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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. 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: $W = mg$, where $g = 9.81 \text{ m/s}^2$ or $32.2 \text{ ft/s}^2$ 2. Force Vectors 2.1. 2D Force Systems Rectangular Components: $F_x = F \cos \theta$, $F_y = F \sin \theta$ Magnitude: $F = \sqrt{F_x^2 + F_y^2}$ Direction: $\theta = \arctan \left( \frac{F_y}{F_x} \right)$ Unit Vector: $\vec{u}_F = \frac{\vec{F}}{F} = \frac{F_x}{F}\hat{i} + \frac{F_y}{F}\hat{j}$ Resultant Force: $\vec{F}_R = \sum \vec{F} = (\sum F_x)\hat{i} + (\sum F_y)\hat{j}$ 2.2. 3D Force Systems Cartesian Components: $\vec{F} = F_x\hat{i} + F_y\hat{j} + F_z\hat{k}$ Magnitude: $F = \sqrt{F_x^2 + F_y^2 + F_z^2}$ Direction Cosines: $\cos \alpha = \frac{F_x}{F}$, $\cos \beta = \frac{F_y}{F}$, $\cos \gamma = \frac{F_z}{F}$ Identity: $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$ Unit Vector: $\vec{u}_F = \cos \alpha \hat{i} + \cos \beta \hat{j} + \cos \gamma \hat{k}$ Position Vector: $\vec{r} = (x_B - x_A)\hat{i} + (y_B - y_A)\hat{j} + (z_B - z_A)\hat{k}$ Force from Position Vector: $\vec{F} = F \vec{u} = F \frac{\vec{r}}{r}$ Dot Product: $\vec{A} \cdot \vec{B} = AB \cos \theta = A_x B_x + A_y B_y + A_z B_z$ Projection of $\vec{A}$ onto $\vec{B}$: $A_B = \vec{A} \cdot \vec{u}_B$ Cross Product: $\vec{C} = \vec{A} \times \vec{B} = (A_y B_z - A_z B_y)\hat{i} - (A_x B_z - A_z B_x)\hat{j} + (A_x B_y - A_y B_x)\hat{k}$ 3. Equilibrium of a Particle Condition: $\sum \vec{F} = \vec{0}$ 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): Essential for identifying all external forces. 4. Force System Resultants Moment of a Force (Scalar - 2D): $M_O = Fd$, where $d$ is perpendicular distance. ($+$ counter-clockwise) Moment of a Force (Vector - 2D/3D): $\vec{M}_O = \vec{r} \times \vec{F}$ 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. Couple Moment: $\vec{M} = \vec{r} \times \vec{F}$ (or $M = Fd$ for 2D, $d$ is perpendicular distance between forces) Resultant Force & Moment: $\vec{F}_R = \sum \vec{F}$ $\vec{M}_{R_O} = \sum \vec{M}_O + \sum \vec{M}_{couple}$ 5. Equilibrium of a Rigid Body Conditions for 2D Equilibrium: $\sum F_x = 0$ $\sum F_y = 0$ $\sum M_O = 0$ (any point $O$) Conditions for 3D Equilibrium: $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$ $\sum M_x = 0$, $\sum M_y = 0$, $\sum M_z = 0$ Supports & Reactions: Pin Support: Two force components ($R_x, R_y$) Roller Support: One force component (normal to surface) Fixed Support: Two force components ($R_x, R_y$) and one moment ($M$) (2D) Three force components ($R_x, R_y, R_z$) and three moments ($M_x, M_y, M_z$) (3D) Cable/Rope: Tension force (pulling) along the cable Smooth Surface: Normal force perpendicular to the surface 6. Trusses, Frames, and Machines 6.1. Trusses 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 the truss into two sections, apply rigid body equilibrium ($\sum F_x = 0, \sum F_y = 0, \sum M_O = 0$) to one section. Zero-Force Members: If only two non-collinear members meet at an unloaded joint, both are zero-force members. If three members meet at an unloaded joint, two of which are collinear, the third non-collinear member is a zero-force member. 6.2. Frames and Machines Consist of multi-force members. Disassemble the structure into its component parts. Apply rigid body equilibrium equations to each part. Remember action-reaction pairs between connected members. 7. Internal Forces Axial Force (N): Normal to the cross-section. Shear Force (V): Tangential to the cross-section. Bending Moment (M): Rotational effect about an axis in the cross-section. Sign Convention (for Beams): Shear: Positive if it tends to rotate the element clockwise. Moment: Positive if it tends to bend the element concave up (compression at top, tension at bottom). Relations: $\frac{dV}{dx} = -w(x)$ (distributed load) $\frac{dM}{dx} = V(x)$ 8. Friction Static Friction: $F_s \le \mu_s N$ (up to impending motion) Kinetic Friction: $F_k = \mu_k N$ (for motion) Typically, $\mu_k < \mu_s$. Angle of Static Friction: $\tan \phi_s = \mu_s$ Wedges: Often involve friction on multiple surfaces. Draw FBDs for each block. Pulleys/Flat Belts: $T_2 = T_1 e^{\mu_s \beta}$, where $\beta$ is the angle of contact in radians. 9. Center of Gravity and Centroid Center of Gravity (CG): Point where the entire weight of the 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: Geometric center of an area or volume. $\bar{x} = \frac{\int x dA}{\int dA}$, $\bar{y} = \frac{\int y dA}{\int dA}$ (for area $A$) For composite bodies: $\bar{x} = \frac{\sum \bar{x}_i A_i}{\sum A_i}$, $\bar{y} = \frac{\sum \bar{y}_i A_i}{\sum A_i}$ Theorems of Pappus and Guldinus: Area of Revolution: $A = \theta \bar{r} L$ (length $L$ revolved by angle $\theta$ about an axis, $\bar{r}$ is centroidal distance to axis) Volume of Revolution: $V = \theta \bar{r} A$ (area $A$ revolved by angle $\theta$ about an axis, $\bar{r}$ is centroidal distance to axis) 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: $I_x = \bar{I}_x + Ad_y^2$ $I_y = \bar{I}_y + Ad_x^2$ $J_O = \bar{J}_C + Ad^2$ ($\bar{I}$ is moment of inertia about centroidal axis, $A$ is area, $d$ is distance between parallel axes) Radius of Gyration: $k = \sqrt{I/A}$ Mass Moment of Inertia: $I = \int r^2 dm$ Parallel-Axis Theorem: $I = \bar{I} + md^2$ Common Shapes (Centroidal): Shape $I_x$ (Rectangle $b \times h$) $I_x$ (Circle radius $r$) Rectangle $\frac{1}{12}bh^3$ - Circle $\frac{1}{4}\pi r^4$ $\frac{1}{4}\pi r^4$ 11. Kinematics of a Particle 11.1. Rectilinear Kinematics Position: $s(t)$ Velocity: $v = \frac{ds}{dt}$ Acceleration: $a = \frac{dv}{dt} = \frac{d^2s}{dt^2}$ Chain Rule: $a = v \frac{dv}{ds}$ Constant Acceleration Formulas: $v = v_0 + a_c t$ $s = s_0 + v_0 t + \frac{1}{2} a_c t^2$ $v^2 = v_0^2 + 2 a_c (s - s_0)$ 11.2. Curvilinear Kinematics Position Vector: $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ Velocity Vector: $\vec{v} = \frac{d\vec{r}}{dt} = \dot{x}\hat{i} + \dot{y}\hat{j} + \dot{z}\hat{k}$ Acceleration Vector: $\vec{a} = \frac{d\vec{v}}{dt} = \ddot{x}\hat{i} + \ddot{y}\hat{j} + \ddot{z}\hat{k}$ Normal and Tangential Components: $v = \dot{s}$ $a_t = \dot{v} = \frac{dv}{dt}$ (tangential acceleration, changes speed) $a_n = \frac{v^2}{\rho}$ (normal acceleration, changes direction, $\rho$ is radius of curvature) $\vec{a} = a_t \vec{u}_t + a_n \vec{u}_n$ $a = \sqrt{a_t^2 + a_n^2}$ Polar Coordinates: $\vec{r} = r \vec{u}_r$ $\vec{v} = \dot{r} \vec{u}_r + r \dot{\theta} \vec{u}_\theta$ $\vec{a} = (\ddot{r} - r \dot{\theta}^2) \vec{u}_r + (r \ddot{\theta} + 2 \dot{r} \dot{\theta}) \vec{u}_\theta$ 12. Kinetics of a Particle 12.1. Newton's Second Law $\sum \vec{F} = m\vec{a}$ Rectangular Coordinates: $\sum F_x = m a_x$ $\sum F_y = m a_y$ $\sum F_z = m a_z$ Normal and Tangential Coordinates: $\sum F_t = m a_t$ $\sum F_n = m a_n = m \frac{v^2}{\rho}$ Polar Coordinates: $\sum F_r = m a_r = m (\ddot{r} - r \dot{\theta}^2)$ $\sum F_\theta = m a_\theta = m (r \ddot{\theta} + 2 \dot{r} \dot{\theta})$ 12.2. Work and Energy Work of a Force: $U = \int \vec{F} \cdot d\vec{r}$ Work-Energy Principle: $T_1 + \sum U_{1-2} = T_2$ Kinetic Energy: $T = \frac{1}{2} m v^2$ Potential Energy: Gravitational: $V_g = W y$ Elastic: $V_e = \frac{1}{2} k s^2$ Conservation of Energy (Conservative Forces Only): $T_1 + V_1 = T_2 + V_2$ Power: $P = \frac{dU}{dt} = \vec{F} \cdot \vec{v}$ Efficiency: $\epsilon = \frac{\text{Power Output}}{\text{Power Input}}$ 12.3. Impulse and Momentum Linear Impulse: $\text{Imp} = \int \vec{F} dt$ 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 (No External Impulse): $m\vec{v}_1 = m\vec{v}_2$ Impact: Coefficient of Restitution: $e = \frac{(v_B)_2 - (v_A)_2}{(v_A)_1 - (v_B)_1}$ For central impact, momentum is conserved in the direction of impact. Angular Momentum: $\vec{H}_O = \vec{r} \times m\vec{v}$ Angular Impulse: $\int \vec{M}_O dt$ Principle of Angular Impulse and Momentum: $(\vec{H}_O)_1 + \sum \int_{t_1}^{t_2} \vec{M}_O dt = (\vec{H}_O)_2$ 13. Planar Kinematics of a Rigid Body Rotation about a Fixed Axis: $\omega = \frac{d\theta}{dt}$ $\alpha = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}$ $\alpha = \omega \frac{d\omega}{d\theta}$ Constant Angular Acceleration: $\omega = \omega_0 + \alpha_c t$, $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha_c t^2$, $\omega^2 = \omega_0^2 + 2\alpha_c(\theta - \theta_0)$ Velocity of point $P$: $v = \omega r$ Acceleration of point $P$: $a_t = \alpha r$, $a_n = \omega^2 r$, $a = \sqrt{a_t^2 + a_n^2}$ Absolute Motion Analysis: Relate position coordinates using geometry, then differentiate to find velocity and acceleration. Relative Motion Analysis: $\vec{v}_B = \vec{v}_A + \vec{v}_{B/A}$ (where $\vec{v}_{B/A} = \vec{\omega} \times \vec{r}_{B/A}$) $\vec{a}_B = \vec{a}_A + \vec{a}_{B/A}$ (where $\vec{a}_{B/A} = \vec{\alpha} \times \vec{r}_{B/A} - \omega^2 \vec{r}_{B/A}$) Instantaneous Center of Zero Velocity (IC): For pure rotation, $\vec{v}_P = \vec{\omega} \times \vec{r}_{P/IC}$. Locate IC by perpendiculars to velocities, or by extending lines collinear with velocities. $v_P = \omega r_{P/IC}$ 14. Planar Kinetics of a Rigid Body Equations of Motion: $\sum F_x = m(\bar{a}_x)$ $\sum F_y = m(\bar{a}_y)$ $\sum M_G = \bar{I} \alpha$ (moment about center of mass $G$) OR $\sum M_P = \sum (\mathcal{M}_k)_P$ (moment about any point $P$, where $\sum (\mathcal{M}_k)_P$ includes moments of $m\bar{a}_x, m\bar{a}_y, \bar{I}\alpha$) Work and Energy: $T = \frac{1}{2} m \bar{v}^2 + \frac{1}{2} \bar{I} \omega^2$ (for general planar motion) $T = \frac{1}{2} I_O \omega^2$ (for rotation about fixed axis $O$) $T_1 + V_1 = T_2 + V_2$ (Conservation of Energy) Impulse and Momentum: $m(\vec{v}_G)_1 + \sum \int \vec{F} dt = m(\vec{v}_G)_2$ $(\bar{H}_G)_1 + \sum \int \vec{M}_G dt = (\bar{H}_G)_2$ (where $\bar{H}_G = \bar{I}\omega$) OR $(H_O)_1 + \sum \int \vec{M}_O dt = (H_O)_2$ (for fixed axis $O$, where $H_O = I_O \omega$)