Day 36 - Lagrangian Examples III

Day 36 - Lagrangian Examples III#

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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#

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 \(\dot{y}_1 = R\dot{\phi}\) to do this.

Holonomic constraints are those that can be expressed as a function of the coordinates and time, \(f(q_1, q_2, ..., q_n, t) = 0\).

Clicker Question 36-1a#

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 36-1b#

For the Atwood’s machine with a rotating pulley, what are the units of the generalized force, \(F_{\phi}\)?

  1. Newtons

  2. Joules

  3. Newton-meters

  4. Joule-seconds

  5. Something else

Clicker Question 36-2a#

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 36-2b#

For the Atwood’s machine with a rotating pulley, what are the units of the generalized momentum, \(p_{\phi}\)?

  1. Newtons

  2. Joules

  3. Newton-meters

  4. Joule-seconds

  5. Something else

Clicker Question 36-3a#

Two blocks (\(m\)) connected by a spring (\(k\), \(L\)) on a horizontal frictionless surface. The first block is described by a coordinate \(x_1\) and the second block is described by a coordinate \(x_2\) from a mutual origin. What is the Lagrangian for this system?

  1. \(\mathcal{L} = T - U = \frac{1}{2}m\dot{x}_1^2 + \frac{1}{2}m\dot{x}_2^2 - \frac{1}{2}k(x_1 - x_2 - L)^2\)

  2. \(\mathcal{L} = T - U = \frac{1}{2}m\dot{x}_1^2 + \frac{1}{2}m\dot{x}_2^2 - \frac{1}{2}k(x_1 - x_2)^2\)

  3. \(\mathcal{L} = T - U = \frac{1}{2}m\dot{x}_1^2 + \frac{1}{2}m\dot{x}_2^2 - \frac{1}{2}k(x_1 - x_2 + L)^2\)

  4. \(\mathcal{L} = T - U = \frac{1}{2}m\dot{x}_1^2 + \frac{1}{2}m\dot{x}_2^2 - \frac{1}{2}k(x_1 + x_2 - L)^2\)

  5. Something else

Clicker Question 36-3b#

For the two block and spring system, what is the equation of motion for the first block?

  1. \(m\ddot{x}_1 = -k(x_1 - x_2)\)

  2. \(m\ddot{x}_1 = -k(x_1 - x_2 + L)\)

  3. \(m\ddot{x}_1 = -k(x_1 - x_2 - L)\)

  4. \(m\ddot{x}_1 = -k(x_1 + x_2 - L)\)

  5. Something else

Clicker Question 36-3c#

For the two block and spring system, what is the equation of motion for the second block?

  1. \(m\ddot{x}_2 = +k(x_1 - x_2)\)

  2. \(m\ddot{x}_2 = -k(x_1 - x_2 - L)\)

  3. \(m\ddot{x}_2 = +k(x_1 - x_2 - L)\)

  4. \(m\ddot{x}_2 = +k(x_1 + x_2 - L)\)

  5. Something else

Clicker Question 36-4a#

Sometimes, we can choose better or more useful generalized coordinates. For the two block and spring system, we can choose the center of mass coordinate, \(x_{cm} = (x_1 + x_2)/2\), and the relative coordinate, \(x = x_1 - x_2\). What is the Lagrangian for this system in these new coordinates?

  1. \(\mathcal{L} = T - U = m\dot{x}_{cm}^2 + \frac{1}{4}m\dot{x}^2 - \frac{1}{2}k(x - L)^2\)

  2. \(\mathcal{L} = T - U = m\dot{x}_{cm}^2 + \frac{1}{4}m\dot{x}^2 - \frac{1}{2}k(x + L)^2\)

  3. \(\mathcal{L} = T - U = m\dot{x}_{cm}^2 + \frac{1}{4}m\dot{x}^2 - \frac{1}{2}k(x)^2\)

  4. \(\mathcal{L} = T - U = m\dot{x}_{cm}^2 + \frac{1}{4}m\dot{x}^2 - \frac{1}{2}k(x + L)^2\)

  5. Something else

Clicker Question 36-4b#

For this new Lagrangian, what is the equation of motion for the relative coordinate, \(x\)?

  1. \(\frac{1}{2}m\ddot{x} = -k(x - L)\)

  2. \(\frac{1}{2}m\ddot{x} = -k(x + L)\)

  3. \(\frac{1}{2}m\ddot{x} = -kx\)

  4. \(\frac{1}{2}m\ddot{x} = -k(x + L)\)

  5. Something else

What is the equation of motion for the center of mass coordinate, \(x_{cm}\)?

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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