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Center of Mass and Momentum in Various Coordinates

Explore the relationship between center of mass and momentum using polar, cylindrical, and spherical coordinates. Work through examples to understand multi-particle mass distributions and continuous mass distributions. Learn how to calculate velocities in different cases and apply concepts to real-world scenarios.

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Center of Mass and Momentum in Various Coordinates

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  1. Lecture #3 • Center of Mass • Defined • Relation to momentum • Polar, Cylindrical and Spherical Coordinates • Worked problems • DVD Demonstration on momentum cons. and CM motion :10

  2. Center of Mass • Center of Mass and Center of gravity happen to be equivalent • For a multi-particle discrete mass-distribution • For a continuous mass-distribution . :15

  3. Worked Example L3-1 – CM Motion Initial Final • Given m1 to m2 m= m Initial Final m = 3m Calculate Vcm Initial and Final for two cases :50

  4. Linear Momentum and CM :20

  5. Spherical Coordinates and Earth • Spherical coordinates • “Phi” or “Fee” (j) – East-west same as longitude • “Theta” (q) – North-south, same as Colatitude • (q) is 0 at north pole, 180 at south pole, 90 at equator • “r” (radius) :60

  6. Cylindrical and Spherical Coordinates :30

  7. Worked Example L3-2 – Discrete masses y 2 units x • Given m1 to m10 O2 ma= m y ma = 3m x O1 Calculate Given origin 1 unit For homework given O2 O1 :50

  8. Worked Example L3-3 – Continuous mass • Given quarter disk with uniform mass-density s and radius 2 km: • Calculate M total • Write r in polar coords • Write out double integral, in r and phi • Solve integral 2 km j r O1 Calculate Given origin O1 • REPEAT for Half disk :60

  9. Lecture #3 Wind-up • . • . • . • Office hours today and tomorrow 4-5:30. • Homework problems in Taylor, + Supplement. • Second homework due in class Thursday 9/4 :72

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