#### Transcript 15-1. principle of linear impulse and momentum

Chap 15 Kinetics of a particle impulse and momentum 15-1. principle of linear impulse and momentum (1) Linear impulse I ,N S The integral I =∫F dt is defined as the linear impulse which measure the effect of a force during the time the force acts . t2 I F (t )dt t1 (2) Linear momentum L , kg/s The form L=m v is defined as the linear momentum of the particle . (3) Principle of linear impulse and momentum the initial momentum of the particle at time T1 plus the vector sum of all the time integral t1 to t2 is equivalent to the linear momentum of the particle at time t2 . t2 m v1 t1 Fdt m v2 Equation of motion for a particle of mass m is ∑F =m a = m dv /dt ∑F dt = m dv Integrating both sides to yield . t2 t1 v2 Fdt m dv m v2 m v1 v1 m v1 t2 t1 Fdt m v2 In x,y,z, components m(v x )1 Fx dt m(v x ) t2 t1 2 m(v y )1 Fy dt m(v y ) 2 t2 t1 m(v z )1 Fz dt m(v z ) 2 t2 t1 15.2 Principle of linear impulse and momentum for a system of particles. Equation of motion for a system of particle is Fi Fi dt t2 t1 dv i m a m i i i dt m dvi i Fi dt ( vi ) 2 ( vi )1 m d v i i ( vi ) 2 mi vi ( vi )1 mi vi 2 mi vi 1 or mi vi t2 1 t1 i ...........1 Fi dt m v The location of the mass G of the system is i 2 mrG m r i i m drG dt m m i dr i i dt m or vG vi ............2 i m Substitute Eq (2) to (1) to get m vG 1 tt 2 1 Fdt m v G 2 15.3 Conservation of linear momentum for a system 1. Internal impulses The impulse occur in equal but opposite collinear pairs. t2 t2 t2 t1 Fdt t1 f12dt t1 f 21dt 2. Non impulsive forces The force cause negligible impulses during the very short time period of the motion studied . (1)weight of a body. (2)force imparted by slightly de formed spring having a relatively small stiffness ( Fs = ks ). (3)Any force that is very small compared to other longer impulsive forces. 3.Impulsive forces The forces are very large , act for a very short period of time and produce a significant charge in momentum. Note :when making the distinction between impulsive and nonimpulsive forces it is important to realize that it only applied during a specific time. Conservation of linear moment. The vector sum of the linear moments for a system of particles remain constant throughout the time period t1 to t2 15.4 Impact Impact (1) Definition. Two bodies collide with each other during a very short internal of time which causes relatively large impulsive forces exerted between the bodies. (2) Types. (A). Central impact. The direction of motion of the mass center of the two colliding particles is along the line of impact. (B). Oblique impact. The motion of one or both the particles is at angle with the line of impact. Central Impact To illustrate the method for analyzing the mechanics of impact , consider the case involving the central of 2 smooth particles A and B. mAvA1 mAvA2 V ∫Pdt -∫Pdt B A Require vA1>vA2 A Effect of A on B B Effect of B on A AB Maximum deformation Deformation impulse Before impact conservation of momentum of the system of panicles A and B mAvA1 mbvb1 m A vA2 mBvB 2 Principle of impulse and momentum for panicle A (or B) (a) Deformation please for A Pdt mAvA Pdt mAv (b) Restitution place for A mv Rdt mv A 2 deformation impulse is Pdt m v A A m Av Restitution impulse is Rdt m v m v A A A2 Define Rdt m v m v Pdt m v m v A A A2 A A1 v v A2 e vA1 v Similarly by considering particles B We have e v v A2 v A1 v (v A1 v)e v v A2 1.v A1e v A2 (1 e)v (v v B1 )e v B 2 v 2.(1 e)v vB 2 vB1e vA1e vA2 vB 2 vB1e vB 2 vA2 e vA1 vB1 =relative velocity just after impact =relative velocity just before impact In general 0 e 1 (1)e 1 Elastic impact Pdt Rdt (1)e 0 Plastic impact Pdt 0 3. Oblique impact Plastic Impact (e = 0):The impact is said to be inelastic or plastic when e = 0. In this case there is no restitution impulse given to the particles (∫Rdt = 0), so that after collision both particles couple or stick together & move with a common velocity. Oblique Impact. When oblique impact occur between2 smooth particles, the particle move away form each other with velocities having unknown direction as well As unknown magnitudes. Provided the initial velocities are known,4 unknown are present in the problem. vA2 vB2 mAvAy2 Line of impact mAvAx1 ∫Fdt + = mAvAx1 vA1 vB1 mAvAy1 Plane of contact mBvBy2 mBvBx1 ∫Fdt + = mBvBx2 mBvBy1 15-5 angular momentum 1. angular momentum Moment of the particle’s linear momentum about point 0 or other point. 2. scalar formulation Assume that the path of motion of the particle lies in x y-plane. Magnitude of angular momentum Ho is Ho= Ho = d(mv) Angular momentum=moment of momentum Ho=r x mv= i j k rx ry rz mvx mvy mvz 15-6 Moment of a force and Angular momentum Equation of motion of a particle is ( m=constant ) F mV By performing a cross-product multiplication of each side of this equation by the position vector r , We have the moments of forces About point O F r F rx(mV ) Ho r xmv Mo or Mo Ho Mo Ho r x(mv ) d Ho (r xm v ) dt r xm v r xm v m(r xv ) r xmv Resultant moment about pt.o of all force Time rate change of angular momentum of the particle about pt.o Recall F L