1. 2008 Physics 2111 Fundamentals of Physics Chapter 9 1 Fundamentals of Physics Chapter 9a Systems of Particles 1.A Special Point 2.The Center of Mass 3.Newton’s 2 nd Law for a System of Particles 4.Linear Momentum 5.Linear Momentum of a System of Particles 6.Conservation of Linear Momentum 7.A Rocket 8.External Forces & Internal Energy Changes Review & Summary Questions Exercises & Problems

2. 2008 Physics 2111 Fundamentals of Physics Chapter 9 2 The Center of Mass The center of mass of a body or system of bodies is the point that moves as though all of the mass were concentrated there and all external forces were applied there.

3. 2008 Physics 2111 Fundamentals of Physics Chapter 9 3 The Center of Mass Consider 2 particles separated by distance d: The center of mass, x com, of the system is the point that moves as though all of the mass were concentrated there and all external forces were applied there.

4. 2008 Physics 2111 Fundamentals of Physics Chapter 9 4 Center of Mass (cm) x com is closer to the larger mass.

5. 2008 Physics 2111 Fundamentals of Physics Chapter 9 5 The Center of Mass Many particles distributed along the x-axis: The center of mass, x com, of the system is the point that moves as though all of the mass were concentrated there and all external forces were applied there. Many particles distributed in 3 dimensions:

6. 2008 Physics 2111 Fundamentals of Physics Chapter 9 6 Center of Mass?

7. 2008 Physics 2111 Fundamentals of Physics Chapter 9 7 CM of Solid Bodies Uniform density case:

8. 2008 Physics 2111 Fundamentals of Physics Chapter 9 8 Newton’s 2 nd Law for a System of Particles Net Force of all external forces Internal forces cancel (Newton’s 3 rd Law) Total mass of the system Acceleration of the center of mass

9. 2008 Physics 2111 Fundamentals of Physics Chapter 9 9 Newton’s 2 nd Law for a System of Particles The center of mass of a system moves like a particle of mass M under the influence of the net external force acting on the system. Net of all external forces Internal forces cancel (Newton’s 3 rd Law) Total mass of the system Acceleration of the center of mass

10. 2008 Physics 2111 Fundamentals of Physics Chapter 9 10 Linear Momentum Linear momentum of a particle: The net force acting on a particle equals the time rate of change of the particle’s linear momentum: Newton’s 2 nd Law

11. 2008 Physics 2111 Fundamentals of Physics Chapter 9 11 Linear Momentum of a System of Particles The linear momentum of a system of particles equals the product of the total mass and the velocity of the center of mass: Total linear momentum of a system of particles: Differentiating:

12. 2008 Physics 2111 Fundamentals of Physics Chapter 9 12 Conservation of Momentum Law of conservation of momentum: If no net external force acts on a system of particles, the total linear momentum of the system cannot change: If the component of the net external force on a closed system is zero along an axis, then the component of linear momentum of the system along that axis cannot change.

13. 2008 Physics 2111 Fundamentals of Physics Chapter 9 13 A shell explodes at its highest point. the com continues on its way! Coordinates of the explosion: Momentum in the x-direction at the explosion: Projectile motion from the explosion: v 0 = 20 m/s  = 60 0

14. 2008 Physics 2111 Fundamentals of Physics Chapter 9 14 Method 2: the com continues on its way!

15. 2008 Physics 2111 Fundamentals of Physics Chapter 9 15 The com continues on its way!

16. 2008 Physics 2111 Fundamentals of Physics Chapter 9 16 no friction “the system” = person + boat When the person walks to the front of the boat, how far is the boat from the pier?

17. 2008 Physics 2111 Fundamentals of Physics Chapter 9 17 Conservation of ?? Initially both are moving with velocity v 0 ; then the man starts to run with velocity v rel with respect to the car. momentum before: momentum after:

18. 2008 Physics 2111 Fundamentals of Physics Chapter 9 18 ROCKETS

19. 2008 Physics 2111 Fundamentals of Physics Chapter 9 19 Rockets Further investigation and experimentation have confirmed the findings of Isaac Newton in the 17th century and it is now definitely established that a rocket can function in a vacuum as well as in an atmosphere. The Times regrets the error for after the rocket quits our air and really starts on its longer journey, its flight would be neither accelerated nor maintained by the explosion of the charges it then might have left. To claim that it would be is to deny a fundamental law of dynamics, and only Dr. Einstein and his chosen dozen, so few and fit, are licensed to do that. …..That Professor Goddard, with his "chair" in Clark College and the countenancing of the Smithsonian Institution, does not know the relation of action to reaction, and of the need to have something better than a vacuum against which to react--to say that would be absurd. Of course he only seems to lack the knowledge ladled out daily in high schools. NYT, January 13, 1920 NYT, July, 1969

20. 2008 Physics 2111 Fundamentals of Physics Chapter 9 20 Rockets The rocket + exhaust gas has constant momentum in the absence of external forces (e.g. in outer space). Constant Total Energy? Constant Mechanical Energy? Burning Fuel Exhaust Gas “The System” Rocket

21. 2008 Physics 2111 Fundamentals of Physics Chapter 9 21 Fundamentals of Physics Chapter 9b Collisions 1.Collisions 2.Impulse & Linear Momentum 3.Momentum & Kinetic Energy 4.Inelastic Collisions in 1-Dimension 5.Elastic Collisions in 1-Dimension 6.Collisions in 2-Dimensions Review & Summary Questions Exercises & Problems

22. 2008 Physics 2111 Fundamentals of Physics Chapter 9 22 Collisions A collision is an isolated event in which two or more bodies exert relatively strong forces on each other for a relatively short time. A collision does not require physical contact. Interacting Moving freely “Isolated” - no significant external forces during the collision.

23. 2008 Physics 2111 Fundamentals of Physics Chapter 9 23 Impulse During a Collision Newton’s 2nd Law: During a collision, the left & right bodies briefly exert a force on each other: F(t) acts for time  t An Impulse Newton’s 3rd Law (eq. 9-23)

24. 2008 Physics 2111 Fundamentals of Physics Chapter 9 24 Impulse – Linear Momentum Theorem (vectors!) equal areas

25. 2008 Physics 2111 Fundamentals of Physics Chapter 9 25 A Series of Collisions A stream of objects (e.g. bullets, a stream of water, …) hitting a target: What is the force on the target?  v depends on what happens to the objects in the stream? Stop, bounce back,…

26. 2008 Physics 2111 Fundamentals of Physics Chapter 9 26 Throw a ball against the wall J=? F avg = ? a) Impulse on the ball? b) Average force on the wall?

27. 2008 Physics 2111 Fundamentals of Physics Chapter 9 27 Momentum & Kinetic Energy in Collisions Collisions in a closed, isolated system: –“Closed” - no mass enters or leaves –“Isolated” - no external forces –Elastic Collision Total Kinetic Energy is conserved. Total Linear momentum is conserved. –Inelastic Collision Total Linear momentum is conserved. Completely Inelastic Collision –The bodies stick together after they collide. The linear momentum of each colliding body may change but the total momentum of the system cannot change, whether the collision is elastic or inelastic.

28. 2008 Physics 2111 Fundamentals of Physics Chapter 9 28 Inelastic Collisions in One Dimension momentum vectors:

29. 2008 Physics 2111 Fundamentals of Physics Chapter 9 29 Completely Inelastic Collisions stuck together

30. 2008 Physics 2111 Fundamentals of Physics Chapter 9 30 Velocity of the Center of mass time For an isolated system, the velocity of the center of mass cannot be changed by a collision: Completely Inelastic Case:

31. 2008 Physics 2111 Fundamentals of Physics Chapter 9 31 The Ballistic Pendulum Velocity of a speeding bullet? Step 1: Bullet quickly stops in block Completely inelastic collision. Little time for block to move. (Tension in cords approximately perpendicular to block’s direction of travel; no work done.) Step 2: Bullet & block swing upward Mechanical Energy Conserved.

32. 2008 Physics 2111 Fundamentals of Physics Chapter 9 32 Colliding Blocks Conservation of momentum: At maximum compression: Conservation of total mechanical energy: m 1 = 2.0 kg, 10 m/s m 2 = 5.0 kg, 3 m/s k = 1120 N/m

33. 2008 Physics 2111 Fundamentals of Physics Chapter 9 33 Elastic Collisions in One Dimension In an elastic collision, the kinetic energy of each colliding body may change, but the total kinetic energy of the system does not change.

34. 2008 Physics 2111 Fundamentals of Physics Chapter 9 34 Elastic Collisions with a Moving Target

35. 2008 Physics 2111 Fundamentals of Physics Chapter 9 35 Planetary Assist (“slingshot”) v = 12 km/s v J = 13 km/s

36. 2008 Physics 2111 Fundamentals of Physics Chapter 9 36 Collisions in Two Dimensions

37. 2008 Physics 2111 Fundamentals of Physics Chapter 9 37 Billiard Ball Collision Pythagorean Theorem! Equal mass elastic scattering. e.g. billiards

38. 2008 Physics 2111 Fundamentals of Physics Chapter 9 38 Equal Mass Elastic Collision proton + proton  proton + proton