Definition of a System

1. Definition of a System Energy of a System Momentum of a System Force a System Work a System Impulse on a System Center-of-Mass

2. What is a System? A system is the particle or group of particles as defined by a problem in physics. It may be as small as a single atom consisting of neutrons, protons and electrons. It may be as large as the entire universe. It may or may not include every object in the problem. This depends on what is being asked?

3. The Work-Energy Theorem The work-energy theorem is true for systems as well as for individual particles. The work by all forces can be found using integration The change in kinetic energy is just the sum of the change kinetic energy for each particle

4. The Work-Energy Theorem Remember that for gravity and elastic forces, we can write where And so, we can write (remembering that there are other kinds of energy like energy of deformation, heat, etc.)

5. Momentum and Collisions There is nothing new here. You already learned that collisions required knowledge of systems and momenta add.

6. The Force on a System The net force on a system is the sum of the net force on every particle in the system. The particles to be considered in the system are given as part of the problem. Example question: What is the net force on the system that includes the two books below? We only need to consider the forces ON each of the books.

7. The Force on a System • The forces on the top book are: • Gravity from the earth • Normal force from the apple • Friction from the apple • Normal force from the bottom book • Friction from the bottom book • Example question: What is the net force on the system that • includes the two books below? The weight of the apple (gravity of the earth on the apple) is acting ON THE APPLE, not ON THE BOOK!!!

8. The Force on a System • The forces on the bottom book are: • Gravity from the earth • Normal force from top book • Friction from the top book • Normal force from the table • Friction from the table • Normal force from the hand • Example question: What is the net force on the system that includes the two books below? The weight of the top book (gravity of the earth on the top book) is acting ON THE TOPBOOK, not ON THE BOTTOM BOOK!!!

9. The Force on a System • Gravity from the earth • Normal force from the apple • Friction from the apple • Normal force from the bottom • book • Friction from the bottom book • Example question: What is the net force on the system that includes the two books below? Top Book Bottom Book • Gravity from the earth • Normal force from top • book • Friction from the top book • Normal force from the table • Friction from the table • Normal force from the hand The force due to gravity on the system is the force of gravity on the top book plus the force of gravity on the bottom book. In other words, it is Mg.

10. The Force on a System • Gravity from the earth (on the system) • Normal force from the apple • Friction from the apple • Normal force from the table • Friction from the table • Normal force from the hand • Example question: What is the net force on the system that includes the two books below? Thus, we are left with the following forces acting on the system (both books)… Look carefully. You will see that we could have treated both books as a single particle of mass M. This is a general rule. System

11. The Work on a System The net work on a system is the sum of the net work on every particle in the system. The particles to be considered in the system are given as part of the problem. Example question: What is the net work over a distance d on the system that includes the two books below? We only need to consider the forces ON each of the books.

12. The Force on a System • Gravity from the earth • Normal force from the apple • Friction from the apple • Normal force from the bottom • book • Friction from the bottom book • Example question: What is the net work over a distance d on the system that includes the two books below? Top Book Bottom Book • Gravity from the earth • Normal force from top • book • Friction from the top book • Normal force from the table • Friction from the table • Normal force from the hand The force due to gravity on the system is the force of gravity on the top book plus the force of gravity on the bottom book. In other words, it is Mg.

13. The Force on a System • Gravity from the earth (on the system) • Normal force from the apple • Friction from the apple • Normal force from the table • Friction from the table • Normal force from the hand • Example question: What is the net work over a distance d on the system that includes the two books below? Thus, we are left with the following forces acting on the system (both books)… Look carefully. You will see that we could have treated both books as a single particle of mass M. This is a general rule. System

14. The Work on a System • Gravity from the earth (on the system) • Normal force from the apple • Friction from the apple • Normal force from the table • Friction from the table • Normal force from the hand • Example question: What is the net work over a distance d on the system that includes the two books below? Thus, we are left with the following forces acting on the system (both books)… We can now do the integral on the total force on the system. If this is difficult, we can do the integral separately for each force and add the results. System

15. The Impulse on a System The net impulse on a system is the sum of the net work on every particle in the system. The particles to be considered in the system are given as part of the problem. Example question: What is the net impulse over a time Dt on the system that includes the two books below? We only need to consider the forces ON each of the books.

16. The Impulse on a System • Gravity from the earth (on the system) • Normal force from the apple • Friction from the apple • Normal force from the table • Friction from the table • Normal force from the hand • Example question: What is the net impulse over a time Dt on the system that includes the two books below? Thus, we are left with the following forces acting on the system (both books)… We can now do the integral on the total force on the system. If this is difficult, we can do the integral separately for each force and add the results. System

17. Definition of the center-of-mass The center-of-mass of a system is that point in the system that follows the laws of physics for a single particle. In other words, it is the “position” of the system. Motion of center-of-mass The center-of-mass of any group of objects or any large single object follows the same principles as particles in the previous chapters. Newton’s laws of motion still apply. The constant acceleration equations still apply. Work-energy theorem still applies.

18. Center-of-mass for individual particles The center of mass of a system of objects is the point in space that behaves like a single point mass under the influence of external forces. It is the “position” of the system. Center of mass for a group of point masses is given by the equation…

19. Example for calculating center of mass y 2 1 4 3 x 2m

20. Fn Example: 2 spheres against One F*n F 1 F fk 2 1 mg Fn F*n F 2 3 fk mg Fn Things to notice The acceleration a1 is not the same as a2. The acceleration a1 is always smaller than a3. F 3 fk mg

21. Example: 2 spheres against One Fn F F 2 1 2 1 fk 2mg F 3 Fn Things to notice For large forces, acm is smaller than a3. For small forces, acm is almost equal to a3. F 3 fk mg

22. Velocity and Acceleration of Center-of-Mass