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Chapter 15: Kinetics of a Particle: Impulse and MomentumTextbook: Engineering MechanicsSTATICS and DYNAMICS- 11th Ed., R. C. Hibbeler and A. Gupta Course Instructor: Miss Saman Shahid Impulse and momentum principles are required to predict the motion of this golf ball. “the product of the average force acting upon a body and the time during which it acts, equivalent to the change in the momentum of the body produced by such a force. Principle of Linear Impulse and Momentum Linear momentum of the particle: L=mv The integral of I is referred to as linear impulse. This term is a vector quantity which measures the effect of a force during the time the force acts. For problem solving: Initial momentum+ sum of all impulses = final momentum Principle of Linear Impulse and Momentum for a System of Particles Conservation of Linear Momentum for a System of Particles When the sum of the external impulses acting on a system of particles is zero, eq., 15.6 reduces to: The conservation of linear momentum is often applied when particles collide or interact. Internal impulses for the system will always cancel out, since they occur in equal but opposite collinear pairs. If the time period over which the motion is studies is very short, some of the external impulses may also neglected or considered approximately equal to zero. Impulsive and Non-Impulsive Forces The forces causing negligible impulses are called nonimpulsive forces. Forces which are very large and act for a very short period of time produce a significant change in momentum and are called impulsive forces. Impulsive forces normally occur due to an explosion or the striking of one body, the force imparted by a slightly deformed spring having a relatively small stiffness, or for that matter is very small compared to other larger (impulsive) forces. Consider the tennis ball with a racket. During the very short time of interaction, the force of the racket on the ball is impulsive since it changes the ball’s momentum drastically. Ball’s weight will have negligible effect on the change in momentum, and therefore it is nonimpulsive. Consequently, it can be neglected from an impulse-momentum analysis during this time. If impulse- momentum analysis is considered, then the impulse of the ball’s weight is important since it, along with air resistance, causes the change in the momentum of the ball. Impact Impact occurs when two bodies collide with each other during a very short period of time, causing relatively large (impulsive) forces to be exerted between the bodies. Examples: 1) String of a hammer on a nail, 2) Golf club on a ball Two types of impact: Central Impact: It occurs when the direction of motion of the mass centers of the particles. This line is called the line of impact. Oblique Impact: When the motion of one or both of the particles is at an angle with the line of impact. 1- Central Impact •Two particles have the initial momenta and vA1 > vB1, collision will eventually occur. •During collision the particles will be deformed or non-rigid. •At the instant of maximum deformation, both particles move with a common velocity. •The particles will return to their original shape and will have equal and opposite restitution impulse. •Just after separation the particles will have the final momenta and vB2>vA2 Momentum for the System of Particles is Conserved, since during collision the internal impulses of deformation and restitution cancel •In order to obtain second equation necessary to solve for vA2 and vB2, we must apply the principle of impulse and momentum to each particle. During the deformation phase for particle A. •The ratio of the restitution impulse to the deformation impulse is called the coefficient of restitution e. •If the unknown v is eliminated from two equations of e then then coefficient of restitution can be expressed in terms of particles’ initial and final velocities. . Coefficient of Restitution: Experimentally it has been found that e varies appreciably with impact velocity as well as with the size and shape of the colliding bodies. In general e has a value between zero and one. Elastic Impact (e=1): If the collision between the two particles is perfectly elastic, the formation impulse (Pdt) is equal and opposite to the restitution impulse (Rdt). Although in reality this can never be achieved e=1 for an elastic collision. If the impact is perfectly elastic, no energy is lost in the collision. 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 and move with a common velocity. If the collision is plastic, the energy lost during collision is a maximum. Oblique Impact When oblique impact occurs between two smooth particles, the particles move away from each other with velocities having unknown directions as well as unknown magnitudes. Provided the initial velocities are known, four unknowns may be represented either as vA2, vB2, θ2 , Ф2. Angular Momentum The angular momentum of a particle about point O is defined as the “moment” of the particle’s linear momentum about O. Since this concept is analogues to finding the moment of a force about a point, the angular momentum Ho, is sometimes referred to as the “moment of momentum”. Scalar Formulation: if a particle moves along a curve lying in the x-y plane, the angular momentum at any instant can be determined about point O (actually the z-axis) by using a scalar formulation. Here d is the moment arm or perpendicular distance from O to the line of action of mv. Right Hand Rule:The curl of the fingers of the right hand indicates the sense of rotation of mv about O, so that in this case the thumb or (Ho) is directed perpendicular to the x-y plane along the+z-axis. Angular Momentum- vector formulation If the particle moves along a space curve, the vector cross product can be used to determine the angular momentum about O. Here r denotes a position vector drawn from point O to the particle. As shown in the figure, Ho is perpendicular to the shaded plane containing r and mv. Relation between Moment of a Force and Angular Momentum Another way of expressing Newton’s Law The moments about point O of all forces acting on the particle may be related to the particle’s angular momentum by using the equation of motion. The moments of the forces about point O can be obtained by performing a cross-product multiplication of each side of this equation by the position vector r, which is measured in the x,y,z inertial frame of reference. The resultant moment about point O of all the forces acting on the particle is equal to the time rate of change of the particle’s angular momentum about point O. L=mv, (F=ma=mdv/dt=d(mv)/dt) , so the resultant force acting on the particle is equal to the time rate of change of the particle’s linear momentum. Angular Impulse and Momentum Principles Equation can be rewritten in the form below and after integrating it we get the principle of angular impulse and momentum. The quanitiy (Modt) is called Angular Impulse. Vector Formulation: Using impulse and momentum principles it is therefore possible to write two equations which define the particle’s motion: Scalar Formulation: In general, the above equations may be expressed in x,y,z component form, yielding a total of six independent scalar equations. First two equations represent “principle of linear impulse” and third equation represents the “principle of angular momentum about the z-axis. Conservation of Angular Momentum: When the angular impulses acting on a particle are all zero during time t1 to t2, we can write: It states that from t1 to t2, the particle’s angular momentum remains constant. Obviously, if no external impulse is applied to the particle, both linear and angular momentum will be conserved. In some cases, however, the particle’s angular momentum will be conserved and linear momentum may not.