The document discusses the center of mass of objects and systems of objects. It defines the center of mass as the point that behaves as if all the mass is concentrated there and external forces are applied there. It provides examples of how changing body position during motions like jumping can affect the height and motion of the center of mass. Equations are presented for calculating the center of mass for systems of particles and continuous objects. Examples of solving for the center of mass in different geometric configurations are also included.

1. Center of Mass Image: http://oregonstate.edu/instruct/exss323/Lecture_06.pdf

2. The center of mass of a body or a 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. Motion of the Center of Mass See animations of projectile motion of rotating and non-rotating objects at: http://www.kettering.edu/~drussell/Demos/COM/com-a.html

4. Influences of Body Position Can use changes in body position to: Increase take-off height of COM (raise arms) Decrease landing height (lift legs) Increase height of individual body parts during flight (lower other parts) http://oregonstate.edu/instruct/exss323/Lecture_06.pdf

5. Center of Mass Motion See animated video of a hammer thrown. Watch the motion of the center of mass: http://www.regentsprep.org/Regents/physics/phys06/acentomas/default.htm

6. High Jump Trajectory of the center of mass is determined when jumper leaves ground (including maximum height of COM) Jumper changes body position in midair to improve performance http://oregonstate.edu/instruct/exss323/Lecture_06.pdf

7. Center of Mass Equation For two masses m1 and m2, the center of mass is at:

8. Center of Mass Equation For many particles, the center of mass can be written as:

9. 1-D Center of Mass exercise Find the center of mass of three particles: 1 kg 2 kg 4 kg

10. Center of Mass 3-D In 3 dimensions the same equations apply:

11. 2-D exercise Find the center of mass of a system of three particles: 1 2 3 121 70 3.4 3 0 140 2.5 2 0 0 1.2 1 y (cm) x (cm) Mass (kg) Particle

12. Answer to 2-D exercise 1 2 3

13. Exercise: non-uniform disk Find the center of mass of a disk of radius 2R from which an off-center disk of radius R is missing: 2R R

14. Non-uniform disk Consider 3 disks: small (filled), large (filled), and non-symmetrical: 2R R m NS ? Non-sym m L =m s +m NS 0 Large m s -R Small mass COM Disk

15. Non-uniform disk The center of mass of a large filled disk is at the origin: Solve for x NS : 2R R

16. Solid Bodies For an infinite number of individual particles: Replace summation with integrals:

17. Solid Bodies: integrate Use density: Then the integral becomes: We will integrate over solid objects when we get to E&M