1. 1 Class #11 Intuitive understanding of curl “Curl-o-meter” Energy Applications Rolling down a ramp Pendulum  Simple  Solid Potential wells 2 nd derivative as a spring constant Test Review

2. 2 Curl as limit of tiny line-integrals

3. 3 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.

4. 4 L9-1 – Line integral and area integral of curl 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

5. 5 The curl-o-meter (by Ronco®) Conservative force a cd b e f

6. 6 Kinetic energy of rolling objects

7. 7 L8-2 Energy problems - Rolling A hoop and a cylinder of equal radius “R” and mass M roll down equivalent ramps a) What is velocity “v” at ramp bottom in each case? b) Which shape “wins” the ramp race. O y x h m y x m

8. 8 Simple and Solid Pendula L m1m1 Approach is same for solid pendulum If replace z with z of CM and If replace with appropriate moment of inertia

9. 9 L m1m1 L is distance from pivot to CM of m1 R is radius of spherical pendulum bob. What is correction to ordinary pendulum frequency if R=L/2? L10-1 Solid Pendulum