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1. General Principles Newton's First Law: A particle originally at rest, or moving in a straight line with constant velocity, will remain in this state provided the particle is not subjected to an unbalanced force. Newton's Second Law: A particle acted upon by an unbalanced force $\vec{F}$ experiences an acceleration $\vec{a}$ that has the same direction as the force and a magnitude that is directly proportional to the force. $ \vec{F} = m\vec{a} $ Newton's Third Law: The mutual forces of action and reaction between two particles are equal, opposite, and collinear. Gravitational Law: $ F = G \frac{m_1 m_2}{r^2} $, where $G = 6.673 \times 10^{-11} \text{ m}^3/(\text{kg} \cdot \text{s}^2)$ Weight: $ W = mg $, where $g = 9.81 \text{ m/s}^2$ or $32.2 \text{ ft/s}^2$ 2. Force Vectors 2.1. Scalar & Vector Quantities Scalar: Mass, volume, length, time. Vector: Force, velocity, position, moment. (Magnitude & Direction) 2.2. Vector Operations Vector Addition (Parallelogram Law): $\vec{R} = \vec{A} + \vec{B}$ Components: $F_x = F \cos \theta$, $F_y = F \sin \theta$ Magnitude: $F = \sqrt{F_x^2 + F_y^2}$ Direction: $\theta = \arctan \left( \frac{F_y}{F_x} \right)$ 2.3. Position Vectors $\vec{r} = (x_B - x_A)\hat{i} + (y_B - y_A)\hat{j} + (z_B - z_A)\hat{k}$ 2.4. Unit Vector $\vec{u}_A = \frac{\vec{A}}{|\vec{A}|}$ 2.5. Dot Product $\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta$ $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z$ Angle between vectors: $\theta = \arccos \left( \frac{\vec{A} \cdot \vec{B}}{|\vec{A}| |\vec{B}|} \right)$ Projection: $A_B = \vec{A} \cdot \vec{u}_B$ 2.6. Cross Product $\vec{A} \times \vec{B} = (A_y B_z - A_z B_y)\hat{i} - (A_x B_z - A_z B_x)\hat{j} + (A_x B_y - A_y B_x)\hat{k}$ Magnitude: $|\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}| \sin \theta$ 3. Equilibrium of a Particle (Statics) 2D Equilibrium: $\sum F_x = 0$, $\sum F_y = 0$ 3D Equilibrium: $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$ Always draw a Free-Body Diagram (FBD). 4. Force System Resultants (Statics) 4.1. Moment of a Force Scalar (2D): $M_O = Fd$ (Perpendicular distance) Vector (3D): $\vec{M}_O = \vec{r} \times \vec{F}$ 4.2. Principle of Transmissibility A force can be moved along its line of action without changing its external effect on a rigid body. 4.3. Couple Moment $\vec{M} = \vec{r} \times \vec{F}$ or $M = Fd$ (scalar) A couple moment is a free vector. 4.4. Resultant of a Force and Couple System $\vec{F}_R = \sum \vec{F}$ $\vec{M}_{R_O} = \sum \vec{M}_O + \sum \vec{M}_{couple}$ 4.5. Distributed Load Resultant Force: $F_R = \int w(x) dx$ (Area under the loading curve) Location: $\bar{x} = \frac{\int x w(x) dx}{\int w(x) dx}$ (Centroid of the area) 5. Equilibrium of a Rigid Body (Statics) 2D Equilibrium: $\sum F_x = 0$, $\sum F_y = 0$, $\sum M_O = 0$ 3D Equilibrium: $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$, $\sum M_x = 0$, $\sum M_y = 0$, $\sum M_z = 0$ Always draw a FBD, showing all applied and reaction forces/moments. 5.1. Common Support Reactions Type of Support Reaction Cable, Rope Force along cable Roller Force normal to surface Pin, Hinge Two force components (e.g., $F_x, F_y$) Fixed Support Two force components and a moment (e.g., $F_x, F_y, M_z$) 6. Structural Analysis (Statics) 6.1. Trusses (Pin-connected members) Assumptions: Members are two-force members, loads applied at joints. Method of Joints: Apply particle equilibrium ($\sum F_x=0, \sum F_y=0$) at each joint. Method of Sections: Cut through members, apply rigid body equilibrium ($\sum F_x=0, \sum F_y=0, \sum M=0$) to a section. Zero-Force Members: Two non-collinear members connected at an unloaded joint: both are zero-force. Three members, two collinear, connected at an unloaded joint: the third is zero-force. 6.2. Frames and Machines Composed of multi-force members. Disassemble the structure into its component parts and draw FBDs for each part. Apply rigid body equilibrium to each part. Remember action-reaction pairs between connected members. 7. Internal Forces (Statics) 7.1. Shear Force and Bending Moment Sign Convention: Axial Force (N): Tension is positive. Shear Force (V): Up on right face / Down on left face is positive. Bending Moment (M): Causes compression in top fibers / tension in bottom fibers (smiles) is positive. Relations: $\frac{dV}{dx} = -w(x)$ (Slope of shear diagram = negative distributed load) $\frac{dM}{dx} = V(x)$ (Slope of moment diagram = shear force) $\Delta V = -\int w(x) dx$ $\Delta M = \int V(x) dx$ 8. Friction (Statics) Static Friction: $F_s \le \mu_s N$ Kinetic Friction: $F_k = \mu_k N$ $\mu_s \ge \mu_k$ Angle of static friction: $\phi_s = \arctan(\mu_s)$ Impending motion occurs when $F_s = F_{s,max} = \mu_s N$. 9. Center of Gravity & Centroid (Statics) Center of Gravity: $ \bar{x} = \frac{\sum \tilde{x} W}{\sum W}$, $ \bar{y} = \frac{\sum \tilde{y} W}{\sum W}$, $ \bar{z} = \frac{\sum \tilde{z} W}{\sum W}$ Centroid of Area: $ \bar{x} = \frac{\sum \tilde{x} A}{\sum A}$, $ \bar{y} = \frac{\sum \tilde{y} A}{\sum A}$ Centroid of Volume: $ \bar{x} = \frac{\sum \tilde{x} V}{\sum V}$, $ \bar{y} = \frac{\sum \tilde{y} V}{\sum V}$, $ \bar{z} = \frac{\sum \tilde{z} V}{\sum V}$ 9.1. Pappus-Guldinus Theorems Area of Surface of Revolution: $A = \theta \bar{r} L$ Volume of Body of Revolution: $V = \theta \bar{r} A$ 10. Moments of Inertia (Statics) Area Moment of Inertia: $I_x = \int y^2 dA$ $I_y = \int x^2 dA$ $J_O = \int r^2 dA = I_x + I_y$ (Polar moment of inertia) Parallel-Axis Theorem: $I_x = \bar{I}_x + Ad_y^2$ $I_y = \bar{I}_y + Ad_x^2$ $J_O = \bar{J}_C + Ad^2$ Radius of Gyration: $k = \sqrt{I/A}$ 11. Kinematics of a Particle (Dynamics) 11.1. Rectilinear Motion Velocity: $v = \frac{ds}{dt}$ Acceleration: $a = \frac{dv}{dt} = \frac{d^2s}{dt^2}$ Chain Rule: $a \, ds = v \, dv$ Constant Acceleration: $v = v_0 + a_c t$ $s = s_0 + v_0 t + \frac{1}{2} a_c t^2$ $v^2 = v_0^2 + 2 a_c (s - s_0)$ 11.2. Curvilinear Motion 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: $\vec{a} = \ddot{x}\hat{i} + \ddot{y}\hat{j} + \ddot{z}\hat{k}$ 11.3. Projectile Motion $a_x = 0$, $v_x = (v_0)_x$, $x = x_0 + (v_0)_x t$ $a_y = -g$, $v_y = (v_0)_y - gt$, $y = y_0 + (v_0)_y t - \frac{1}{2}gt^2$ 11.4. Normal and Tangential Components $v = \dot{s}$ $a_t = \dot{v} = \frac{dv}{dt}$ (tangential acceleration) $a_n = \frac{v^2}{\rho}$ (normal acceleration, $\rho$ = radius of curvature) $a = \sqrt{a_t^2 + a_n^2}$ 11.5. Cylindrical Components $\vec{r} = r\hat{u}_r + z\hat{k}$ $\vec{v} = \dot{r}\hat{u}_r + r\dot{\theta}\hat{u}_\theta + \dot{z}\hat{k}$ $\vec{a} = (\ddot{r} - r\dot{\theta}^2)\hat{u}_r + (r\ddot{\theta} + 2\dot{r}\dot{\theta})\hat{u}_\theta + \ddot{z}\hat{k}$ 11.6. Relative Motion $\vec{r}_B = \vec{r}_A + \vec{r}_{B/A}$ $\vec{v}_B = \vec{v}_A + \vec{v}_{B/A}$ $\vec{a}_B = \vec{a}_A + \vec{a}_{B/A}$ 12. Kinetics of a Particle (Dynamics) 12.1. 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: $\sum F_t = ma_t$, $\sum F_n = ma_n$ Cylindrical: $\sum F_r = ma_r$, $\sum F_\theta = ma_\theta$, $\sum F_z = ma_z$ 12.2. 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: $T_1 + U_{1-2} = T_2$ Potential Energy: Gravity: $V_g = Wy$ Elastic: $V_e = \frac{1}{2}ks^2$ Conservation of Energy: $T_1 + V_1 = T_2 + V_2$ (for conservative forces) 12.3. Impulse and Momentum Linear Momentum: $\vec{L} = m\vec{v}$ Linear Impulse: $\text{Imp}_{1-2} = \int_{t_1}^{t_2} \vec{F} dt$ 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: $m_A(\vec{v}_A)_1 + m_B(\vec{v}_B)_1 = m_A(\vec{v}_A)_2 + m_B(\vec{v}_B)_2$ (when impulses are zero or cancel) Impact: Coefficient of Restitution (e): $e = \frac{(v_B)_2 - (v_A)_2}{(v_A)_1 - (v_B)_1}$ $e=1$ (elastic), $e=0$ (plastic) 13. Planar Kinematics of a Rigid Body (Dynamics) 13.1. Angular Motion Angular Velocity: $\omega = \frac{d\theta}{dt}$ Angular Acceleration: $\alpha = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}$ Chain Rule: $\alpha \, d\theta = \omega \, d\omega$ Constant Angular Acceleration: $\omega = \omega_0 + \alpha_c t$ $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha_c t^2$ $\omega^2 = \omega_0^2 + 2 \alpha_c (\theta - \theta_0)$ 13.2. Absolute Motion Analysis Relate position of points using geometry, then differentiate with respect to time. $v = r\omega$ (for point on rotating body) $a_t = r\alpha$, $a_n = r\omega^2$ 13.3. Relative Motion Analysis (Translating Axes) $\vec{v}_B = \vec{v}_A + \vec{v}_{B/A}$ $\vec{a}_B = \vec{a}_A + \vec{a}_{B/A}$ For a rigid body: $\vec{v}_{B/A} = \vec{\omega} \times \vec{r}_{B/A}$ For a rigid body: $\vec{a}_{B/A} = \vec{\alpha} \times \vec{r}_{B/A} - \omega^2 \vec{r}_{B/A}$ 13.4. Relative Motion Analysis (Rotating Axes) $\vec{v}_B = \vec{v}_A + \vec{\Omega} \times \vec{r}_{B/A} + (\vec{v}_{B/A})_{xyz}$ $\vec{a}_B = \vec{a}_A + \dot{\vec{\Omega}} \times \vec{r}_{B/A} + \vec{\Omega} \times (\vec{\Omega} \times \vec{r}_{B/A}) + 2\vec{\Omega} \times (\vec{v}_{B/A})_{xyz} + (\vec{a}_{B/A})_{xyz}$ Last term is Coriolis acceleration: $2\vec{\Omega} \times (\vec{v}_{B/A})_{xyz}$ 13.5. Instantaneous Center of Zero Velocity (IC) Locate IC for a body undergoing general planar motion. $v = r\omega$, where $r$ is distance from IC to point. 14. Planar Kinetics of a Rigid Body (Dynamics) 14.1. Mass Moment of Inertia $I = \int r^2 dm$ Parallel-Axis Theorem: $I = \bar{I} + md^2$ Radius of Gyration: $k = \sqrt{I/m}$ 14.2. Equations of Motion $\sum F_x = m(\bar{a}_x)$ $\sum F_y = m(\bar{a}_y)$ $\sum M_G = \bar{I}\alpha$ (Sum moments about center of mass G) Alternatively, $\sum M_P = \sum (\mathcal{M}_k)_P$ (Sum moments about any point P, including kinetic moments) 14.3. Work and Energy Kinetic Energy: $T = \frac{1}{2}m\bar{v}^2 + \frac{1}{2}\bar{I}\omega^2$ Work of a Couple Moment: $U_M = \int_{\theta_1}^{\theta_2} M d\theta$ Principle of Work and Energy: $T_1 + U_{1-2} = T_2$ Conservation of Energy: $T_1 + V_1 = T_2 + V_2$ 14.4. Impulse and Momentum Linear Impulse and Momentum: $m(\vec{v}_G)_1 + \sum \int_{t_1}^{t_2} \vec{F} dt = m(\vec{v}_G)_2$ Angular Momentum: $(H_G)_1 + \sum \int_{t_1}^{t_2} M_G dt = (H_G)_2$ $(H_G) = \bar{I}\omega$ (for planar motion) Conservation of Angular Momentum: $(H_G)_1 = (H_G)_2$ or $(H_O)_1 = (H_O)_2$ (if no external moments about G or O)