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Simple Pendulum

Simple Pendulum

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Simple Pendulum

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  1. Simple Pendulum Q Rcm q L cos q Equation of Motion mg Simple Pendulum

  2. Compound Pendulum Q Center of mass L q Rcm cos q Equation of Motion mg Compound Pendulum

  3. Pendulum Simple Pendulum Compound Pendulum Approximation

  4. Different Centers Center of mass = Center of gravity: Center of gyration: (Should be called radius of gyration) Center of oscillation: Parallel Axis Theorem Center of percussion: -- Later

  5. Compound PendulumEffective Length G leff h b h’ for us a

  6. Exact SolutionEnergy Method For when the angles are not small kis the largest angle for an energy E

  7. Energy Method (continued) Elliptical Integral

  8. Approximate Solution To find period, set:

  9. Exact Solution (by definition) See Mathematica notebook Incomplete Elliptical Integral of the first kind (EllipticF[φ,a] in Mathematica) Period: Exact Approx. Complete Elliptical Integral of the first kind (EllipticK[a] in Mathematica

  10. Inverse Function Jacobi Amplitude function Jacobi Sine function k=p-.1 k=.01 k=3p/4 k=p/2