Class 11 Intuitive understanding of curl n Curlometer

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

Curl as limit of tiny line-integrals 2

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. 3

L 9 -1 – Line integral and area integral of curl y Q(0, 1) c b O a P(1, 0) x Calculate, along a, c Calculate, along a, b Calculate, inside a, c Calculate, inside a, b 4

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

Kinetic energy of rolling objects 6

L 8 -2 Energy problems - Rolling y y m m h O x x 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. 7

Simple and Solid Pendula L m 1 Approach is same for solid pendulum If replace z with z of CM and If replace with appropriate moment of inertia 8

L 10 -1 Solid Pendulum L m 1 L is distance from pivot to CM of m 1 R is radius of spherical pendulum bob. What is correction to ordinary pendulum frequency if R=L/2? 9