on a second object, the second object applies an equal and opposite force on the first. component of force causing rotation (angular movement) center of mass of ball τθ = ½Δ(Iω2) If one object applies a moment/torque on a second object, the second object applies an equal and opposite moment/torque on the first. component of force causing translation (linear movement) Fd = mgδh + ½Δ(mv2) racket Conservation of Angular Reaction Moment Smarter Every Day video #58: watch?v=rtwbpyjjqru 13

14 Conservation of Angular Conservation of Angular Hang Hitch Local and Remote Angular The angular momentum of a body segment about its own center of mass is called local angular momentum. Local and Remote Angular H total = H local + H remote The angular momentum of a body segment about the whole body s center of mass is called remote momentum. The total angular momentum of a body segment is composed of both local and remote angular momentum: H total = H local + H remote 69 [Text, Fig ] = I cm,seg ω cm,seg + (mr 2 )ω cm,body m = mass of segment this comes from the radius of gyration r = distance between segment center of mass and body center of mass 70 14