1. 1 Class #12 Energy Applications Pendulum  Simple  Solid Potential wells 2 nd derivative as a spring constant

2. 2 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

3. 3 ODE Summary Math Physics

4. 4 Analogy of 1-D system to roller coaster E1E1 K(x)=E-U(x) <- General 1-D system K(x)=E-mgx <- Roller Coaster E2E2 E3E3 E4E4 x

5. 5 Potential Wells Mass m Spring constant k K > 0 K < 0

6. 6 Series Expansion of potential around a critical point Taylor series -- Generic Taylor series -- Potential Can be ignored or set to zero … “gauge invariance” Is already zero for potential evaluated about a critical point

7. 7 L10-2 Potential Wells What is equation of parabola of equivalent curvature? What is resonant frequency of a 1 kg mass operating in this potential?

8. 8 L 10-3 Potential Wells Calculate the Equilibrium separation of two water molecules Calculate the vibration frequency assuming M=18 amu 1 amu=