1 Lecture #8 of 24 Study skills workshop – Cramer 121 – 7-9 pm Prof. Bob Cormack – The illusion of understanding Homework #3 – Moment of Inertia of disk w/ hole Buoyancy Energy and Conservative forces Force as gradient of potential Curl  Stokes Theorem and Gauss’s Theorem Line Integrals Energy conservation problems Demo – Energy Conservation :02

2 Disk with Hole :10 Axis 1 (through CM) Axis 2 (Parallel to axis 1) d R, M

3 Buoyancy Archimedes  Taking bath Noticing water displaced by his body as he got in the tub Running starkers thru the streets shouting “Eureka” The buoyant force on an object is equal to the WEIGHT of the fluid displaced by that object :15 Impure crown? King Hiero commissions ”Mission oriented Research”

4 Force as the gradient of potential :20

5 Gravitational Potential :25

6 Curl as limit of tiny line-integrals :25

7 Stokes and Gauss’s theorem’s :30 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.

8 The curl-o-meter (by Ronco®) :35 Conservative force a cd b e f

9 Maxwell’s Equations :35  Curl is zero EXCEPT Where there is a current I.

10 L8-1 – Area integral of curl :50 Taylor 4.3 (modified) O y x P (1,0) Q (0,1) a c b Calculate, along a,c Calculate, along a,b Calculate, inside a,c Calculate, inside a,b

11 L8-2 Energy problems :65 A block of mass “m” starts from rest and slides down a ramp of height “h” and angle “theta”. a) Calculate velocity “v” at bottom of ramp b) Do the same for a rolling disk (mass “m”, radius r) c) Do part “a” again, in the presence of friction “  ” O y x m h m y x

12 Retarding forces summary :72  Linear Drag on a sphere (Stokes) Falling from rest w/ gravity  Quadratic Drag on a Sphere (Newton) Falling from rest w/ gravity Decelerating from v without gravity

13 Lecture #8 Wind-up. Read Chapter 4 9/24 Office hours 4-6 pm First test 9/26 HW posted mid- afternoon :72  Stokes’ Theorem  For conservative forces