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Fundamental 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, i.e., $\vec{F} = m\vec{a}$. Newton's Third Law: The mutual forces of action and reaction between two particles are equal, opposite, and collinear. Newton's Law of Gravitational Attraction: $F = G \frac{m_1 m_2}{r^2}$, where $G = 66.73 \times 10^{-12} \text{ m}^3/(\text{kg} \cdot \text{s}^2)$. Weight: $W = mg$. On Earth, $g = 9.81 \text{ m/s}^2$ or $32.2 \text{ ft/s}^2$. Vector Operations Cartesian Vector: $\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$ Magnitude: $A = \sqrt{A_x^2 + A_y^2 + A_z^2}$ Unit Vector: $\hat{u}_A = \frac{\vec{A}}{A} = \frac{A_x}{A}\hat{i} + \frac{A_y}{A}\hat{j} + \frac{A_z}{A}\hat{k} = \cos\alpha \hat{i} + \cos\beta \hat{j} + \cos\gamma \hat{k}$ Dot Product: $\vec{A} \cdot \vec{B} = AB \cos\theta = A_x B_x + A_y B_y + A_z B_z$ Used to find angle between vectors or project one vector onto another. $\vec{A} \cdot \vec{B} = 0$ if $\vec{A} \perp \vec{B}$. Cross Product: $\vec{C} = \vec{A} \times \vec{B}$ Magnitude: $C = AB \sin\theta$. Direction by right-hand rule. Matrix form: $\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix}$ $\vec{A} \times \vec{B} = -\vec{B} \times \vec{A}$. $\vec{A} \times \vec{B} = 0$ if $\vec{A} \parallel \vec{B}$. Equilibrium of a Particle (2D & 3D) Free-Body Diagram (FBD): Crucial first step. Show all external forces acting on the particle. Equations of Equilibrium: 2D: $\sum F_x = 0$, $\sum F_y = 0$ 3D: $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$ Spring Force: $F_s = ks$, where $k$ is spring stiffness, $s$ is deformation. Force System Resultants Moment of a Force (Scalar): $M_O = F d$, where $d$ is perpendicular distance from $O$ to line of action of $F$. Moment of a Force (Vector): $\vec{M}_O = \vec{r} \times \vec{F}$, where $\vec{r}$ is position vector from $O$ to any point on line of action of $\vec{F}$. Varignon's Theorem: The moment of a force about a point is equal to the sum of the moments of its components about the point. Moment of a Couple: $\vec{M} = \vec{r} \times \vec{F}$. The magnitude is $M = Fd$. It's a free vector. Resultant of a Force System: $\vec{R} = \sum \vec{F}$ $\vec{M}_{R_O} = \sum \vec{M}_O + \sum \vec{M}_{\text{couples}}$ A force-couple system can be reduced to a single resultant force if $\vec{R} \cdot \vec{M}_{R_O} = 0$. Equilibrium of a Rigid Body FBD: Show all external forces and couple moments. Identify knowns and unknowns. Equations of Equilibrium: $\sum \vec{F} = \vec{0} \implies \sum F_x = 0, \sum F_y = 0, \sum F_z = 0$ $\sum \vec{M}_O = \vec{0} \implies \sum M_x = 0, \sum M_y = 0, \sum M_z = 0$ (moment about any point $O$) Common Supports and Reactions: Support Type 2D Reactions 3D Reactions Cable/Rope 1 force (tension) 1 force (tension) Roller 1 force $\perp$ surface 1 force $\perp$ surface Pin/Hinge 2 forces ($F_x, F_y$) 3 forces ($F_x, F_y, F_z$) Fixed Support 2 forces, 1 moment 3 forces, 3 moments Smooth Surface 1 force $\perp$ surface 1 force $\perp$ surface Trusses, Frames, and Machines Trusses: Members are two-force members (only axial force). Assumed pinned connections. 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 section. Zero-Force Members: If only two non-collinear members connect at a joint and no external load/reaction, both are zero-force. If three members connect at a joint, two are collinear, and no external load/reaction, the third is zero-force. Frames & Machines: Contain multi-force members. Analyze by disassembling the structure into its component parts. Apply rigid body equilibrium to each part. Remember action-reaction pairs between connected parts. Internal Forces Axial Force (N): Perpendicular to cross-section. $\sum F_x = 0$ (for internal force) Shear Force (V): Parallel to cross-section. $\sum F_y = 0$ (for internal force) Bending Moment (M): Causes rotation about an axis in the cross-section plane. $\sum M = 0$ (for internal moment) Shear and Moment Diagrams: Graph $V(x)$ and $M(x)$ along a beam. $dV/dx = w(x)$ (load intensity) $dM/dx = V(x)$ Jump in V: concentrated force. Jump in M: concentrated moment. Friction Static Friction: $F_s \le \mu_s N$, where $\mu_s$ is coefficient of static friction. Max friction occurs at impending motion. Kinetic Friction: $F_k = \mu_k N$, where $\mu_k$ is coefficient of kinetic friction. $\mu_k Angle of Static Friction: $\tan\phi_s = \mu_s$. Wedges: Use FBDs for wedge and object. Friction on Flat Belts: $T_2 = T_1 e^{\mu_s \beta}$, where $\beta$ is angle of contact in radians. Center of Gravity & Centroid Center of Gravity: Point where entire weight acts. $\bar{x} = \frac{\sum W_i x_i}{\sum W_i}$, $\bar{y} = \frac{\sum W_i y_i}$, $\bar{z} = \frac{\sum W_i z_i}{\sum W_i}$ Centroid (Area): Geometric center of an area. $\bar{x} = \frac{\sum A_i \bar{x}_i}{\sum A_i}$, $\bar{y} = \frac{\sum A_i \bar{y}_i}{\sum A_i}$ For composite areas, divide into simple shapes. Theorem of Pappus and Guldinus: Area of surface of revolution: $A = \theta \bar{r} L$ (for arc rotated through angle $\theta$) or $A = 2\pi \bar{r} L$ (full revolution). Volume of body of revolution: $V = \theta \bar{r} A$ (for area rotated through angle $\theta$) or $V = 2\pi \bar{r} A$ (full revolution). Moments of Inertia Area Moment of Inertia: $I_x = \int y^2 dA$, $I_y = \int x^2 dA$ Polar moment of inertia: $J_O = I_x + I_y = \int r^2 dA$ Parallel-Axis Theorem: $I_x = \bar{I}_{x'} + A d_y^2$ $I_y = \bar{I}_{y'} + A d_x^2$ $J_O = \bar{J}_C + A d^2$ $\bar{I}$ is moment of inertia about centroidal axis. Radius of Gyration: $k = \sqrt{I/A}$ Mass Moment of Inertia: $I = \int r^2 dm$ Parallel-Axis Theorem: $I = \bar{I} + md^2$ Kinematics of a Particle Rectilinear Kinematics Velocity: $v = ds/dt$ Acceleration: $a = dv/dt = d^2s/dt^2$ Relation: $a ds = v dv$ Constant Acceleration: $v = v_0 + a_c t$ $s = s_0 + v_0 t + \frac{1}{2} a_c t^2$ $v^2 = v_0^2 + 2 a_c (s - s_0)$ Curvilinear Kinematics Position: $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ Velocity: $\vec{v} = d\vec{r}/dt = v_x\hat{i} + v_y\hat{j} + v_z\hat{k}$ Acceleration: $\vec{a} = d\vec{v}/dt = a_x\hat{i} + a_y\hat{j} + a_z\hat{k}$ Normal and Tangential Components: $v = \dot{s}$ $a_t = \dot{v} = v dv/ds$ (rate of change of speed) $a_n = v^2/\rho$ (rate of change of direction, $\rho$ is radius of curvature) $\vec{a} = a_t \hat{u}_t + a_n \hat{u}_n$ Cylindrical Components: $\vec{r} = r \hat{u}_r + z \hat{k}$ $\vec{v} = \dot{r} \hat{u}_r + r\dot{\theta} \hat{u}_\theta + \dot{z} \hat{k}$ $\vec{a} = (\ddot{r} - r\dot{\theta}^2) \hat{u}_r + (r\ddot{\theta} + 2\dot{r}\dot{\theta}) \hat{u}_\theta + \ddot{z} \hat{k}$ Relative Motion (Translating Axes): $\vec{r}_{B/A} = \vec{r}_B - \vec{r}_A$, $\vec{v}_B = \vec{v}_A + \vec{v}_{B/A}$, $\vec{a}_B = \vec{a}_A + \vec{a}_{B/A}$ 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: $\sum F_t = ma_t$, $\sum F_n = ma_n$ Cylindrical: $\sum F_r = ma_r$, $\sum F_\theta = ma_\theta$, $\sum F_z = ma_z$ Work and Energy: Work of a Force: $U_{1-2} = \int \vec{F} \cdot d\vec{r}$ Kinetic Energy: $T = \frac{1}{2}mv^2$ Principle of Work and Energy: $T_1 + \sum U_{1-2} = T_2$ Conservative Forces: Gravity: $V_g = mgy$ 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: $\vec{p} = m\vec{v}$ Linear Impulse: $\text{Imp} = \int \vec{F} dt$ Principle of Linear Impulse and Momentum: $m\vec{v}_1 + \sum \int \vec{F} dt = m\vec{v}_2$ Conservation of Linear Momentum: $m_A \vec{v}_{A1} + m_B \vec{v}_{B1} = m_A \vec{v}_{A2} + m_B \vec{v}_{B2}$ (for isolated systems) Angular Momentum: $\vec{H}_O = \vec{r} \times m\vec{v}$ Principle of Angular Impulse and Momentum: $(\vec{H}_O)_1 + \sum \int \vec{M}_O dt = (\vec{H}_O)_2$ Coefficient of Restitution: $e = \frac{(v_B)_2 - (v_A)_2}{(v_A)_1 - (v_B)_1}$ (for impact) Planar Kinematics of a Rigid Body Translation: All points have same velocity and acceleration. $\vec{v}_B = \vec{v}_A$ $\vec{a}_B = \vec{a}_A$ Rotation about a Fixed Axis: Angular Velocity: $\omega = d\theta/dt$ Angular Acceleration: $\alpha = d\omega/dt$ Relation: $\alpha d\theta = \omega d\omega$ Constant Angular Acceleration: $\omega = \omega_0 + \alpha_c t$ $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha_c t^2$ $\omega^2 = \omega_0^2 + 2 \alpha_c (\theta - \theta_0)$ Point on Body: $v = r\omega$, $a_t = r\alpha$, $a_n = r\omega^2 = v^2/r$ Absolute General Plane Motion Analysis: Use position vectors and time derivatives. Relative Motion Analysis (Rotating Axes): $\vec{v}_B = \vec{v}_A + \vec{\omega} \times \vec{r}_{B/A}$ (for rigid body) $\vec{a}_B = \vec{a}_A + \vec{\alpha} \times \vec{r}_{B/A} - \omega^2 \vec{r}_{B/A}$ (for rigid body) For particles on rotating frame: $\vec{v}_B = \vec{v}_A + (\vec{v}_{B/A})_{xyz} + \vec{\Omega} \times \vec{r}_{B/A}$ $\vec{a}_B = \vec{a}_A + (\vec{a}_{B/A})_{xyz} + \dot{\vec{\Omega}} \times \vec{r}_{B/A} + \vec{\Omega} \times (\vec{\Omega} \times \vec{r}_{B/A}) + 2\vec{\Omega} \times (\vec{v}_{B/A})_{xyz}$ (Coriolis) Instantaneous Center of Zero Velocity (IC): Point on body with zero velocity. Locate IC at intersection of perpendiculars to velocity vectors. $v = r_{IC} \omega$. Planar Kinetics of a Rigid Body Equations of Motion: $\sum F_x = m(\bar{a}_x)$ $\sum F_y = m(\bar{a}_y)$ $\sum M_G = \bar{I} \alpha$ (moment about center of mass G) Alternatively, $\sum M_P = \sum (\vec{M}_k)_P$ (moment about any point P) $\sum (\vec{M}_k)_P = \bar{I} \alpha + m a_G d$ (where $d$ is perpendicular distance from P to $a_G$). Work and Energy: Kinetic Energy: $T = \frac{1}{2} m \bar{v}^2 + \frac{1}{2} \bar{I} \omega^2$ Principle of Work and Energy: $T_1 + \sum U_{1-2} = T_2$ Conservation of Energy: $T_1 + V_1 = T_2 + V_2$ Impulse and Momentum: Linear: $m(\vec{v}_G)_1 + \sum \int \vec{F} dt = m(\vec{v}_G)_2$ Angular: $(\vec{H}_G)_1 + \sum \int \vec{M}_G dt = (\vec{H}_G)_2$ (about G) $(\vec{H}_O)_1 + \sum \int \vec{M}_O dt = (\vec{H}_O)_2$ (about fixed point O)