Day 36 - Lagrangian Examples III

Day 36 - Lagrangian Examples III#

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Announcements#

  • Last “Class” Week

  • Homework 8 due Friday, Nov 21st (late after Nov 26th)

  • Next Week: Project Work and Discussion

  • Last Week: Presentations

  • Final Project Due Dec 8th (no later than 11:59 pm)

  • No Final Exam

Complete Google Form#

By November 21st#

Reporting your group members for the final project and a short summary of your project idea for sharing with the class.

https://forms.gle/iPKR9EDAaHW3GirN7

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Reminders#

We found the Lagrangian for the Atwood’s machine with a rotating pulley to be:

\[\mathcal{L} = \dfrac{1}{2}(M+m)R^2\dot{\phi}^2 + \dfrac{1}{4}M_pR^2\dot{\phi}^2 - (M-m)gR\phi\]

where \(M\) is the mass of the left block, \(m\) is the mass of the right block, \(M_p\) is the mass of the pulley, \(R\) is the radius of the pulley, and \(\phi\) is the angle of rotation of the pulley.

We used the scleronomic constraint \(y_1 = R\phi\) to do this.

Clicker Question 36-1#

We derived the Lagrangian for the Atwood’s machine with a rotating pulley to be:

\[\mathcal{L} = \dfrac{1}{2}(M+m)R^2\dot{\phi}^2 + \dfrac{1}{4}M_pR^2\dot{\phi}^2 - (M-m)gR\phi\]

What is generalized force, \(F_{\phi} = \partial \mathcal{L} / \partial \dot{\phi}\)?

  1. \(+(M-m)gR\)

  2. \(-(M-m)gR\)

  3. \(+(M+m)R^2\dot{\phi}\)

  4. \(-(M+m)R^2\dot{\phi}\)

  5. Something else

Clicker Question 36-2#

We derived the Lagrangian for the Atwood’s machine with a rotating pulley to be:

\[\mathcal{L} = \dfrac{1}{2}(M+m)R^2\dot{\phi}^2 + \dfrac{1}{4}M_pR^2\dot{\phi}^2 - (M-m)gR\phi\]

What is the generalized momentum, \(p_{\phi} = \partial \mathcal{L} / \partial \dot{\phi}\)?

  1. \(+(M-m)gR\)

  2. \(-(M-m)gR\)

  3. \(+(M+m)R^2\dot{\phi}\)

  4. \(-(M+m)R^2\dot{\phi}\)

  5. Something else

Clicker Question 36-3#

For the constraint for the bead in a parabolic bowl (\(z=c\rho^2\)), what are the units of \(c\)?

  1. \([L^2]\)

  2. \([L^{-2}]\)

  3. \([L]\)

  4. \([L^{-1}]\)

  5. Something else

Clicker Question 36-4#

For the bead in a parabolic bowl, there is a generic Lagrangian:

\[\mathcal{L}(\rho, \dot{\rho}, \phi, \dot{\phi}, z, \dot{z}, t)\]

How many coordinates are there, truly? here, each variable is a coordinate

A. 2 B. 3 C. 4 D. 5 E. None of these

Which coordinates are independent?

Clicker Question 36-5#

The Lagrangian for the bead in a parabola does not depend on which of the following?

  1. \(\rho\)

  2. \(\phi\)

  3. \(z\)

  4. More than one of these

  5. None of these

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