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Statics: Force Systems Scalar Notation Resultant Force: $\sum F_x = R_x$, $\sum F_y = R_y$ Magnitude: $R = \sqrt{R_x^2 + R_y^2}$ Direction: $\theta = \arctan\left(\frac{R_y}{R_x}\right)$ Vector Notation Position Vector: $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ Unit Vector: $\vec{u} = \frac{\vec{r}}{|\vec{r}|}$ Force Vector: $\vec{F} = F\vec{u} = F\left(\frac{x}{r}\hat{i} + \frac{y}{r}\hat{j} + \frac{z}{r}\hat{k}\right)$ Resultant Force: $\sum \vec{F} = (\sum F_x)\hat{i} + (\sum F_y)\hat{j} + (\sum F_z)\hat{k}$ Dot Product $\vec{A} \cdot \vec{B} = |\vec{A}||\vec{B}|\cos\theta$ $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z$ Projection of $\vec{A}$ onto $\vec{B}$: $A_B = \vec{A} \cdot \vec{u}_B$ Statics: Equilibrium of a Particle 2D: $\sum F_x = 0$, $\sum F_y = 0$ 3D: $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$ Statics: Moments & Couples Moment of a Force (Scalar) $M_O = Fd$ (where $d$ is perpendicular distance) Moment of a Force (Vector) $\vec{M}_O = \vec{r} \times \vec{F}$ Cartesian: $\vec{M}_O = (r_y F_z - r_z F_y)\hat{i} + (r_z F_x - r_x F_z)\hat{j} + (r_x F_y - r_y F_x)\hat{k}$ Moment about an Axis $M_{axis} = \vec{u}_{axis} \cdot (\vec{r} \times \vec{F})$ Couple Moment $\vec{M} = \vec{r} \times \vec{F}$ (where $\vec{r}$ is from negative to positive force) Statics: Rigid Body Equilibrium 2D: $\sum F_x = 0$, $\sum F_y = 0$, $\sum M_O = 0$ 3D: $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$, $\sum M_x = 0$, $\sum M_y = 0$, $\sum M_z = 0$ Common Support Reactions Support Reactions (2D) Roller 1 normal force Pin 2 force components Fixed Support 2 force components, 1 moment Statics: Trusses Method of Joints Assume tension and apply $\sum F_x = 0$, $\sum F_y = 0$ at each pin. Method of Sections Cut through members to expose forces, apply rigid body equilibrium equations. Statics: Frames & Machines Separate structure into components. Draw FBD for each component. Apply equilibrium equations to each component. Internal forces are equal and opposite on connecting members. Statics: Centroid & Moment of Inertia Centroid of Area $\bar{x} = \frac{\int x dA}{\int dA}$, $\bar{y} = \frac{\int y dA}{\int dA}$ Composite Areas: $\bar{x} = \frac{\sum \bar{x}_i A_i}{\sum A_i}$, $\bar{y} = \frac{\sum \bar{y}_i A_i}{\sum A_i}$ Moment of Inertia (Area) $I_x = \int y^2 dA$, $I_y = \int x^2 dA$, $J_O = \int r^2 dA = I_x + I_y$ (Polar) Parallel-Axis Theorem $I_x = \bar{I}_x + Ad_y^2$, $I_y = \bar{I}_y + Ad_x^2$ Dynamics: Kinematics of a Particle Rectilinear Motion Velocity: $v = \frac{ds}{dt}$ Acceleration: $a = \frac{dv}{dt} = v\frac{dv}{ds}$ Constant Acceleration: $v = v_0 + at$ $s = s_0 + v_0 t + \frac{1}{2}at^2$ $v^2 = v_0^2 + 2a(s - s_0)$ Curvilinear Motion (Rectangular Components) Position: $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ Velocity: $\vec{v} = \dot{x}\hat{i} + \dot{y}\hat{j} + \dot{z}\hat{k}$ Acceleration: $\vec{a} = \ddot{x}\hat{i} + \ddot{y}\hat{j} + \ddot{z}\hat{k}$ Curvilinear Motion (Normal-Tangential Components) Velocity: $\vec{v} = v\hat{u}_t$ Acceleration: $\vec{a} = \dot{v}\hat{u}_t + \frac{v^2}{\rho}\hat{u}_n$ Radius of Curvature: $\rho = \frac{[1 + (dy/dx)^2]^{3/2}}{|d^2y/dx^2|}$ Curvilinear Motion (Cylindrical Components) Position: $\vec{r} = r\hat{u}_r + z\hat{k}$ Velocity: $\vec{v} = \dot{r}\hat{u}_r + r\dot{\theta}\hat{u}_{\theta} + \dot{z}\hat{k}$ Acceleration: $\vec{a} = (\ddot{r} - r\dot{\theta}^2)\hat{u}_r + (r\ddot{\theta} + 2\dot{r}\dot{\theta})\hat{u}_{\theta} + \ddot{z}\hat{k}$ Dynamics: Kinetics of a Particle Newton's Second Law $\sum \vec{F} = m\vec{a}$ Rectangular: $\sum F_x = ma_x$, $\sum F_y = ma_y$, $\sum F_z = ma_z$ Normal-Tangential: $\sum F_t = ma_t$, $\sum F_n = ma_n = m\frac{v^2}{\rho}$ Cylindrical: $\sum F_r = ma_r$, $\sum F_{\theta} = ma_{\theta}$, $\sum F_z = ma_z$ Work and Energy Work of a Force: $U_{1-2} = \int_{s_1}^{s_2} \vec{F} \cdot d\vec{r}$ Kinetic Energy: $T = \frac{1}{2}mv^2$ Principle of Work and Energy: $T_1 + U_{1-2} = T_2$ Conservative Forces: Gravity: $V_g = W y$ Spring: $V_e = \frac{1}{2}ks^2$ Conservation of Energy: $T_1 + V_1 = T_2 + V_2$ (only for conservative forces) Impulse and Momentum Linear Momentum: $\vec{L} = m\vec{v}$ Linear Impulse: $\int_{t_1}^{t_2} \vec{F} dt$ Principle of Linear Impulse and Momentum: $m\vec{v}_1 + \sum \int_{t_1}^{t_2} \vec{F} dt = m\vec{v}_2$ Conservation of Linear Momentum: $\sum (m\vec{v})_1 = \sum (m\vec{v})_2$ (if $\sum \vec{F}_{ext} = 0$) Coefficient of Restitution: $e = \frac{(v_B)_2 - (v_A)_2}{(v_A)_1 - (v_B)_1}$ Dynamics: Kinematics of a Rigid Body Rotation about a Fixed Axis Angular Velocity: $\omega = \frac{d\theta}{dt}$ Angular Acceleration: $\alpha = \frac{d\omega}{dt} = \omega\frac{d\omega}{d\theta}$ Constant Angular Acceleration: $\omega = \omega_0 + \alpha t$ $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$ $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$ Velocity of a Point: $v = \omega r$ (magnitude), $\vec{v} = \vec{\omega} \times \vec{r}$ (vector) Acceleration of a Point: $a_t = \alpha r$, $a_n = \omega^2 r = v^2/r$ $\vec{a} = \vec{\alpha} \times \vec{r} - \omega^2 \vec{r}$ Absolute General Plane Motion $\vec{v}_P = \vec{v}_A + \vec{\omega} \times \vec{r}_{P/A}$ $\vec{a}_P = \vec{a}_A + \vec{\alpha} \times \vec{r}_{P/A} - \omega^2 \vec{r}_{P/A}$ Relative-Motion Analysis (Translating Axes) $\vec{v}_B = \vec{v}_A + \vec{v}_{B/A}$ $\vec{a}_B = \vec{a}_A + \vec{a}_{B/A}$ Relative-Motion Analysis (Rotating Axes) $\vec{v}_B = \vec{v}_A + (\vec{\Omega} \times \vec{r}_{B/A}) + (\vec{v}_{B/A})_{xyz}$ $\vec{a}_B = \vec{a}_A + (\dot{\vec{\Omega}} \times \vec{r}_{B/A}) + (\vec{\Omega} \times (\vec{\Omega} \times \vec{r}_{B/A})) + 2(\vec{\Omega} \times (\vec{v}_{B/A})_{xyz}) + (\vec{a}_{B/A})_{xyz}$ Coriolis Acceleration: $2(\vec{\Omega} \times (\vec{v}_{B/A})_{xyz})$ Dynamics: Kinetics of a Rigid Body Equations of Motion (Plane Motion) $\sum F_x = m(\bar{a}_x)$ $\sum F_y = m(\bar{a}_y)$ $\sum M_G = I_G \alpha$ (Moment about center of mass G) $\sum M_P = I_P \alpha$ (Moment about fixed point P) Moment of Inertia (Mass) $I = \int r^2 dm$ Parallel-Axis Theorem: $I = \bar{I} + md^2$ Radius of Gyration: $k = \sqrt{I/m}$ Work and Energy (Rigid Body) Kinetic Energy: $T = \frac{1}{2}m\bar{v}^2 + \frac{1}{2}\bar{I}\omega^2$ Principle of Work and Energy: $T_1 + U_{1-2} = T_2$ Conservation of Energy: $T_1 + V_1 = T_2 + V_2$ Impulse and Momentum (Rigid Body) Linear Impulse-Momentum: $m(\vec{v}_G)_1 + \sum \int \vec{F} dt = m(\vec{v}_G)_2$ Angular Impulse-Momentum (about G): $\bar{I}\omega_1 + \sum \int M_G dt = \bar{I}\omega_2$ Angular Impulse-Momentum (about fixed point O): $(H_O)_1 + \sum \int M_O dt = (H_O)_2$ Conservation of Angular Momentum: $(H_O)_1 = (H_O)_2$ (if $\sum M_O = 0$)