Upcoming SlideShare
×

# Center Of Mass

15,908
-1

Published on

physics description of center of mass

Published in: Technology, Education
1 Comment
1 Like
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
• that really helped thanks so much

Are you sure you want to  Yes  No
Views
Total Views
15,908
On Slideshare
0
From Embeds
0
Number of Embeds
0
Actions
Shares
0
140
1
Likes
1
Embeds 0
No embeds

No notes for slide

### Center Of Mass

1. 1. Center of Mass Image: http://oregonstate.edu/instruct/exss323/Lecture_06.pdf
2. 2. <ul><li>The center of mass of a body or a system </li></ul><ul><li>of bodies is the point that moves as </li></ul><ul><li>though all of the </li></ul><ul><li>mass were </li></ul><ul><li>concentrated there </li></ul><ul><li>and all external </li></ul><ul><li>forces were </li></ul><ul><li>applied there. </li></ul>
3. 3. Motion of the Center of Mass <ul><li>See animations of projectile motion of rotating and non-rotating objects at: </li></ul><ul><li>http://www.kettering.edu/~drussell/Demos/COM/com-a.html </li></ul>
4. 4. Influences of Body Position <ul><li>Can use changes in body position to: </li></ul><ul><ul><li>Increase take-off height of COM (raise arms) </li></ul></ul><ul><ul><li>Decrease landing height (lift legs) </li></ul></ul><ul><ul><li>Increase height of individual body parts during flight (lower other parts) </li></ul></ul>http://oregonstate.edu/instruct/exss323/Lecture_06.pdf
5. 5. Center of Mass Motion <ul><li>See animated video of a hammer thrown. </li></ul><ul><li>Watch the motion of the center of mass: </li></ul><ul><li>http://www.regentsprep.org/Regents/physics/phys06/acentomas/default.htm </li></ul>
6. 6. High Jump <ul><li>Trajectory of the center of mass is determined when jumper leaves ground (including maximum height of COM) </li></ul><ul><li>Jumper changes body position in midair to improve performance </li></ul>http://oregonstate.edu/instruct/exss323/Lecture_06.pdf
7. 7. Center of Mass Equation <ul><li>For two masses m1 and m2, the center of mass is at: </li></ul>
8. 8. Center of Mass Equation <ul><li>For many particles, the center of mass can be written as: </li></ul>
9. 9. 1-D Center of Mass exercise <ul><li>Find the center of mass of three particles: </li></ul>1 kg 2 kg 4 kg
10. 10. Center of Mass 3-D <ul><li>In 3 dimensions the same equations apply: </li></ul>
11. 11. 2-D exercise <ul><li>Find the center of mass of a system of three particles: </li></ul>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. 12. Answer to 2-D exercise 1 2 3
13. 13. Exercise: non-uniform disk <ul><li>Find the center of mass of a disk of radius 2R from which an off-center disk of radius R is missing: </li></ul>2R R
14. 14. Non-uniform disk <ul><li>Consider 3 disks: small (filled), large (filled), and non-symmetrical: </li></ul>2R R m NS ? Non-sym m L =m s +m NS 0 Large m s -R Small mass COM Disk
15. 15. Non-uniform disk <ul><li>The center of mass of a large filled disk is at the origin: </li></ul><ul><li>Solve for x NS : </li></ul>2R R
16. 16. Solid Bodies <ul><li>For an infinite number of individual particles: </li></ul><ul><li>Replace summation with integrals: </li></ul>
17. 17. Solid Bodies: integrate <ul><li>Use density: </li></ul><ul><li>Then the integral becomes: </li></ul><ul><li>We will integrate over solid objects when we get to E&M </li></ul>