Chap 15 Kinetics of a particle impulse and

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.

(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 T 1 plus the vector sum of all the time integral t 1 to t 2 is equivalent to the linear momentum of the particle at time t 2. 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.

In x, y, z, components

15. 2 Principle of linear impulse and momentum for a system of particles. Equation of motion for a system of particle is or The location of the mass G of the system is

Substitute Eq (2) to (1) to get

15. 3 Conservation of linear momentum for a system 1. Internal impulses The impulse occur in equal but opposite collinear pairs.

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 t 1 to t 2

15. 4 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. m. Av. A 1 m. Av. A 2 V ∫Pdt -∫Pdt B A Require v. A 1>v. A 2 A Effect of A on B B Effect of B on A Deformation impulse AB Maximum deformation Before impact conservation of momentum of the system of panicles A and B

Principle of impulse and momentum for panicle A (or B) (a) Deformation please for A (b) Restitution place for A

deformation impulse is Restitution impulse is Define Similarly by considering particles B We have

=relative velocity just after impact =relative velocity just before impact In general Elastic impact

Plastic impact 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 between 2 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. v. A 2 v. B 2 m. Av. Ay 2 Line of impact m. Av. Ax 1 ∫Fdt + = m. Av. Ax 1 v. A 1 v. B 1 m. Av. Ay 1 Plane of contact m. Bv. By 2 m. Bv. Bx 1 ∫Fdt + = m. Bv. Bx 2 m. Bv. By 1

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 rx mvx j ry mvy k rz mvz

15 -6 Moment of a force and Angular momentum Equation of motion of a particle is ( m=constant ) 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 Resultant moment about pt. o of all force Time rate change of angular momentum of the particle about pt. o

Recall