1 Topics ? Newton?s Laws ? Mechanical Behaviors of Bodies in Contact ? Kinetic Relationship Mechanical Behavior of Bodies in Contact ? Friction ? Static, dynamic and rolling ? Momentum ? Conservation of momentum ? Impulse ? Impulse and momentum relationship ? Impact ? Coefficient of elasticity & restitution Friction ? A force acting at the interface of surfaces in contact ? Opposite to the direction of motion or impending motion Fa W Fs motion impending motion ? F s = ? s R ? ? s : coefficient of static friction ? R: normal force (reaction to the object?s weight) R 2 Types of Friction ? Static friction ? Kinetic friction Rolling friction ? friction Static Friction ? No applied force (F a ) ? No friction, no motion ? F a <= F s,max Fa W Fs R ? F a = F s ? No motion or impending ? F a > F s,max ? Coefficient of friction ? ? Kinetic friction ? F k < F s,max Applied force F s,max F k F s Static Friction ? F s,max = ? s R ? ? s = F s,max /R Nt f f Fa W F ? Nature o sur aces ? Interactions of molecules ? Dry surfaces ? Constant, regardless of contact area s R 3 Kinetic Friction ? F k = ? k R ? ? k = F k / R ? F 375 N Fs R Example ? Moving a box ? Push or Pull? ? Direction of force application N F NF F Ff f ?Push ? More difficult? ?Pul ?Easier? N F Ff N F Ff 5 Example ? Race Car Tires ? Is a wider tire better in race cars? ? Increased friction? ? Not really ? Fs = ? R ? Why wider tire? ? Increased weight Other Applications ? Friction good or bad? ? Benefits ? Increased stabilityIncreased st bilit ? Performance enhancement ? Acceleration & sudden change of direction ? Drawbacks ? Injuries on artificial turf Other Applications ? Shoe design ? Material ? Stud types 6 Determination of Coefficient ? Force platform/force sensors C ffi i t f f i ti ? oe c en o r ction ? Shoes on diff. surfaces Momentum ? The quantity of motion that an object possesses ? At an instant V m ? M = mv ? Unit: kg m/s ? Vector Conservation of Momentum ? The total momentum in a particular direction does not change unless an external force acts on the m 2 Block at rest external acts on the system in the direction of motion ? M 1 + M 2 = M 1+2 frictionless m 2 m 1 v 1 Block at rest Before impact 7 Example ? M 1 + M 2 = M 1+2 ? Before impact ? M 1 = m 1 v 1 ? M =0 frictionless m 2 m 1 v 1 Block at rest 2 = 0 Before impact frictionless v f m 1 + m 2 After impact ?After impact ? M 1+2 = (m 1 +m 2 )v f Example ? A 90 kg hockey player traveling at a velocity of 6 m/s collides head-on with an 120 kg player traveling at - 7m/s What is their m a m b . What is combined velocity after collision? v a v b Example ? Cont. ? Known ? m a = 90 kg, V a = 6 m/s ? m b = 120 kg,V b = -7 m/s ? Unknown V a+b = ? m a m b ? M a + M b = M a+b ? m a V a + m b V b = m a+b V a+b ()()( )()( )90 6 120 7 90 120kg kg kg V m s m s a+b +?=+ 540 840 210kg kg kg V m s m s a+b ?? ?= ? V kg 210 a+b m s m s =? ? =? 300 143. In the direction of player B. v a v b 8 Impulse ? An external force applied over a period of time ? An accumulative event Impulse: I=F t F vm ? I = ? ? T: amount of time ? F: force ? Unit: Ns ? Vector Example 0 200 400 600 800 1000 Fo rc e (N) 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time (sec) Typical Vertical GRF - Gait F1 F2 0 200 400 600 800 1000 Fo rc e (N) Time (sec) Typical Vertical GRF - Gait Impulse = F?t Example - Impulse 400 600 800 1000 o rce (N) Typical Vertical GRF - Gait ? = n ii tFI ? Impulse = F?t ? Impulse ? Area under the curve ? Total amount of effort 0 200 F o Time (sec) =i 1 9 Example - Impulse ? Breaking impulse ? Decelerating ? Negative -300 -200 -100 0 100 200 Forc e (N ) Typical A/P GRF - Gait Peak braking force Peak propulsive force ? Propulsive impulse ? Accelerating ? Positive -300 -200 -100 0 100 200 Forc e (N ) Time (sec) Typical A/P GRF - Gait Propulsive Impulse Breaking Impulse Impulse of Horizontal GRF 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time (sec) Relationship: Impulse & Momentum ? An external impulse applied to a system causes a change of the system?s momentum ? ?I = ?M ? F?t= m?v ? Derived from the 2 nd law ? F = ma Example ? Discussion! ? Impulse = momentum ? ?I = ?M ? F?t= m?v ? F and ?t ? Pitching ? To do what? ? ? Catching ? To do what? ? ? Any other examples? 10 Comments about Impulse and Momentum ? Impulse ? An accumulative event ? Momentum An instantaneous event ? instantaneous event ? Impulse of entire takeoff ? Change of velocity at the takeoff Vertical GRF in jumping Example ? A golf ball of 0.08 kg is struck by a golf club with an average force of 300 N for a duration of 0.01 second. What is the momentum and the velocity of the club head at the time of release? ? Known: m = 0.08 kg, ? t = 0.01s, F = 300 N ? Unknown: ?M = ? V f = ? Example ? cont ? Known: m = 0.08 kg, ? t = 0.01s, F = 300 N ? ?M = ? V f = ? ? Solution: ? F?t = m?v ? 1) ?M = Ft = (300 N)(0.01 s) = 3 kg m/s ? 2) Ft = m(V f -V i ) m(V F t V Ft m (300 N)(0.01s) 0.08 kg 37.5 f f m s ?=? = ? == V i ) 11 Impact ? Collision of 2 bodies over a very short period of time ? Fast force application ? High loading rate = F/?t Impact Impact ? Landing ? Running ? Jumping 0 500 1000 1500 2000 2500 0 0.1 0.2 0.3 0.4 0.5 V e rt ical G R F ( N ) -500 Time (sec) 12 Types of Impact ? Direct impact ? Billiard m 1 m 2 u 1 u2 Before impact After impact ? Oblique impact ? Basketball: bounce pass ? Tennis: volley v 1 v 2 After impact incidence angle Reflection angle Coefficient of Restitution ? Under the direct impact ? Ratio of the difference of velocities before and after impact m 1 m 2 u 1 u2 Before impact After impact v 1 v 2 After impact impactbeforevrelative impactaftervrelative e=? 21 21 uu vv e ? ? =? Coefficient of Restitution ? Coefficient of Elasticity ? Perfectly elastic impact ? e = 1 Perfectly elastic Perfectly inelastic e=1 e = 0 ? Perfectly inelastic impact ? e = 0 ? Most impacts ? 0< e < 1 e = 13 Coefficient of Restitution ? Impact of ball with the ground ? A special case with ground stationary ? Coefficient of r h d u 1 h b v 1 d b h h e =? 1 1 u v e =? szhang Microsoft PowerPoint - Linear Kinetics.pptx