Exam #1 review

Today's lecture was a review for the first midterm, coming up! Most of the lecture was spent on going over problems, which I won't reproduce here. However, I will reproduce my condensed list of physics formulas here.

(Note: this list focuses on physics, not math. I give no guarantee that every single formula you'll need is on the list! Making your own formula sheet with whatever information you find useful could be a good way to study.)

Formula reference, midterm #1


Chapter 1: Newtonian physics

Newton's three laws:

  1. If \( \vec{F} = 0 \), then \( v \) is constant.
  2. \( \vec{F} = \dot{\vec{p}} = m\vec{a} \) (if \( m \) is constant.)
  3. \( \vec{F}{12} = -\vec{F}{21} \).

These are true in inertial (non-accelerating) reference frames.

Important curvilinear coordinates:

Cartesian: \( (x,y,z) \)

Cylindrical: \( (\rho, \phi, z) \)

Newton's second law in cylindrical components:

\[ \begin{aligned} F_\rho = m(\ddot{\rho} - \rho \dot{\phi}^2) \\ F_\phi = m(\rho \ddot{\phi} + 2\dot{\rho} \dot{\phi}) \end{aligned} \]

Spherical: \( (r, \theta, \phi) \)


Chapter 2: Drag forces

Drag force is velocity-dependent. Most general form: \( \vec{f}(\vec{v}) = -f(v) \hat{v} \).

For motion small compared to the speed of sound, \( f(v) = bv + cv^2 \).

Microscopic origin:

For an object with linear size \( D \), and a medium with viscosity \( \eta \),

For an object with cross-sectional area \( A \) and drag coefficient \( C_D \), and a medium with density \( \rho \),

Usually we can ignore either linear or quadratic drag, if one is much larger. Determine relative size through Reynolds number,

\[ \begin{aligned} R = \frac{f_{\rm quad}}{f_{\rm lin}} = \frac{D\rho v}{\eta}. \end{aligned} \]

\( R \ll 1 \): keep linear force. \( R \gg 1 \): keep quadratic force.

Linear drag:

Horizontal motion, no other forces:

\[ \begin{aligned} v_x(t) = v_0 e^{-t/\tau}, \\ \end{aligned} \]

where natural time \( \tau = m/b \).

Vertical motion w/gravity (note that \( +y \) is up):

\[ \begin{aligned} v_y(t) = v_0 e^{-t/\tau} - v_{\rm ter} (1 - e^{-t/\tau}), \end{aligned} \]

where terminal velocity \( v_{\rm ter} = mg/b = g\tau \).

Quadratic drag:

Horizontal motion, no other forces:

\[ \begin{aligned} v_x(t) = \frac{v_0}{1+t/\tau_c}, \end{aligned} \]

where quadratic natural time \( \tau_c = m/(cv_0) \).

Vertical motion w/gravity (note that \( +y \) is up, assume \( v_y < 0 \)):

\[ \begin{aligned} v_y(t) = -v_{\rm ter,c} \tanh \left( \frac{t}{\tau} \right) \end{aligned} \]

where quadratic terminal velocity \( v_{\rm ter,c} = \sqrt{mg/c} \) and \( \tau = v_{\rm ter,c}/g \).


Chapter 3: Momentum conservation

Conservation of linear momentum: for a collection of \( N \) masses \( {m_\alpha} \),

\[ \begin{aligned} \vec{F}_{\rm net, ext} = \frac{d\vec{P}}{dt} \end{aligned} \]

where

\[ \begin{aligned} \vec{P} = \sum_\alpha \vec{p}_\alpha = \sum_{\alpha=1}^N m_\alpha \dot{\vec{r}}_\alpha. \end{aligned} \]

No net external force implies \( \vec{P} \) is constant.