Day 23 - Homework Session

Day 23 - Homework Session#

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Announcements#

  • Homework 3 is graded

  • Midterm 1 is still being graded

    • Re: Problem 2 \(\rightarrow\) HOLY CRAP YOU ALL ARE AMAZING

  • Homework 5 is due tonight

  • Homework 6 is due next Friday

  • Danny will be out next Wednesday

    • Class will be on zoom (usual link)

Reminders#

We solved the damped harmonic oscillator equation:

\[\ddot{x} + 2 \beta \dot{x} + \omega_0^2 x = 0\]

We chose a solution (ansatz) of the form

\[x(t) = C_1 e^{r t} + C_2 e^{r t}\]

and computed the roots of the characteristic equation:

\[r^2 + 2 \beta r + \omega_0^2 = 0\]

We found the roots to be:

\[r = -\beta \pm \sqrt{\beta^2 - \omega_0^2}\]

Weak Damping#

We found that when \(\beta^2 < \omega_0^2\), the roots are complex:

\[r = -\beta \pm i \sqrt{\omega_0^2 - \beta^2}\]

This means that the solution is oscillatory:

\[x(t) = e^{-\beta t} \left( C_1 \cos(\sqrt{\omega_0^2 - \beta^2} t) + C_2 \sin(\sqrt{\omega_0^2 - \beta^2} t) \right)\]

The solution is a damped oscillation with frequency \(\omega_1 = \sqrt{\omega_0^2 - \beta^2}\).

Strong Damping#

When \(\beta^2 > \omega_0^2\), the roots are real:

\[r = -\beta \pm \sqrt{\beta^2 - \omega_0^2}\]

This means that the solution is not oscillatory: $\(x(t) = C_1 e^{r_1 t} + C_2 e^{r_2 t}\)\( where \)r_1 = -\beta + \sqrt{\beta^2 - \omega_0^2} < 0\( and \)r_2 = -\beta - \sqrt{\beta^2 - \omega_0^2} < 0$.

The solution is the sum of two exponentials with different decay rates.

Critical Damping#

When \(\beta^2 = \omega_0^2\), the roots are real and equal (repeated roots): $\(r = -\beta\)$

This means that the solution is not oscillatory, but also that our ansatz is not sufficient. The correct form of the solution is:

\[x(t) = (C_1 + C_2 t) e^{-\beta t}\]

In most cases, we will work with weak damping.