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Free Oscillations and Rotations of a Rigid Pendulum

Questions you should answer

  1. Apply the laws of dynamics to a rigid body that can rotate or swing about a horizontal axis in the gravitational field and derive the differential equation for the physical pendulum. What do we mean saying that the physical and mathematical pendulums are dynamically equivalent systems?
  2. How does the reduced (effective) length of the physical pendulum relate to the physical parameters of the pendulum? What is the meaning of the reduced length?
  3. Why is the pendulum referred to as an oscillatory system with a soft restoring force?
  4. How many parameters characterize the model of the pendulum used in the simulation program?
  5. How can we obtain the equation of the phase trajectory for the pendulum without integrating its differential equation of motion?
  6. To which property of the system is the symmetry of its phase trajectories about the abscissa axis related? The symmetry about the ordinate axis?
  7. What is the shape of the phase trajectories for small oscillations of the pendulum in the absence of friction? How do these trajectories modify when the amplitude of oscillations is increased?
  8. How does the ratio of the potential and kinetic energies change as the amplitude of oscillations is increased?
  9. Deduce the equation of the separatrix, that is, of the curve that delimits the trajectories of oscillations and rotations in the phase plane.
  10. What is the value of the initial angular velocity which we should impart to the pendulum in the equilibrium position in order the pendulum could reach the inverted position?
  11. How much time is required for the pendulum to reach the inverted position in conditions of the limiting motion (that is, the motion with minimal initial velocity), starting from the lower position of equilibrium?
  12. Deduce approximate (linearized) equations of motion that describe the motion of the pendulum in a small vicinity of the inverted equilibrium position. What is the general solution to this equation?

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