Today's lecture was a review for the final exam, covering everything since the first midterm. Like my last review, this is a review focused on physics over math, and I give no warranty that my list contains every single formula you might want to know. Making your own formula sheet with whatever info you want to have readily available would be a great way to review!
Definition of work done by force \( \vec{F} \) from \( \vec{r}_1 \) to \( \vec{r}_2 \):
\[ \begin{aligned} W = \int_{\vec{r}_1}^{\vec{r}_2} \vec{F} \cdot d\vec{r} \end{aligned} \]
Work-KE theorem:
\[ \begin{aligned} \Delta T = W_{\rm net} \end{aligned} \]
Kinetic energy: \( T = \frac{1}{2} mv^2 \) Potential energy (only if \( \vec{F} \) is conservative):
\[ \begin{aligned} U(\vec{r}) = -\int_{\vec{r}_0}^{\vec{r}} \vec{F} \cdot d\vec{r} \end{aligned} \]
Conservation of energy: \( E = T + U \) is time-independent if only conservative forces are present. If we have non-conservative forces, then
\[ \begin{aligned} \Delta E = W_{\rm non-cons} \end{aligned} \]
Force from potential:
\[ \begin{aligned} \vec{F} = -\vec{\nabla} U \end{aligned} \]
A force is conservative if \( \vec{\nabla} \times \vec{F} = 0 \) everywhere.
Energy and 1-d systems: equilibrium points where \( dU/dx = 0 \). Taylor expand at equilibrium point \( x_0 \) to test stability:
Stable equilibrium: \( U(x) \approx -k(x-x_0) \) - oscillating solutions.
Unstable equilibrium: \( U(x) \approx +k(x-x_0) \) - exponential solutions.
Definition of central force, with source at origin:
\[ \begin{aligned} \vec{F}(\vec{r}) = f(\vec{r}) \hat{r} \end{aligned} \]
A conservative, central force has \( f(\vec{r}) = f(r) \), i.e. force only depends on distance.
Gravitational force, on mass \( m \) due to source \( M \) at origin:
\[ \begin{aligned} \vec{F}_g = -\frac{GMm}{r^2} \hat{r} \end{aligned} \]
Potential energy:
\[ \begin{aligned} U(r) = -\frac{GMm}{r} \end{aligned} \]
Gravitational field:
\[ \begin{aligned} \vec{g} = \frac{\vec{F}_g}{m} = -\frac{GM}{r^2} \hat{r} \end{aligned} \]
Gravitational potential:
\[ \begin{aligned} \Phi(r) = \frac{U(r)}{m} = -\frac{GM}{r} \end{aligned} \]
Dealing with extended sources: gravitational field at \( \vec{r} \) due to source at \( \vec{r}' \) (overall source mass density \( \rho(\vec{r}') \)) is
\[ \begin{aligned} \vec{g} = -G \int \frac{dV' \rho(\vec{r}')}{|\vec{r} - \vec{r}'|^3} (\vec{r} - \vec{r}') \\ = -G \int \frac{dm}{s^2} \hat{s} \end{aligned} \]
where \( dm = \rho(\vec{r}') dV' \) and \( \vec{s} = \vec{r} - \vec{r}' \).
Or, calculate gravitational potential directly:
\[ \begin{aligned} \Phi(\vec{r}) = -G \int \frac{dm}{s} = -G \int \frac{dV' \rho(\vec{r}')}{|\vec{r} - \vec{r}'|}. \end{aligned} \]
Gauss' law: given a volume of space \( V \) with closed boundary surface \( \partial V \),
\[ \begin{aligned} \oint_{\partial V} \vec{g} \cdot d\vec{A} = -4\pi G \int_V \rho(\vec{r}) dV \\ = -4\pi G M_{\rm enc} \end{aligned} \]
(where \( M_{\rm enc} \) is the enclosed mass inside of \( V \).)
To use in practice, use symmetry to find a Gaussian surface over which \( \vec{g} \) points in the same direction as \( d\vec{A} \), the normal vector to the surface. Then the integral on the left-hand side becomes the surface area of \( \partial V \).
Simple harmonic oscillator:
\[ \begin{aligned} \ddot{x} + \omega_0^2 x = 0 \end{aligned} \]
where \( \omega_0^2 = k/m \). Solution takes the form
\[ \begin{aligned} x(t) = A \cos(\omega_0 t - \delta) \end{aligned} \]
with \( A, \delta \) unknown constants. Total energy:
\[ \begin{aligned} E = \frac{1}{2} kA^2 \end{aligned} \]
Damped harmonic oscillator: linear damping force of the form \( F = -b\dot{x} \). Equation of motion is
\[ \begin{aligned} \ddot{x} + 2\beta \dot{x} + \omega_0^2 x = 0 \end{aligned} \]
where \( \beta = b/(2m) \) is the damping constant. General solution:
\[ \begin{aligned} x(t) = e^{-\beta t} \left(C_1 e^{\sqrt{\beta^2 - \omega_0^2} t} + C_2 e^{-\sqrt{\beta^2 - \omega_0^2} t} \right) \end{aligned} \]
with \( C_1, C_2 \) unknown constants. Solution can be simplified to certain forms based on relative size of \( \beta, \omega_0 \):
\[ \begin{aligned} x(t) = A e^{-\beta t} \cos(\omega_1 t - \delta) \end{aligned} \]
where \( \omega_1 \equiv \sqrt{\omega_0^2 - \beta^2} \).
\[ \begin{aligned} x(t) = C_1 e^{-\beta t} + C_2 t e^{-\beta t} \end{aligned} \]
\[ \begin{aligned} x(t) = C_1 e^{-(\beta - \sqrt{\beta^2 - \omega_0^2})t} + C_2 e^{-(\beta + \sqrt{\beta^2 - \omega_0^2}) t}. \end{aligned} \]
Driven, damped oscillator: adding a driving force \( F(t) \). Equation of motion is
\[ \begin{aligned} \ddot{x} + 2\beta \dot{x} + \omega_0^2 x = \frac{F(t)}{m}. \end{aligned} \]
Driving force adds a particular solution \( x_p(t) \) to complementary solution from regular damped oscillator. Long-time behavior is only \( x_p(t) \).
Constant force \( F(t) = F_0 \): constant offset at long times,
\[ \begin{aligned} x_p(t) = \frac{F_0}{m\omega_0^2}. \end{aligned} \]
Sinusoidal driving force: \( F(t) = F_0 \cos(\omega t) \) gives
\[ \begin{aligned} x_p(t) = A \cos(\omega t - \delta), \end{aligned} \]
where
\[ \begin{aligned} A = \frac{F_0/m}{\sqrt{(\omega_0^2 - \omega^2)^2 + 4\beta^2 \omega^2}} \end{aligned} \]
and
\[ \begin{aligned} \tan \delta = \frac{2\beta \omega}{\omega_0^2 - \omega^2}. \end{aligned} \]
Resonance: peak amplitude
\[ \begin{aligned} A_{\rm peak} \approx \frac{F_0}{2m\omega_0 \beta}. \end{aligned} \]
at \( \omega_{\rm peak} = \sqrt{\omega0^2 - 2\beta^2} \), or \( \omega{0,{\rm peak}} = \omega \) if varying natural frequency.
Full width half-max, \( \omega_{\rm HM} \approx \omega_0 \pm \beta \). \( |A|^2 \) reduced by half from peak.
Q-factor: \( Q = \omega_0 / (2\beta) \). Large \( Q \) gives high and narrow resonance peak with driving. Without driving, large \( Q \) implies an oscillator that will ring for a long time before decaying.
If \( F(t+\tau) = F(t) \), then we can write a Fourier series
\[ \begin{aligned} F(t) = \sum_{n=0}^\infty \left[ a_n \cos(n\omega t) + b_n \sin (n \omega t) \right] \end{aligned} \]
where \( \omega = 2\pi/\tau \). Formulas for Fourier coefficients:
\[ \begin{aligned} a_n = \frac{2}{\tau} \int_{-\tau/2}^{\tau/2} F(t) \cos(n \omega t) dt \\ b_n = \frac{2}{\tau} \int_{-\tau/2}^{\tau/2} F(t) \sin(n \omega t) dt \\ a_0 = \frac{1}{\tau} \int_{-\tau/2}^{\tau/2} F(t) dt, \end{aligned} \]
and \( b_0 = 0 \). These rely on the orthogonality relations,
\[ \begin{aligned} \frac{2}{\tau} \int_{-\tau/2}^{\tau/2} \cos(m \omega t) \cos (n \omega t) dt = \delta_{mn} \end{aligned} \]
where \( \delta_{mn} \) is the Kronecker delta symbol, equal to \( 1 \) if \( m=n \) and \( 0 \) otherwise.
Solving the damped, driven oscillator with Fourier series: particular solution takes the form
\[ \begin{aligned} x_p(t) = \sum_{n=0}^\infty \left[ A_n \cos (n \omega t - \delta_n) + B_n \sin (n \omega t - \delta_n) \right] \end{aligned} \]
where we use the same amplitude/phase shift formulas as above, but with the Fourier coefficients as the force magnitude and \( n \omega \) as frequency. Explicitly,
\[ \begin{aligned} A_n = \frac{a_n/m}{\sqrt{(\omega_0^2 - n^2\omega^2)^2 + 4\beta^2 n^2 \omega^2}} \\ B_n = \frac{b_n/m}{\sqrt{(\omega_0^2 - n^2\omega^2)^2 + 4\beta^2 n^2 \omega^2}} \\ \delta_n = \tan^{-1} \left( \frac{2\beta n \omega}{\omega_0^2 - n^2 \omega^2} \right) \end{aligned} \]