Homework Help is Here – Start Your Trial Now!
Science
Advanced Physics
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

icon
Related questions
Question

I need help with this physics question.

Consider a pendulum of length \( \ell \) with all the mass \( m \) at its end. The pendulum is allowed to swing freely in both directions. Using \( \phi \) to describe the azimuthal angle about the \( z \) axis and \( \theta \) 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 \( v_0 \) and angle \( \theta_0 = 90^\circ \) (horizontal), use energy and angular momentum conservation to find the minimum angles of \( \theta_{\text{min}} \) subtended by the pendulum. (Note that the angle will oscillate between \( 90^\circ \) and the minimum angle. (b) Write the Lagrangian using \( \theta \) and \( \phi \) as generalized coordinates. (c) Write the equations of motion for \( \theta \) and \( \phi \). (d) Rewrite the equations of motion for \( \theta \) using angular momentum conservation to eliminate reference to \( \phi \). (e) Find the value of \( L \) required for the stable orbit to be at \( \theta = 45^\circ \). (f) For the steady orbit found in (e) consider small perturbations of the orbit. Find the frequency with which the pendulum oscillates around \( \theta = 45^\circ \).
Transcribed Image Text:Consider a pendulum of length \( \ell \) with all the mass \( m \) at its end. The pendulum is allowed to swing freely in both directions. Using \( \phi \) to describe the azimuthal angle about the \( z \) axis and \( \theta \) 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 \( v_0 \) and angle \( \theta_0 = 90^\circ \) (horizontal), use energy and angular momentum conservation to find the minimum angles of \( \theta_{\text{min}} \) subtended by the pendulum. (Note that the angle will oscillate between \( 90^\circ \) and the minimum angle. (b) Write the Lagrangian using \( \theta \) and \( \phi \) as generalized coordinates. (c) Write the equations of motion for \( \theta \) and \( \phi \). (d) Rewrite the equations of motion for \( \theta \) using angular momentum conservation to eliminate reference to \( \phi \). (e) Find the value of \( L \) required for the stable orbit to be at \( \theta = 45^\circ \). (f) For the steady orbit found in (e) consider small perturbations of the orbit. Find the frequency with which the pendulum oscillates around \( \theta = 45^\circ \).
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 5 steps with 9 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer