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Consider a pendulum of length & with all the mass m at its end. The pendulum is allowed to swing freely in both directions. Using to describe the azimuthal angle about the z axis and to measure the angular deviation of the pendulum from the downward direction, address the following questions: (a) If the pendulum is initially moving horizontally with velocity vo and angle 0o = 90° (horizontal), use energy and angular momentum conservation to find the minimum angles of min subtended by the pendulum. (Note that the angle will oscillate between 90° and the minimum angle. (h) Write the Lagrangian using A and d as generalized coordinates

Consider a pendulum of length & with all the mass m at its end. The pendulum is allowed to swing freely in both directions. Using to describe the azimuthal angle about the z axis and to measure the angular deviation of the pendulum from the downward direction, address the following questions: (a) If the pendulum is initially moving horizontally with velocity vo and angle 0o = 90° (horizontal), use energy and angular momentum conservation to find the minimum angles of min subtended by the pendulum. (Note that the angle will oscillate between 90° and the minimum angle. (h) Write the Lagrangian using A and d as generalized coordinates

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Consider a pendulum of length & with all the mass m at its end. The pendulum is allowed to swing freely in both directions. Using to describe the azimuthal angle about the z axis and 0 to measure the angular deviation of the pendulum from the downward direction, address the following questions: = 90° (a) If the pendulum is initially moving horizontally with velocity vo and angle o (horizontal), use energy and angular momentum conservation to find the minimum angles of min subtended by the pendulum. (Note that the angle will oscillate between 90° and the minimum angle. (b) Write the Lagrangian using ¤ and & as generalized coordinates. (c) Write the equations of motion for 0 and 6. (d) Rewrite the equations of motion for using angular momentum conservation to elim- inate and reference to p. (e) Find the value of L required for the stable orbit to be at 0 = 45°. (f) For the steady orbit found in (e) consider small perturbations of the orbit. Find the frequency with which the pendulum oscillates around 0 = 45°.
Transcribed Image Text:Consider a pendulum of length & with all the mass m at its end. The pendulum is allowed to swing freely in both directions. Using to describe the azimuthal angle about the z axis and 0 to measure the angular deviation of the pendulum from the downward direction, address the following questions: = 90° (a) If the pendulum is initially moving horizontally with velocity vo and angle o (horizontal), use energy and angular momentum conservation to find the minimum angles of min subtended by the pendulum. (Note that the angle will oscillate between 90° and the minimum angle. (b) Write the Lagrangian using ¤ and & as generalized coordinates. (c) Write the equations of motion for 0 and 6. (d) Rewrite the equations of motion for using angular momentum conservation to elim- inate and reference to p. (e) Find the value of L required for the stable orbit to be at 0 = 45°. (f) For the steady orbit found in (e) consider small perturbations of the orbit. Find the frequency with which the pendulum oscillates around 0 = 45°.
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