Day 37 - Homework Session

Day 37 - Homework Session#

Routhian Mechanics

https://en.wikipedia.org/wiki/Routhian_mechanics

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Announcements#

  • 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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Exercise 5 - Changing Coordinates#

Consider two particles with masses \(m_1=m_2=m\) connected by a spring with spring constant \(k\). They sit on a frictionless surface but are constrained to the \(x\)-axis. The location of the first particle is \(x_1\) and the location of the second particle is \(x_2\). The spring is at equilibrium when the particles are a distance \(L\) apart. Assume the spring is massless and that the left mass never changes position with the right mass (i.e., the spring is always horizontal).

  • 5a. Write down the kinetic and potential energy for this system in terms of \(x_1\) and \(x_2\) and their associated velocities.

In this problem we are changing the coordinates to explore a different aspect of the problem (and to make the math easier). This is a common technique that will not always be obvious. But practice will help us identify when it is useful.

Exercise 5 - Changing Coordinates#

  • 5b. Write down the Lagrangian for this system in terms of \(x_1\) and \(x_2\) and their associated velocities \(\mathcal{L}(x_1, x_2, \dot{x}_1, \dot{x}_2)\). Find the 2 equations of motion.

  • 5c. Now consider a new coordinate system where the center of mass is at \(x_{cm} = \frac{1}{2}(x_1 + x_2)\) and the relative coordinate is \(x = x_1 - x_2\). Write down the kinetic and potential energy in terms of \(x_{cm}\) and \(x\) and their associated velocities.

Exercise 5 - Changing Coordinates#

  • 5d. Write down the Lagrangian for this system in terms of the center of mass coordinate (\(x_{cm}\)) and the spring’s extension (\(x\)) and the associated velocities \(\mathcal{L}(x_{cm}, x, \dot{x}_{cm}, \dot{x})\). Find the 2 equations of motion.

  • 5e. Find \(x_{cm}(t)\) and \(x(t)\) for the system for some generic initial conditions of your choosing. Describe the motion of the system.

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