Day 37 - Help Session

Day 37 - Help Session#

Routhian Mechanics#

Routh

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Announcements#

  • Homework 8 is “Late” 24 Apr

    • Last Exercise 0: Reflect Learning Outcomes

  • Final Project is posted

    • Video Presentations due 27 Apr

    • Computational Essay due 1 May

    • Rubric for both are posted

  • No class (20 Apr - 24 Apr) - DC out of country

    • Make appointment for project help (clicker extra credit)

Announcements#

Rest of Semester Schedule#

  • CW16 - Examples of Lagrangian Dynamics (HW8)

  • CW17 - Project Prep (DC out of country)

  • CW18 - Final Project Due

    • Video Presentations due 27 Apr

    • Computational Essay due 1 May

NO IN-CLASS FINAL EXAM#

Clicker Question 36-5a#

Consider a bead sliding on a parabolic bowl described by the constraint \(z = c\rho^2\) where \(\rho\) is the distance from the vertical axis. The Lagrangian for this system in Cartesian coordinates is:

\[\mathcal{L} = \frac{1}{2}m(\dot{x}^2 + \dot{y}^2 + \dot{z}^2) - mgz\]

Don’t use the constraint, what are the equations of motion for this system? Do they seem correct?

Click anything to indicate you are ready to see the answer.

Clicker Question 36-5b#

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 36-5c#

Now use the constraint to write the Lagrangian for the bead in a parabolic bowl in cylindrical coordinates, \((\rho, \phi, z)\). What is the Lagrangian for this system?

  1. \(\mathcal{L} = \frac{1}{2}m(\dot{\rho}^2 + \rho^2\dot{\phi}^2 + 4c^2\rho^2) - mgc\rho^2\)

  2. \(\mathcal{L} = \frac{1}{2}m(\dot{\rho}^2 + \rho^2\dot{\phi}^2 + 4c^2{\rho}^2) - mgc\rho^2\)

  3. \(\mathcal{L} = \frac{1}{2}m(\dot{\rho}^2 + \rho^2\dot{\phi}^2 + 4c^2\rho^2\dot{\rho}^4) - mgc\rho^2\)

  4. \(\mathcal{L} = \frac{1}{2}m(\dot{\rho}^2 + \rho^2\dot{\phi}^2 + 4c^2\rho^2\dot{\rho}^2) - mgc\rho^2\)

  5. Something else

Hint: \(v^2(\rho,\phi,z) = \dot{\rho}^2 + \rho^2\dot{\phi}^2 + \dot{z}^2\)

Clicker Question 36-5d#

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 36-5e#

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

When a coordinate does not appear in the Lagrangian, it is called a cyclic or ignorable coordinate. This means that the generalized momentum associated with that coordinate is conserved.

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