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Center Of Mass


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physics description of center of mass

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Center Of Mass

  1. 1. Center of Mass Image:
  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> </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>
  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> </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>
  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>