Day 10 - Integrating EOMs Numerically

Day 10 - Integrating EOMs Numerically#

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Announcements#

  • PERMANENT CHANGE: OFFICE HOURS

    • DC Office hours 10:00-12:00 on Fridays (No Monday office hours)

  • CHANGES THIS WEEK (DC has a conflict):

Seminars this week#

Astronomy Seminar, Wednesday Feb 5th at 1:30pm in 1400 BPS

  • Lia Corrales, Univ. of Michigan, The cosmic journey of the elements, from dust to life

Physics & Astronomy Colloquium, Thursday Feb 6th at 3:30pm in 1415 BPS

  • Andreas Jung, Purdue University, Entangled Titans: unraveling the mysteries of Quantum Mechanics with top quarks

Reminder: email me your extra credit seminar write-ups

Goals for this week#

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  • Establish a model for drag forces

  • Develop an understanding of the process for modeling forces

  • Produce equations of motion that can be investigated

  • Start probing the behavior of these systems with math and computing

Model-to-EOM Pipeline for Classical Mechanics#

  1. ✅ Develop conceptual description of the system; make justifiable assumptions

  2. ✅ Using a framework of physics (i.e., Newton’s Laws, Lagrangian Dynamics), develop a mathematical model of the system

  3. ✅ Produce the equations of motion (EOM) by following the framework (i.e., ordinary differential equations)

  4. 😵 Solve for trajectories of the system (e.g., \(x(t)\), \(v(t)\), \(v(x)\))

Most EOMs are nonlinear#

We need approximate methods to produce trajectories#

Clicker Question 6-5#

For the system of Quadratic Drag in 1D, we found a solution for the velocity as a function of time, with \(v = 0\) at \(t = 0\).

\[v(t) = v_{term}\tanh(gt/v_{term})\]

where \(v_{term} = \sqrt{mg/c}\). What happens when \(t \rightarrow \infty\)?

  1. The object stops moving.

  2. The object travels at a constant velocity.

  3. The object travels at an increasing velocity.

  4. The object travels at a decreasing velocity.

  5. I’m not sure.

Clicker Question 6-6#

For the gravitational interaction, I want to compute the force acting on body B, located at \(\vec{r}_B\), by body A, located at \(\vec{r}_A\).

The gravitational force is given by:

\[\vec{F} = -G\dfrac{m_1 m_2}{r^2}\hat{r}\]

What is the appropriate form of \(\vec{r}\)?

  1. \(\vec{r} = \vec{r}_A - \vec{r}_B\)

  2. \(\vec{r} = \vec{r}_B - \vec{r}_A\)

  3. Either is ok

Clicker Question 6-7#

We found that the equation of motion for the spring-mass system was:

\[\ddot{x} = -\dfrac{k}{m}x = -\omega^2 x\]

Your friends have proposed the following general solutions:

\[1.\;x(t) = A\cos(\omega t) \qquad 2.\;x(t) = B\sin(\omega t) \qquad 3.\;x(t) = A\cos(\omega t) + B\sin(\omega t) \]
\[ 4.\;x(t) = A\cos(\omega t + \phi) \qquad 5.\;x(t) = B\sin(\omega t + \phi) \qquad 6.\;x(t) = A\cos(\omega t + \phi) + B\sin(\omega t + \phi) \]

How many of them are correct? (1) Only one (2) Two (3) Three (4) Four (5) All of them