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

6. Rotation Density and C of M 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. Rotation Density and C of M 7 The Force on a System The forces on the top book are: 1.Gravity from the earth 2.Normal force from the apple 3.Friction from the apple 4.Normal force from the bottom book 5.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. Rotation Density and C of M 8 The Force on a System The forces on the bottom book are: 1.Gravity from the earth 2.Normal force from top book 3.Friction from the top book 4.Normal force from the table 5.Friction from the table 6.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 TOP BOOK, not ON THE BOTTOM BOOK!!!

9. Rotation Density and C of M 9 The Force on a System 1.Gravity from the earth 2.Normal force from the apple 3.Friction from the apple 4.Normal force from the bottom book 5.Friction from the bottom book Example question: What is the net force on the system that includes the two books below? 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. 1.Gravity from the earth 2.Normal force from top book 1.Friction from the top book 2.Normal force from the table 3.Friction from the table 4.Normal force from the hand Top Book Bottom Book

10. Rotation Density and C of M 10 The Force on a System 1.Gravity from the earth (on the system) 2.Normal force from the apple 3.Friction from the apple 4.Normal force from the table 5.Friction from the table 6.Normal force from the hand Example question: What is the net force on the system that includes the two books below? Look carefully. You will see that we could have treated both books as a single particle of mass M. This is a general rule. Thus, we are left with the following forces acting on the system (both books)… System

11. Rotation Density and C of M 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. Rotation Density and C of M 12 The Force on a System 1.Gravity from the earth 2.Normal force from the apple 3.Friction from the apple 4.Normal force from the bottom book 5.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? 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. 1.Gravity from the earth 2.Normal force from top book 1.Friction from the top book 2.Normal force from the table 3.Friction from the table 4.Normal force from the hand Top Book Bottom Book

13. Rotation Density and C of M 13 The Force on a System 1.Gravity from the earth (on the system) 2.Normal force from the apple 3.Friction from the apple 4.Normal force from the table 5.Friction from the table 6.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? Look carefully. You will see that we could have treated both books as a single particle of mass M. This is a general rule. Thus, we are left with the following forces acting on the system (both books)… System

14. Rotation Density and C of M 14 The Work on a System 1.Gravity from the earth (on the system) 2.Normal force from the apple 3.Friction from the apple 4.Normal force from the table 5.Friction from the table 6.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? 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. Thus, we are left with the following forces acting on the system (both books)… System

15. Rotation Density and C of M 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  t on the system that includes the two books below? We only need to consider the forces ON each of the books.

16. Rotation Density and C of M 16 The Impulse on a System 1.Gravity from the earth (on the system) 2.Normal force from the apple 3.Friction from the apple 4.Normal force from the table 5.Friction from the table 6.Normal force from the hand Example question: What is the net impulse over a time  t on the system that includes the two books below? 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. Thus, we are left with the following forces acting on the system (both books)… System

17. Rotation Density and C of M 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. Rotation Density and C of M 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. Rotation Density and C of M 19 Example for calculating center of mass x y 1 2 3 4 2m

20. Rotation Density and C of M 20 Example: 2 spheres against One 1 2 3 F F 1 F mg FnFn fkfk F* n 2 mg FnFn fkfk 3 F FnFn fkfk Things to notice The acceleration a 1 is not the same as a 2. The acceleration a 1 is always smaller than a 3.

21. Rotation Density and C of M 21 Example: 2 spheres against One 1 2 3 F F 3 F mg FnFn fkfk Things to notice For large forces, a cm is smaller than a 3. For small forces, a cm is almost equal to a 3. 2 F 2mg FnFn fkfk 1

22. Rotation Density and C of M 22 Velocity and Acceleration of Center-of-Mass