1. Chapter 9, System of Particles Center of Mass Linear Momentum and Conservation Impulse Rocket

2. Center of Mass for a System of Particles The center of mass of a body or a system of bodies moves as though all of the mass were concentrated there and all external forces were applied there.

3. Where is the center of mass of this meter stick? (a) 30cm(b) 50cm(c) 90cm The center of mass of a symmetric, homogeneous body must lie on its geometric center. How to locate the center of mass of an irregular planar object? Suspend the object from two different points. Drop a plumb line and mark on the object. The center of mass coincides with the intersection of two lines. com

4. 2 bodies, 1 dimension x com com For example, if x 1 =0, x 2 =d, and m 2 =2m 1, we find that x 2 =d x com com x 1 =0

5. Example 1 Sun Earth X 1 =0 X 2 =d

6. Center of Mass for a System of Particles 2 particles, 1 dimension n particles, 3 dimensions n particles, 3 dimensions, vector equation

7. Example 2

8. Center of Mass for a Solid Body dm is the differential mass element Uniform density

9. Newton’s 2 nd Law for a System of Particles A firework rocket explodes The center of mass moves like an imaginary particle of mass M under the influence of the net external force on the system. system of particles particle

10. Linear Momentum and Impulse Particle System

11. Impulse–Linear Momentum Theorem: Impulse = Force  Duration of the force = Change in Momentum Collision of two particle-like bodies

12. Conservation of Linear Momentum If no net external force acts on a system of particles, the total linear momentum P 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 the linear momentum along that axis cannot change.  Closed system (no mass enters or leaves)  Isolated system (no external net force)

13. Before collision:After collision: Sum of initial momentums = Sum of final momentums

14. Collisions in closed, isolated systems Elastic Inelastic Completely inelastic both colliding objects does not change their shapes total linear momentum is conserved total kinetic energy is conserved the shape of one or both of object changes during the collision total linear momentum is conserved total kinetic energy is NOT conserved the colliding objects stick together after the collision total linear momentum is conserved total kinetic energy is NOT conserved  Closed system (no mass enters or leaves)  Isolated system (no external net force)

15. . In a closed, isolated system containing a collision, the linear momentum of each colliding body may change but the total momentum P of the system cannot change, whether the collision is elastic or inelastic.

16. Completely Inelastic Collisions in 1D Velocity of Center of Mass

17. Sample Problem 9-8 Step 1 total linear momentum is conserved Step 2 total mechanical energy is conserved m+M ? >

18. Elastic Collisions in 1D Equ.s (9-67), (9-68)

19. Sample Problem 9-10 Step 1 Step 2

20. A Rocket System (rocket + its ejected combustion products) is a closed, isolated system. The rocket is accelerated as a result of the thrust it receives from the ejected gases. Rocket Propulsion

21. First rocket equation: where R is the fuel consumption rate, v rel is the velocity of ejected fuel with respect to the rocket M is the instantaneous mass of the rocket a is the acceleration of the rocket T is referred to as the thrust of the rocket engine Second rocket equation: where v i and v f are initial and final velocities of the rocket M i and M f are the initial and final masses of the rocket Rocket Equations

22. Summary #1 Linear momentum: Newton’s second law: Impulse-momentum theorem: Center of mass: System of n particles Solid object Solid object with uniform density

23. Conservation of momentum: (closed, isolated system) Summary #2 Rocket equations: Elastic collisions: Inelastic collisions: Completely inelastic collisions: