Class 7 of 30 Integration of vector quantities

Class #7 of 30 Integration of vector quantities n CM problems revisited Energy and Conservative forces n n Stokes Theorem and Gauss’s Theorem Line Integrals Curl Work done by a force over a path Angular momentum demo 1 : 03

Test #1 of 4 Thurs. 9/26/02 – in class Four problems n n Bring an index card 3”x 5”. Use both sides. Write anything you want that will help. All calculations to be written out and numbers plugged in BEFORE solving with a calculator. Full credit requires a units check. Linear and Angular momentum / Impulse Moment of Inertia / Center of Mass Retarding forces (Stokes / Newton etc. ) Conservative forces / Line integrals / curls / energy conservation 2 : 08

Integration in Different Coordinates 3 : 12

Worked Example L 7 -1 – Continuous mass Given hemisphere with uniform mass-density r and radius 5 m: n n Calculate M total Write r in polar coords Write out triple integral, . components in terms of r and phi Solve integral Calculate Given origin O 1 4 : 20

Worked Example L 3 -2 – Continuous mass 2 km Given quarter disk with uniform mass-density s and radius 2 km: j O 1 r n n Calculate M total Write r in polar coords Write out double integral, components in terms of r and phi. Solve integral Calculate Given origin O 1 5 : 30

Conservative Forces A force is conservative iff: 1. The force depends only on 2. For any two points P 1, P 2 the work done by the force is independent of the path taken between P 1 and P 2. 6 : 35

Line integral and Closed loop integral Conservative force n n n a) P 1 and P 2 with two possible integration paths. b) and c) P 1 and P 2 are brought closer together. d) P 1 and P 2 brought together to an arbitrarily small distance e. Geometric argument that conservative force implies zero closed-loop path integral. 7 : 40

Integration by eyeball a b c d e f Conservative force 8 : 50

y L 7 -2 – Path integrals Q(0, 1) Taylor 4. 3 (modified) c b O a P(1, 0) x Calculate, along (a) Calculate, along (b) Calculate, along (c) Calculate 9 : 65

Angular Momentum and Central Forces Taylor 3 -25 10 : 70

Lecture #7 Wind-up. For conservative forces Read Chapter 4 First test 9/26 11 : 72

Retarding forces summary. Linear Drag on a sphere (Stokes) Quadratic Drag on a Sphere (Newton) 12 : 72

Falling raindrops L 6 -2 A small raindrop falls through a cloud. It has a 1000 mm radius. The density of water is 1 g/cc. The viscosity of air is 180 m. Poise. The density of air is 1. 3 g/liter at STP. a) Draw the free-body diagram. b) What should be the terminal velocity of the raindrop, using quadratic drag? c) What should be the terminal velocity of the raindrop, using linear drag? d) Which of the previous of two answers should we use? ? e) What is the Reynolds number of this 13 raindrop? : 70

Curl as limit of tiny line-integrals 14 : 40

Stokes and Gauss’s theorem’s Gauss – Integrating divergence over a volume is equivalent to integrating function over a surface enclosing that volume. Stokes – Integrating curl over an area is equivalent to integrating function around a path enclosing that area. 15 : 60

The curl-o-meter (by Ronco®) a b c d e f Conservative force 16 : 50

y L 8 -1 – Area integral of curl Q(0, 1) Taylor 4. 3 (modified) c b O a P(1, 0) x Calculate, along a, c Calculate, along a, b Calculate, inside a, c Calculate, inside a, b 17 : 65