Phy 211 General Physics I Chapter 9 Center of Mass Linear Momentum Lecture Notes Rene Descartes 1596 1650 Prominent French mathematician philosopher Active toward end of Galileo s career Studied the nature of collisions between objects First introduced the concept of momentum he defined momentum vis vis as the product of weight times speed Demonstrated the Law of Conservation of Momentum Linear Momentum r Linear momentum p represents inertia in motion Newton momentum as the quantity of motion Conceptually the effort required to bring a moving object to rest depends not only on its mass inertia but also on how r fast rit is moving p mv Definition Momentum is a vector quantity with the same direction as the object s velocity SI units are kg m s Newton s 1st Law revisited The momentum of an object will remain constant unless it is acted upon by a net force or impulse Center of Mass r Center of Mass rcm refers to the average location of mass for a defined mass To determine the center of mass take the sum of each mass multiplied by its position vector and divide by the total mass of the system or n r r r r r mr i i r m1r1 m2r2 m3r3 mnrn rcm i 1 m1 m2 m3 mn msys Note if the objects in the system are in motion the velocity of the system center of mass is n r r r r r m v m2v2 mnvn vcm 1 1 m1 m2 mn m v i i i 1 msys When psystem 0 i e Fext 0 then vcm constant The motion of all bodies even if they are changing individually will always have values such that vcm constant Impulse Momentum Theorem Newton s 2nd Law can be rewritten r as r r r d mv r dp dv Fnet ma m dt dt dt r Rearranging rterms r p t r r dp Fnet dt F dt r dp po to net t r r Dp Fnet dt this is the Net Impulse to Definition of Impulse associated with an applied force r r J Dp r Fdt t to The SI units for impulse are N s For a constant force or average force impulse simplifies to Therefore Impulse represents simultaneously r 1 The product of the force times the time FavgDt r r r J Dp FavgDt r r r 2 The change in linear momentum of the object Dp mv f mv Notes on Impulse Impulses always occur as action reaction pairs The force time relationship is observed in many real world examples Automobile safety Dashboards Airbags Crumple zones Product packaging Styrofoam spacers Sports Tennis racket string tension Baseball juiced baseballs baseball bats corked aluminum vs wood Golf the spring like effect of golf club heads Boxing gloves lower impulsive forces in the hands A Superman Problem It is well known that bullets and missiles bounce off Superman s chest Suppose a bad guy sprays Superman s chest with 0 003 kg bullets traveling at a speed of 300 m s fired from a machine gun at a rate of 100 rounds min Each bullet bounces straight back with no loss in speed Problems a What is the impulse exerted on Superman s chest by a single bullet b What is the average force exerted by the stream of bullets on Superman s chest Collisions A specific type of interaction between 2 objects The basic assumptions of a collision 1 Interaction is short lived compared to the time of observation 2 A relatively large force acts on each colliding object 3 The motion of one or both objects changes abruptly following collision 4 There is a clean separation between the state of the objects before collision vs after collision 3 classifications for collisions Perfectly elastic colliding objects bounce off each other and no energy is lost due to heat formation or deformation Ksystem is conserved Perfectly inelastic colliding objects stick together Ksystem is not conserved Somewhat inelastic basically all other type of collisions KE is not conserved Conservation of Linear Momentum The total linear momentum of a system will remain constant when r rno external net force acts r upon r the system or p1 p2 before collision p1 p2 after collision Note Individual momentum vectors may change due to collisions etc but the linear momentum for the system remains constant Useful for solving collision problems Where all information is not known given To simplify the problem Conservation of Momentum is even more fundamental than Newton s Laws Conservation of Momentum Examples The ballistic pendulum 2 body collisions we can t solve 3 body systems Perfectly inelastic Epre collision Epost collision Perfectly elastic Epre collision Epost collision Collisions in 2 D or 3 D Linear momentum is conserved by components r r r r p1 p2 before collision p1 p2 after collision By Components p1x p2x i p1x p2x i before collision after collision p1y p2y j p1y p2y j before collision after collision Notes on Collisions Force During collisions the forces generated Are short in duration Are called impulsive forces or impact forces or collision forces Often vary in intensity magnitude during the event Can be described by an average collision force r r r Dp impulse FNet Favg i e Dt time Example a golf club collides with a 0 1 kg golf ball initially at rest t 0 01s The velocity of the ball following the impact is 25 m s The impulse exerted on the ball is r r Dp mDv 0 1 kg 25 m s 0 m s i 2 5 N s i The average impulsive force exerted on the ball is r r Dp 2 5 N s Favg i 250 N i Dt 0 01 s r exerted on the club is The average impulsive force r Dp 2 5 N s Favg i 250 N i Dt 0 01 s