Mechanics

In intertial frames. i.e. when the observer is stationary

1. Draw a sketch
2. Draw a free body diagram, labelling all forces ($F_{A \text{ on } B}$ etc.)
3. Define a coordinate system (try to have one axis in the same direction as an acceleration vector and the origin at the position of the particle at t=0)
4. Resolve forces, apply Newton's laws and solve

$f_{s, \text{max}} = \mu_{s} \, n$

Valid when the object is stationary

$f_k = \mu_{k} \, n$

Valid when the object is moving

$\underline{A} \cdot \underline{B} = |\underline{A}| |\underline{B}| \cos \theta$

$ \underline{A} \times \underline{B} = \underline{C} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a & b & c \\ d & e & f \end{vmatrix} $

|$\underline{A}$||$\underline{B}$| sin($\theta$)

Use the right hand screw rule (place right hand on $\underline{A}$ and curl fingers towards $\underline{B}$, the direction of your thumb gives the direction of $\underline{C}$)

$\hat{i}$ in the direction of the +ve x-axis
$\hat{j}$ in the direction of the +ve y-axis
$\hat{k}$ in the direction of the +ve z-axis

$\underline{r} = x \hat{i} + y \hat{j} + z \hat{k}$

$\underline{v} = \frac{dx}{dt} \hat{i} + \frac{dy}{dt} \hat{j} + \frac{dz}{dt} \hat{k}$

$\underline{a} = \frac{d^2x}{dt^2} \hat{i} + \frac{d^2y}{dt^2} \hat{j} + \frac{d^2z}{dt^2} \hat{k}$

$\underline{F} = m \left( \frac{d^2x}{dt^2} \hat{i} + \frac{d^2y}{dt^2} \hat{j} + \frac{d^2z}{dt^2} \hat{k} \right)$

A force that acts between two particles along the imaginary line joining the two particles

$x = r \cos \theta$
$y = r \sin \theta$
$\frac{y}{x} = \tan \theta$
$x^2 + y^2 = r^2$

1. Take r($\theta$) and find cooridinate of points on the graph by subbing in values of $\theta$
2, Plot the points then draw the graph

$\hat{r}$ in the direction of the line connecting the particle and the origin
$\hat{\theta}$ perpendicular to $\hat{r}$ and in the direction the particle is traveling in

r = r$\hat{r}$

$\underline{v} = \dot{r} \, \hat{r} + r \, \dot{\theta} \, \hat{\theta}$

$\underline{a} = \left( \ddot{r} - r \dot{\theta}^2 \right) \hat{r} + \left( r \ddot{\theta} + 2 \dot{r} \dot{\theta} \right) \hat{\theta}$

$\underline{F} = m \left( \ddot{r} - r \dot{\theta}^2 \right) \hat{r} + m \left( r \ddot{\theta} + 2 \dot{r} \dot{\theta} \right) \hat{\theta}$

$a_{\theta}$ = 0

$a_\theta = 0$
$r^2 {\theta} = h$ (where h is the angular momentum per unit mass)
$\frac{dL}{d\theta} = 0$ (where L is the angular momentum)

1. Find h
2. Use $a_r = \ddot{r} - r \dot{\theta}^2$
3. Solve for what the question is asking for

$W = \underline{F} \cdot \underline{r} = |\underline{F}| |\underline{r}| \cos \theta$

$W = \int_{x_1}^{x_2} F_x \, dx + \int_{y_1}^{y_2} F_y \, dy + \int_{z_1}^{z_2} F_z \, dz$

1. Decide on a path between the points
2. Get equation relating x, y and maybe z, along that path
3. get $F_x$ in terms of only x get $F_y$ in terms of only y and maybe get $F_z$ in terms of only z
4. Use equation for work

$W = \int_{r_1}^{r_2} F_r \, dr + \int_{\theta_1}^{\theta_2} F_{\theta} \, d\theta$

W = Δ$E_k$ + Δ$E_p$

$P = \frac{dW}{dt} = \underline{F} \cdot \underline{v}$ $\therefore \text{W = } \int_{t_1}^{t_2} P \, dt$

$\underline r = r \hat r$ $\underline \theta = \theta \hat \theta$

$ \oint \mathbf{F} \cdot d\mathbf{r} = 0 $

$V(\underline{r}) = - \int \mathbf{F} \cdot d\mathbf{\underline{r}}$

$-\frac{dV}{d\underline{r}} = \mathbf{F}(r)$

$\underline{\tau} = \underline{r} \times \underline{F}$

$\underline{L} = \underline{r} \times \underline{p}$

$\underline{\tau} = \frac{d\underline{L}}{dt}$

$x_{\text{cm}} = \frac{1}{M} \int x \, dm$

$ dm = \sigma y \, dx $

$y_{\text{cm}} = \frac{1}{M} \int y \, dm$

$ dm = \sigma x \, dy $

$ M = \int \sigma \cdot \text{Area} \, (dx \, or \,dy) $

$ \text{CM} = \frac{ m_1 (CM_1) + m_2 (CM_2)}{m_1 + m_2} $

$\int r^2 \, dm = \frac{\underline{L}}{\omega}$

dm ....

$I_{comp} = I_1 + I_2 ...$

An axis which has even mass distributed on either side

$\omega_f^2 = \omega_i^2 + 2 \underline{\alpha} \Delta \theta$

$\omega_f = \omega_i + \underline{\alpha} t$

$\Delta \theta = \omega_i t + \frac{1}{2} \underline{\alpha} t^2$

$E_k = \frac{1}{2} {I} \, {\omega}^2$

$\tau = I \, \underline{\alpha}$