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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 $F$ experiences an acceleration $a$ that has the same direction as the force and a magnitude that is directly proportional to the force. $F = ma$ 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 $G = 6.673 \times 10^{-11} \text{ m}^3/(\text{kg} \cdot \text{s}^2)$ 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: Magnitude only (e.g., mass, length, time) Vector: Magnitude and direction (e.g., force, velocity, acceleration) 2.2. Vector Operations Vector Addition (Parallelogram Law): Resultant $\vec{R} = \vec{A} + \vec{B}$ Vector Subtraction: $\vec{R}' = \vec{A} - \vec{B} = \vec{A} + (-\vec{B})$ Components: $F_x = F \cos \theta$, $F_y = F \sin \theta$ Unit Vector: $\vec{u}_A = \frac{\vec{A}}{|\vec{A}|}$ Dot Product: $\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta = A_x B_x + A_y B_y + A_z B_z$ Cross Product: $\vec{A} \times \vec{B} = (A_y B_z - A_z B_y)\vec{i} - (A_x B_z - A_z B_x)\vec{j} + (A_x B_y - A_y B_x)\vec{k}$ 2.3. Position & Force Vectors Position Vector: $\vec{r} = (x_B - x_A)\vec{i} + (y_B - y_A)\vec{j} + (z_B - z_A)\vec{k}$ Force Vector (from position): $\vec{F} = F \vec{u} = F \left( \frac{\vec{r}}{|\vec{r}|} \right)$ 3. Equilibrium of a Particle 2D Equilibrium: $\sum F_x = 0$, $\sum F_y = 0$ 3D Equilibrium: $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$ Free-Body Diagram (FBD): Crucial for problem-solving. Isolate the particle and show all external forces acting on it. 4. Force System Resultants 4.1. Moment of a Force Scalar (2D): $M_O = Fd$, where $d$ is perpendicular distance from $O$ to line of action of $F$. Vector (2D/3D): $\vec{M}_O = \vec{r} \times \vec{F}$ Principle of Transmissibility: A force can be applied anywhere along its line of action without changing its external effect on a rigid body. 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. 4.2. Couple Moment Definition: Two parallel forces with the same magnitude, opposite direction, and separated by a perpendicular distance $d$. Magnitude: $M = Fd$ Vector: $\vec{M} = \vec{r} \times \vec{F}$ (where $\vec{r}$ connects points on the lines of action of the two forces) 4.3. Simplification of a Force and Couple System Any system of forces and couple moments acting on a rigid body can be reduced to an equivalent single resultant force $\vec{F}_R$ acting at a point $O$ and a resultant couple moment $\vec{M}_{R_O}$. $\vec{F}_R = \sum \vec{F}$ $\vec{M}_{R_O} = \sum \vec{M}_O + \sum \vec{M}_{couple}$ 5. Equilibrium of a Rigid Body Conditions for Equilibrium: $\sum \vec{F} = \vec{0}$ (Sum of forces is zero) $\sum \vec{M}_O = \vec{0}$ (Sum of moments about any point $O$ is zero) 2D Equilibrium Equations: $\sum F_x = 0$ $\sum F_y = 0$ $\sum M_O = 0$ 3D Equilibrium Equations: $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$ $\sum M_x = 0$, $\sum M_y = 0$, $\sum M_z = 0$ Supports and Reactions: Roller: One unknown force, perpendicular to surface. Pin: Two unknown forces (x and y components). Fixed Support: Three unknown forces (x, y, moment). Ball and Socket: Three unknown forces (x, y, z components). Fixed (3D): Three forces (x, y, z) and three moments (Mx, My, Mz). 6. Structural Analysis 6.1. Trusses Assumptions: Members are connected by pins, loads applied at joints, members are two-force members. Method of Joints: Apply particle equilibrium ($\sum F_x=0, \sum F_y=0$) at each joint. Method of Sections: Cut the truss and apply rigid body equilibrium ($\sum F_x=0, \sum F_y=0, \sum M=0$) to a section. Zero-Force Members: Identify members with no force to simplify analysis. 6.2. Frames and Machines Frames: Stationary structures designed to support loads. Machines: Structures containing moving parts designed to transmit and alter the effect of forces. Analysis: Disassemble the structure into its component parts and apply equilibrium equations to each part. Forces at internal connections are action-reaction pairs. 7. Internal Forces Normal Force (N): Perpendicular to the cross-section. Shear Force (V): Parallel to the cross-section. Bending Moment (M): Moment about the centroidal axis of the cross-section. Sign Convention: Normal: Tension (+), Compression (-) Shear: Up on LHS (+), Down on LHS (-) Moment: Causes compression in top fibres (+) Relations: $\frac{dV}{dx} = -w(x)$ (Distributed load) $\frac{dM}{dx} = V(x)$ 8. Friction Static Friction: $F_s \le \mu_s N$, where $\mu_s$ is coefficient of static friction. Max static friction occurs at impending motion. Kinetic Friction: $F_k = \mu_k N$, where $\mu_k$ is coefficient of kinetic friction. $\mu_k Angle of Friction: $\tan \phi_s = \mu_s$ 9. Center of Gravity and Centroid Center of Gravity ($G$): Point where the entire weight of a body acts. $\bar{x} = \frac{\sum W_i x_i}{\sum W_i}$, $\bar{y} = \frac{\sum W_i y_i}{\sum W_i}$, $\bar{z} = \frac{\sum W_i z_i}{\sum W_i}$ Centroid ($C$): Geometric center of an area or volume. $\bar{x} = \frac{\int x dA}{\int dA} = \frac{\sum \bar{x}_i A_i}{\sum A_i}$ $\bar{y} = \frac{\int y dA}{\int dA} = \frac{\sum \bar{y}_i A_i}{\sum A_i}$ 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 Area Moment of Inertia: $I_x = \int y^2 dA$, $I_y = \int x^2 dA$ $J_O = I_x + I_y = \int r^2 dA$ (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: $k = \sqrt{\frac{I}{A}}$ Mass Moment of Inertia: $I = \int r^2 dm$ 11. Kinematics of a Particle 11.1. Rectilinear Motion Velocity: $v = \frac{ds}{dt}$ Acceleration: $a = \frac{dv}{dt} = v \frac{dv}{ds}$ 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: $\vec{r}(t) = x(t)\vec{i} + y(t)\vec{j} + z(t)\vec{k}$ Velocity: $\vec{v} = \frac{d\vec{r}}{dt} = \dot{x}\vec{i} + \dot{y}\vec{j} + \dot{z}\vec{k}$ Acceleration: $\vec{a} = \frac{d\vec{v}}{dt} = \ddot{x}\vec{i} + \ddot{y}\vec{j} + \ddot{z}\vec{k}$ Normal and Tangential Components: $v = \dot{s}$ $a_t = \dot{v} = \frac{dv}{dt}$ $a_n = \frac{v^2}{\rho}$ (where $\rho$ is radius of curvature) $a = \sqrt{a_t^2 + a_n^2}$ Cylindrical Components: $v_r = \dot{r}$, $v_\theta = r\dot{\theta}$, $v_z = \dot{z}$ $a_r = \ddot{r} - r\dot{\theta}^2$ $a_\theta = r\ddot{\theta} + 2\dot{r}\dot{\theta}$ $a_z = \ddot{z}$ 12. Kinetics of a Particle 12.1. Equation of Motion Newton's Second Law: $\sum \vec{F} = m\vec{a}$ Rectangular Components: $\sum F_x = ma_x$, $\sum F_y = ma_y$, $\sum F_z = ma_z$ Normal and Tangential Components: $\sum F_t = ma_t$, $\sum F_n = ma_n$ Cylindrical Components: $\sum F_r = ma_r$, $\sum F_\theta = ma_\theta$, $\sum F_z = ma_z$ 12.2. Work and Energy Work of a Force: $U_{1-2} = \int_{s_1}^{s_2} F \cos \theta ds$ Work of Weight: $U_{1-2} = -W \Delta y = -mg(y_2 - y_1)$ Work of Spring: $U_{1-2} = \frac{1}{2} k (s_1^2 - s_2^2)$ Principle of Work and Energy: $T_1 + U_{1-2} = T_2$, where $T = \frac{1}{2} mv^2$ (Kinetic Energy) Conservation of Energy: $T_1 + V_1 = T_2 + V_2$, where $V = V_g + V_e$ (Potential Energy) $V_g = W y$ (Gravitational Potential Energy) $V_e = \frac{1}{2} k s^2$ (Elastic Potential Energy) 12.3. Impulse and Momentum Linear Impulse: $\text{Imp} = \int_{t_1}^{t_2} F dt$ Linear Momentum: $\vec{L} = m\vec{v}$ 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: If $\sum \vec{F} = 0$, then $\sum (m\vec{v})_1 = \sum (m\vec{v})_2$ Impact: Coefficient of Restitution: $e = \frac{(v_B')_n - (v_A')_n}{(v_A)_n - (v_B)_n}$ $e=1$ (elastic), $e=0$ (plastic) Angular Momentum: $(H_O)_1 + \sum \int_{t_1}^{t_2} M_O dt = (H_O)_2$, where $H_O = (\vec{r} \times m\vec{v})$ 13. Planar Kinematics of a Rigid Body Translation: All points have same velocity and acceleration. Rotation about a Fixed Axis: $\omega = \frac{d\theta}{dt}$ $\alpha = \frac{d\omega}{dt} = \omega \frac{d\omega}{d\theta}$ $v = \omega r$ $a_t = \alpha r$, $a_n = \omega^2 r = v^2/r$ General Plane Motion: Combination of translation and rotation. $\vec{v}_B = \vec{v}_A + \vec{\omega} \times \vec{r}_{B/A}$ $\vec{a}_B = \vec{a}_A + \vec{\alpha} \times \vec{r}_{B/A} - \omega^2 \vec{r}_{B/A}$ Instantaneous Center of Zero Velocity (IC): Point on body with zero velocity. $\vec{v} = \vec{\omega} \times \vec{r}_{P/IC}$ 14. Planar Kinetics of a Rigid Body Equations of Motion: $\sum F_x = m(\bar{a}_x)$ $\sum F_y = m(\bar{a}_y)$ $\sum M_G = \bar{I}\alpha$ (about center of mass $G$) $\sum M_P = \sum (\mathcal{M}_k)_P$ (about any point $P$) Work and Energy: $T = \frac{1}{2} m \bar{v}^2 + \frac{1}{2} \bar{I} \omega^2$ $T_1 + V_1 + U_{1-2}' = T_2 + V_2$ (where $U_{1-2}'$ includes non-conservative work) Impulse and Momentum: $m(\vec{v}_G)_1 + \sum \int_{t_1}^{t_2} \vec{F} dt = m(\vec{v}_G)_2$ $(\vec{H}_G)_1 + \sum \int_{t_1}^{t_2} \vec{M}_G dt = (\vec{H}_G)_2$, where $\vec{H}_G = \bar{I}\vec{\omega}$