1. Physics 321 Hour 8 Potential Energy in Three Dimensions Gradient, Divergence, and Curl

2. Bottom Line

3. A Problem We’ll solve a simple problem using different methods. A sphere rolls without slipping down an incline. We are given m, R, and θ.

4. Newton’s Laws A sphere rolls without slipping down an incline. Given m, R, and θ, find the acceleration.

5. Conservation of Energy I A sphere rolls without slipping down an incline. Given m, R, and θ, find the velocity. Identify all Ts, Us. ΣT+ΣU = E = E 0. Gives v(y).

6. Conservation of Energy II

7. Conservation of Energy III (a) A sphere rolls without slipping down an incline. Given m, R, and θ, find x(t). 1)Write T and U. 2)Write equations of constraint among variables.

8. Conservation of Energy III (b)

9. A Pendulum Problem R m (a) Write T and U as functions of theta.

10. A Pendulum Problem R m (a) Write T and U as functions of theta.

11. A Pendulum Problem R m (b) Initial conditions: θ(0)=θ 0, θ(0)=0 Find θ(t) = ω(t).

12. A Pendulum Problem R m (c) Using this equation in Mathematica, solve for θ(t).

15. A Pendulum Problem R m (d) Find an equation of motion using T+U = 0.

16. A Pendulum Problem R m (e) Use Mathematica to solve this problem.