Day 35 - Lagrangian Examples II

Day 35 - Lagrangian Examples II#

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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 35-1a#

For a hypothetical system, we have the Lagrangian that depends on two generalized coordinates, \(\mathcal{L}(\rho,\phi, \dot{\rho},\dot{\phi})\).

Here \(\rho\) has units of length and \(\phi\) has units of angle.

Which of the following derivatives give the generalized force associated with \(\rho\)?

\(1. \dfrac{\partial \mathcal{L}}{\partial \rho} \qquad 2. \dfrac{\partial \mathcal{L}}{\partial \dot{\rho}}\) \(3. \dfrac{d}{dt} \left( \dfrac{\partial \mathcal{L}}{\partial \rho} \right) \qquad 4. \dfrac{d}{dt} \left( \dfrac{\partial \mathcal{L}}{\partial \dot{\rho}} \right)\) 5. None of these

Clicker Question 35-1b#

For a hypothetical system, we have the Lagrangian that depends on two generalized coordinates, \(\mathcal{L}(\rho,\phi, \dot{\rho},\dot{\phi})\).

Here \(\rho\) has units of length and \(\phi\) has units of angle.

What are the units of the generalized force associated with \(\rho\)?

  1. Newtons (N)

  2. Joules (J)

  3. Newton-meters (N m)

  4. Meters (m)

  5. None of these

Clicker Question 35-1c#

For a hypothetical system, we have the Lagrangian that depends on two generalized coordinates, \(\mathcal{L}(\rho,\phi, \dot{\rho},\dot{\phi})\).

Here \(\rho\) has units of length and \(\phi\) has units of angle.

Which of the following derivatives give the generalized momentum associated with \(\rho\)?

\(1. \dfrac{\partial \mathcal{L}}{\partial \rho} \qquad 2. \dfrac{\partial \mathcal{L}}{\partial \dot{\rho}}\) \(3. \dfrac{d}{dt} \left( \dfrac{\partial \mathcal{L}}{\partial \rho} \right) \qquad 4. \dfrac{d}{dt} \left( \dfrac{\partial \mathcal{L}}{\partial \dot{\rho}} \right)\) 5. None of these

Clicker Question 35-1d#

For a hypothetical system, we have the Lagrangian that depends on two generalized coordinates, \(\mathcal{L}(\rho,\phi, \dot{\rho},\dot{\phi})\).

Here \(\rho\) has units of length and \(\phi\) has units of angle.

What are the units of the generalized momentum associated with \(\rho\)?

  1. kg m/s

  2. kg m\(^2\)/s

  3. kg m\(^2\)/s\(^2\)

  4. kg/s

  5. None of these

Clicker Question 35-1e#

For a hypothetical system, we have the Lagrangian that depends on two generalized coordinates, \(\mathcal{L}(\rho,\phi, \dot{\rho},\dot{\phi})\).

Which of the following derivatives give the generalized force associated with \(\phi\)?

\(1. \dfrac{\partial \mathcal{L}}{\partial \phi} \qquad 2. \dfrac{\partial \mathcal{L}}{\partial \dot{\phi}}\) \(3. \dfrac{d}{dt} \left( \dfrac{\partial \mathcal{L}}{\partial \phi} \right) \qquad 4. \dfrac{d}{dt} \left( \dfrac{\partial \mathcal{L}}{\partial \dot{\phi}} \right)\) 5. None of these

Clicker Question 35-1f#

For a hypothetical system, we have the Lagrangian that depends on two generalized coordinates, \(\mathcal{L}(\rho,\phi, \dot{\rho},\dot{\phi})\).

Here \(\rho\) has units of length and \(\phi\) has units of angle.

What are the units of the generalized force associated with \(\phi\)?

  1. Newtons (N)

  2. Joules (J)

  3. Newton-meters (N m)

  4. Radians (rad)

  5. None of these

Clicker Question 35-1g#

For a hypothetical system, we have the Lagrangian that depends on two generalized coordinates, \(\mathcal{L}(\rho,\phi, \dot{\rho},\dot{\phi})\).

Here \(\rho\) has units of length and \(\phi\) has units of angle.

Which of the following derivatives give the generalized momentum associated with \(\phi\)?

\(1. \dfrac{\partial \mathcal{L}}{\partial \phi} \qquad 2. \dfrac{\partial \mathcal{L}}{\partial \dot{\phi}}\) \(3. \dfrac{d}{dt} \left( \dfrac{\partial \mathcal{L}}{\partial \phi} \right) \qquad 4. \dfrac{d}{dt} \left( \dfrac{\partial \mathcal{L}}{\partial \dot{\phi}} \right)\) 5. None of these

Clicker Question 35-1h#

For a hypothetical system, we have the Lagrangian that depends on two generalized coordinates, \(\mathcal{L}(\rho,\phi, \dot{\rho},\dot{\phi})\).

Here \(\rho\) has units of length and \(\phi\) has units of angle.

What are the units of the generalized momentum associated with \(\phi\)?

  1. kg m/s

  2. kg m\(^2\)/s

  3. kg m\(^2\)/s\(^2\)

  4. kg/s

  5. None of these

Clicker Question 35-2#

For the Atwood’s machine, \(M\) is connected to \(m\) by a string of length \(l\). Each mass has a length of string extended as measured from the center of the pulley (\(R\)) of \(y_1\) and \(y_2\), respectively. The string wraps around half the pulley.

Which of the following represents the equation of constraint for the system?

  1. \(y_1 + y_2 = l - R \phi\)

  2. \(y_1 - y_2 = l + R \phi\)

  3. \(y_1 + y_2 = l - \pi R\)

  4. \(y_1 - y_2 = l + \pi R\)

  5. None of these

Take the time derivative of the constraint equation. What do you notice?

Clicker Question 35-3#

With a Lagrangian of the form \(\mathcal{L} = \frac{1}{2}(M+m)\dot{y}^2_1 - (M-m)gy_1\), we can find the generalized forces and generalized momenta.

\[F_{y_1} = \frac{\partial \mathcal{L}}{\partial y_1} = -\frac{\partial V}{\partial y_1} \quad p_{y_1} = \frac{\partial \mathcal{L}}{\partial \dot{y}_1} = \frac{\partial T}{\partial \dot{y}_1}\]

What are \(F_{y_1}\) and \(p_{y_1}\) for the Atwood’s machine?

  1. \(F_{y_1} = -mg\) and \(p_{y_1} = m\dot{y}_1\)

  2. \(F_{y_1} = -Mgy_1\) and \(p_{y_1} = M\dot{y}_1\)

  3. \(F_{y_1} = -(M-m)g\) and \(p_{y_1} = (M+m)\dot{y}_1\)

  4. \(F_{y_1} = -(M+m)g\) and \(p_{y_1} = (M-m)\dot{y}_1\)

  5. None of these

Clicker Question 35-4#

Now, we allow the pulley (mass, \(M_p\)) to rotate. The Lagrangian is given by: $\(\mathcal{L} = \frac{1}{2}(M+m)\dot{y}_1^2 + \frac{1}{2}I\dot{\phi}^2 - (M-m)gy_1\)$

Where \(I\) is the moment of inertia of the pulley. What is the moment of inertia of the pulley?

  1. \(I = \frac{1}{2}M_pR^2\)

  2. \(I = \frac{1}{3}M_pR^2\)

  3. \(I = M_pR^2\)

  4. \(I = \frac{1}{4}M_pR^2\)

  5. None of these

Clicker Question 35-5#

The rope moves without slipping on the pulley. A rotation of \(R d\phi\) corresponds to a displacement of \(dy_1\) for the first mass, \(M\). What is the new equation of constraint for the system?

  1. \(y_1 + y_2 = l - R \phi\)

  2. \(dy_1 = R d\phi\)

  3. \(y_1 = R\phi\)

  4. \(\dot{y}_1 = R \dot{\phi}\)

  5. More than one of these