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 Given m 1 to m2 m= m m = 3m Calculate Vcm Initial and Final for two cases :50 InitialFinal InitialFinal

4 Linear Momentum and CM :20

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

6 Cylindrical and Spherical Coordinates :30 Coord. System Area dA Volume dV Cartesian Spherical Cylindrical Polar

7 Worked Example L3-2 – Discrete masses Given m 1 to m 10 m  = m m  = 3m y x y x O1O1 O2O2 1 unit 2 units Calculate Given origin O1O1 For homework given O 2 :50

8 Worked Example L3-3 – Continuous mass Given quarter disk with uniform mass-density  and radius 2 km: Calculate M total Write r in polar coords Write out double integral, in r and phi Solve integral  r O1O1 2 km Calculate Given origin O1O1 :60 REPEAT for Half disk

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