System of objects (particles)

Transcription

1 Today Ch 6, Momentum and Collisions System of particles Elastic vs. inelastic collision Elastic collision in 1D Collision in 2D Center of mass Motion of system of particles (Motion of center of mass) 1 System of objects (particles) We consider system of multiple objects, which we can choose any way convenient for us. For example, The system:, and m3; however m4 and m5 are external objects! m3 m5 m4 2 1

2 Internal vs. External Forces Internal forces: Forces between objects within the system External forces: Forces from outside the system m3 m5 m4 3 P p + p +... mv + m v +... Net Momentum of System of Particles (or, Objects) net The system:, and m3; however m4 and m5 are external objects! v v 2 1 v 5 m5 m3 v v 4 3 m4 4 2

3 Net external force, Net external impulse Net external force : F F F net, ext ext,1 + ext, sum of all external forces acting on the system m3 system Net external impulse: m4 m5 I F Δt net, ext net, ext 5 Impulse-Momentum Theorem for System of Particles I P P net, ext net, f net, i Net external impulse is equal to net momentum m m change. Proof: use Newton s 3rd law (see textbook) Note: Internal forces do not matter for net momentum change. 6 3

4 If I net, ext 0 Conservation of momentum, for example, P P net, f net, i F net, ext 0 or Δt is small If the net external impulse is zero, for example, if the net external force is zero, the net momentum of a system is conserved. 7 Elastic vs. Inelastic Collisions In a collision within a system of particles Before collision After collision Net kinetic energy: Knet K1+ K mv1 + mv If Knet, i Knet, f elastic collision. If If K K K net, i net, f inelastic collision. K net, i net, f and objects move together after collision perfectly inelastic collision. 8 4

5 Elastic Collisions in 1-Dimension Both total momentum & total kinetic energy are conserved , i + 2 2, i 1 1, f + & mv + mv mv + mv 2 2, f v1i v2i mv mv mv mv i 2 2i 1 1f 2 2 f Before collision v1f? v2f? +x direction After collision m m 2m 2 + v v + v and m + m m + m v1, v1, + v f i 2, m i 1+ + (see text for proof) 2, f 1, i 2, i iclicker Quiz In 1D elastic collision, if, vi1+10 m/s, v2i0, then, after collision v1f and v2f. a) 5 m/s; 5 m/s b) -5 m/s; 5 m/s c) 0 m/s; 10 m/s d) 0 m/s; -10 m/s e) -10 m/s; 0 m/s V1i10m/s V2i0 Before collision After collision v1f? v2f? +x direction m m m 2 2m , f 1, i + 2, i v m and 2, f m v1, i + m m v2, i v v v 5

6 Collisions in Two Dimensions If net external force is zero, Or P P P P net, i net, f P net,, i x net, f, x P net,, i y net, f, y, where net net momentum is conserved. P mv + m v y V1 If net kinetic energy is conserved Elastic collision not conserved Inelastic collision If moving together after collision Perfectly Inelastic collision (Kinetic energy is not conserved.) v1 V2 v2 x Example 2: Collision at an intersection A 1500 kg car traveling east with a speed of 25 m/s collides at an intersection with a 2500 kg van traveling north at a speed of 20 m/s. Find the direction and magnitude of the velocity of the wreckage after the collision, assuming the vehicles stick together after the collision. 6

7 Example 1: 90 degree deflection rule in a game of pool v 2, f m m v 1,i v 1, f Assume that the collision is elastic, and the two balls have the same mass. Show that the angle between the outgoing balls is 90 degree. (No forward, back or side spin is in effect.) How could we analyze the motion of extended objects, or system of particles? 7

8 Concept of Center of Mass 3 kg m310 kg 6 kg For a system of particles or an extended object, Center of mass is an average position for mass distribution. ib ti Definition of center of mass () in 1D m3 x1 x2 x3 In 1D, x mx m + mx m , where x is position of mass i m i 8

9 Definition of center of mass () in 2D x mx 1 1+ mx m + m y r my 1 1+ my m + m , where ( x, y ) i m i mr i 11+ mr m + m +... i is the position of Example 3 Three particles of masses 1.1 kg, 2.5 kg, and m3 3.4 kg are located as shown in the figure: is at (0,0), is at (140 m,0), and m3 is at (70 m, 120 m). Find the coordinate of the center of mass. 9

10 iclicker Quiz A two-section piece, represented by the gray area on the figure, is cut from a metal plate of uniform thickness. The point that corresponds to the center of mass of this piece is closest to (a) 1 (b) 2 (c) 3 (d) 4 (e) 5 Velocity of center of mass () v x, Δx mv 1 x,1+ mv 2 x, Δ t m + m ( m + m +...) v mv + m v +... P 1 2 x, 1 x,1 2 x,2 net, x Similar for y ponent net v mv m + mv m P mv + mv + Mv where M

11 Example 1 A 2.0 kg particle has a velocity (2.0 m/s, -3.0 m/s), and a 3.0 kg particle has a velocity (1.0 m/s, 6.0 m/s). Find (a) velocity of the center of mass and (b) the total momentum of the system. Acceleration of center of mass () a x, Similar for y ponent. In vector form Δ vx, ma 1 x,1+ ax, Δ t m + m ma 1 1+ ma a ( ) a ma a Ma m a + m a

12 If we have only two objects in the system Ma ma + m a ma F F + F ma F F + F 1 1 1, net 1, ext 1int 1,int 2 2 2, net 2, ext 2,int 1,int F F 2,int F 1,ext F 2,ext ma ma 2 2 F1, ext + F2, ext + F1,int + F2,int Newton s 3 rd law for internal forces: F + F Ma m a + m a F + F F system 1,int 2,int , ext 2, ext net, ext m3 23 F Newton s second law for center of mass of a system of objects F net, ext M a net, ext Ma : Sum of all external forces that act on the system (Internal forces are not included) : Total mass of the system : Acceleration of the center of mass Internal forces do NOT change the motion of C.O.M.!! Only external forces matter for the motion of C.O.M.!!!!!! 12

13 Motion of center of mass under gravity force Newton s second law for C.O.M.: Under F gravity, F net, ext Ma Fnet, ext (0, g ) + (0, g ) +... (0, mg 1 mg 2...) (0, ( ) g) (0, Mg) (0, Mg) Ma a (0, g) Usual projectile motion Center of mass moves like a particle of mass M under the net external force. Motion of COM is simple! 13

14 iclicker Quiz Two objects with unknown mass and velocity collide and stick together moving at 3 m/s along x direction. Assuming that net external force on the two objects is zero, what is the velocity of the center of mass before the collision? (a) 0 (b) 3 m/s along x (c) -3 m/s along x (d) Not enough information 14