Chapter 9 Center of mass and linear momentum I The center of mass System of particles II Solid body Newton s Second law for a system of particles III Linear Momentum System of particles Conservation IV Collision and impulse Single collision Series of collisions V Momentum and kinetic energy in collisions VI Inelastic collisions in 1D Completely inelastic collision Velocity of COM VII Elastic collisions in 1D VIII Collisions in 2D IX Systems with varying mass X External forces and internal energy changes I Center of mass The center of mass of a body or a system of bodies is a point that moves as though all the mass were concentrated there and all external forces were applied there System of particles General x m1 x1 m2 x2 m1 x1 m2 x2 com m1 m2 M M total mass of the system The center of mass lies somewhere between the two particles Choice of the reference origin is arbitrary Shift of the coordinate system but center of mass is still at the same relative distance from each particle I Center of mass System of particles xcom m2 d m1 m2 Origin of reference system coincides with m1 3D xcom 1 n mi xi M i 1 ycom 1 n mi yi M i 1 1 n rcom mi ri M i 1 zcom 1 n mi zi M i 1 Solid bodies Continuous distribution of matter Particles dm differential mass elements 3D xcom 1 x dm M ycom 1 y dm M zcom 1 z dm M M mass of the object Assumption Uniform objects uniform density xcom 1 x dV V ycom 1 y dV V Linear density M L dm dx zcom 1 z dV V M dm dV V Volume density Surface density M A dm dA The center of mass of an object with a point line or plane of symmetry lies on that point line or plane The center of mass of an object does not need to lie within the object Examples doughnut horseshoe Problem solving tactics 1 Use object s symmetry 2 If possible divide object in several parts Treat each of these parts as a particle located at its own center of mass 3 Chose your axes wisely Use one particle of the system as origin of your reference system or let the symmetry lines be your axis II Newton s second law for a system of particles Motion of the center of mass It moves as a particle whose mass is equal to the total mass of the system Fnet Macom Fnet is the net of all external forces that act on the system Internal forces from one part of the system to another are not included Closed system no mass enters or leaves the system during movement M total mass of system acom is the acceleration of the system s center of mass Fnet x Macom x Fnet y Macom y Fnet z Macom z Proof Mrcom m1r1 m2 r2 m3 r3 mn rn drcom M Mvcom m1v1 m2 v2 m3v3 mn vn dt d 2 rcom dv M M Macom m1a1 m2 a2 m3 a3 mn an F1 F2 F3 Fn dt 2 dt includes forces that the particles of the system exert on each other internal forces and forces exerted on the particles from outside the system external Newton s third law internal forces from third law force pairs cancel out in the sum Only external forces III Linear momentum Vector magnitude Linear momentum of a particle p mv The time rate of change of the momentum of a particle is equal to the net force acting on the particle and it is in the direction of that force dp d mv ma Fnet dt dt Equivalent to Newton s second law System of particles The total linear moment P is the vector sum of the individual particle s linear momenta P p1 p2 p3 pn m1v1 m2v2 m3v3 mn vn P Mvcom The linear momentum of a system of particles is equal to the product of the total mass M of the system and the velocity of the center of mass dv dP dP M com Macom Fnet dt dt dt Net external force acting on the system Conservation If no external force acts on a closed isolated system of particles the total linear momentum P of the system cannot change P cte Fnet Closed isolated system dP 0 Pf Pi dt Closed no matter passes through the system boundary in any direction Isolated the net external force acting on the system is zero If it is not isolated each component of the linear momentum is conserved separately if the corresponding component of the net external force is zero If the component of the net external force on a closed system is zero along an axis component of the linear momentum along that axis cannot change The momentum is constant if no external forces act on a closed particle system Internal forces can change the linear momentum of portions of the system but they cannot change the total linear momentum of the entire system IV Collision and impulse Collision isolated event in which two or more bodies exert relatively strong forces on each other for a relatively short time Impulse Measures the strength and duration of the collision force Vector magnitude Third law force pair FR FL JR JL Single collision p dp t F dp F t dt dp t F t dt dt p t J t F t dt p f pi p f f i i f i Impulse linear momentum theorem The change in the linear momentum of a body in a collision is equal to the impulse that acts on that body p p f pi J Units kg m s p fx pix p x J x p fy piy p y J y p fz piz p z J z Favg such that Area under F t t curve Area under Favg t J Favg t Series of collisions Target fixed in place n projectiles n p Total change in linear momentum projectiles Impulse on the target J t arg et J projectiles n p Favg J n n p m v t t t m nm in t Favg m v t J and p have opposite directions pf pi p left J to the right n t Rate at which the projectiles collide with the target m t Rate at which mass collides with the target a Projectiles stop upon impact v vf vi 0 v v b Projectiles bounce v vf vi v v 2v V Momentum and kinetic energy in collisions Assumptions Closed systems no mass enters or leaves them Isolated systems no external forces act on the bodies within the system Elastic collision If the total kinetic energy of the system of two colliding bodies is unchanged conserved by the collision Example Superball into hard floor Inelastic collision The kinetic energy of the system is not conserved some goes into thermal energy sound etc Completely inelastic collision After the collision the bodies loose energy and stick together Example Ball of wet putty into floor …