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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. Vector Operations 2.1. Vector Addition/Subtraction Parallelogram Law: Place tails together, form parallelogram, diagonal is resultant. Triangle Rule: Head-to-tail, resultant from first tail to last head. Component Addition: $\vec{R} = \vec{A} + \vec{B} = (A_x+B_x)\hat{i} + (A_y+B_y)\hat{j} + (A_z+B_z)\hat{k}$. 2.2. Dot Product (Scalar Product) $\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta$ $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z$ Applications: Angle between two vectors: $\cos \theta = \frac{\vec{A} \cdot \vec{B}}{|\vec{A}| |\vec{B}|}$ Projection of $\vec{A}$ onto $\vec{B}$: $A_B = \vec{A} \cdot \hat{u}_B$, where $\hat{u}_B = \frac{\vec{B}}{|\vec{B}|}$. 2.3. Cross Product (Vector Product) $\vec{C} = \vec{A} \times \vec{B}$ Magnitude: $|\vec{C}| = |\vec{A}| |\vec{B}| \sin \theta$ Direction: Right-hand rule. Determinant form: $\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix}$ Applications: Calculating moment of a force. 3. Equilibrium of a Particle Free-Body Diagram (FBD): Essential for problem-solving. Isolate particle and show all external forces acting on it. Equations of Equilibrium: 2D: $\sum F_x = 0$, $\sum F_y = 0$ 3D: $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$ 4. Force System Resultants 4.1. Moment of a Force Scalar Formulation (2D): $M_O = Fd$, where $d$ is the perpendicular distance from $O$ to the line of action of $F$. Use sign convention (e.g., counter-clockwise positive). Vector Formulation (3D): $\vec{M}_O = \vec{r} \times \vec{F}$, where $\vec{r}$ is a position vector from $O$ to any point on the line of action of $F$. 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. $\vec{M}_O = \vec{r} \times (\vec{F}_1 + \vec{F}_2) = (\vec{r} \times \vec{F}_1) + (\vec{r} \times \vec{F}_2)$. 4.2. Couple Moment $\vec{M} = \vec{r} \times \vec{F}$ (where $\vec{r}$ connects the lines of action of the two forces of the couple). Magnitude: $M = Fd$, where $d$ is the perpendicular distance between the forces. A couple moment is a free vector (can be moved anywhere without changing its effect). 4.3. Resultant of a Force System Resultant Force: $\vec{F}_R = \sum \vec{F}$ Resultant Moment: $\vec{M}_{R_O} = \sum \vec{M}_O + \sum (\vec{r} \times \vec{F})$. If the system reduces to a single resultant force, it acts at a distance $d = M_{R_O} / F_R$ from $O$. 5. Equilibrium of a Rigid Body 5.1. Free-Body Diagram (FBD) Isolate the body. Show all external forces and couple moments. Support Reactions: Roller/Smooth Surface: Normal force $\perp$ surface. Pin/Hinge: Two force components (e.g., $F_x, F_y$). Fixed Support: Two force components and a moment (e.g., $F_x, F_y, M$). Cable/Rope: Tension force along the cable. Smooth Pin in Slot: Normal force $\perp$ slot, and a moment if pin is fixed to body. 5.2. Equations of Equilibrium 2D: $\sum F_x = 0$, $\sum F_y = 0$, $\sum M_O = 0$ 3D: $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$, $\sum M_x = 0$, $\sum M_y = 0$, $\sum M_z = 0$ 6. Trusses and Frames 6.1. Trusses Assumptions: Members are two-force members (only axial force), forces 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 meeting at an unloaded joint $\implies$ both are zero-force. Three members, two collinear, meeting at an unloaded joint $\implies$ the third is zero-force. 6.2. Frames and Machines Consist of multi-force members. Disassemble the structure into its component members. Draw FBD for each member and for the entire structure. Apply rigid body equilibrium equations to each part. Internal forces at connections are equal and opposite (Newton's 3rd Law). 7. Centroid and Moment of Inertia 7.1. Centroid of an Area $\bar{x} = \frac{\int_A x dA}{\int_A dA} = \frac{\sum \bar{x}_i A_i}{\sum A_i}$ $\bar{y} = \frac{\int_A y dA}{\int_A dA} = \frac{\sum \bar{y}_i A_i}{\sum A_i}$ For volumes, replace $A$ with $V$. For lines, replace $A$ with $L$. 7.2. Moment of Inertia for Area $I_x = \int_A y^2 dA$ $I_y = \int_A x^2 dA$ $J_O = I_x + I_y = \int_A r^2 dA$ (Polar Moment of Inertia) 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, $d$ is perpendicular distance between axes. 8. Kinematics of a Particle 8.1. Rectilinear Kinematics (Straight Line Motion) Velocity: $v = \frac{ds}{dt}$ Acceleration: $a = \frac{dv}{dt} = \frac{d^2s}{dt^2}$ Also: $a ds = v dv$ Constant Acceleration: $v = v_0 + a_c t$ $s = s_0 + v_0 t + \frac{1}{2} a_c t^2$ $v^2 = v_0^2 + 2 a_c (s - s_0)$ 8.2. Curvilinear Kinematics Rectangular Components: $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ $\vec{v} = \dot{x}\hat{i} + \dot{y}\hat{j} + \dot{z}\hat{k}$ $\vec{a} = \ddot{x}\hat{i} + \ddot{y}\hat{j} + \ddot{z}\hat{k}$ Normal and Tangential Components: $\vec{v} = v \hat{u}_t$ $\vec{a} = a_t \hat{u}_t + a_n \hat{u}_n$ $a_t = \dot{v}$ or $v \frac{dv}{ds}$ (tangential acceleration) $a_n = \frac{v^2}{\rho}$ (normal acceleration, $\rho$ is radius of curvature) Cylindrical Components: $\vec{r} = r\hat{u}_r + z\hat{k}$ $\vec{v} = \dot{r}\hat{u}_r + r\dot{\theta}\hat{u}_\theta + \dot{z}\hat{k}$ $\vec{a} = (\ddot{r} - r\dot{\theta}^2)\hat{u}_r + (r\ddot{\theta} + 2\dot{r}\dot{\theta})\hat{u}_\theta + \ddot{z}\hat{k}$ 9. Kinetics of a Particle 9.1. 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: $\sum F_t = ma_t$, $\sum F_n = ma_n = m\frac{v^2}{\rho}$ Cylindrical: $\sum F_r = m(\ddot{r} - r\dot{\theta}^2)$, $\sum F_\theta = m(r\ddot{\theta} + 2\dot{r}\dot{\theta})$, $\sum F_z = m\ddot{z}$ 9.2. Work and Energy Work of a Force: $U_{1-2} = \int_{s_1}^{s_2} \vec{F} \cdot d\vec{r}$ Work of a spring: $U_s = -\frac{1}{2}k(s_2^2 - s_1^2)$ Kinetic Energy: $T = \frac{1}{2}mv^2$ Principle of Work and Energy: $T_1 + \sum U_{1-2} = T_2$ Conservative Forces: Gravity ($V_g = W y$), Elastic Spring ($V_e = \frac{1}{2}ks^2$). Conservation of Energy: $T_1 + V_1 = T_2 + V_2$ (if only conservative forces do work). 9.3. Impulse and Momentum Linear Momentum: $\vec{L} = m\vec{v}$ Linear Impulse: $\text{Imp} = \int_{t_1}^{t_2} \vec{F} dt$ 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 of external impulses is zero). Coefficient of Restitution (e): $e = \frac{(v_B)_2 - (v_A)_2}{(v_A)_1 - (v_B)_1}$ (for impact along line of impact). Elastic impact: $e=1$ Plastic impact: $e=0$ 10. Kinematics of a Rigid Body 10.1. Rotation About a Fixed Axis Angular Velocity: $\omega = \frac{d\theta}{dt}$ Angular Acceleration: $\alpha = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}$ Also: $\alpha d\theta = \omega d\omega$ Constant Angular Acceleration: $\omega = \omega_0 + \alpha_c t$ $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha_c t^2$ $\omega^2 = \omega_0^2 + 2 \alpha_c (\theta - \theta_0)$ Velocity of a point P: $\vec{v}_P = \vec{\omega} \times \vec{r}_P$ or $v = \omega r$ (magnitude). Acceleration of a point P: $\vec{a}_P = \vec{\alpha} \times \vec{r}_P + \vec{\omega} \times (\vec{\omega} \times \vec{r}_P)$ $a_t = \alpha r$ (tangential) $a_n = \omega^2 r = v^2/r$ (normal) 10.2. General Plane Motion Absolute Motion Analysis: Use position coordinates and differentiate. Relative Velocity: $\vec{v}_B = \vec{v}_A + \vec{v}_{B/A} = \vec{v}_A + \vec{\omega} \times \vec{r}_{B/A}$ Instantaneous Center (IC) of Zero Velocity: For a body in plane motion, there is a point (IC) about which the body appears to rotate at that instant. $v = \omega r_{IC}$ Locate IC by drawing perpendiculars to velocities of two points. Relative Acceleration: $\vec{a}_B = \vec{a}_A + \vec{a}_{B/A} = \vec{a}_A + \vec{\alpha} \times \vec{r}_{B/A} - \omega^2 \vec{r}_{B/A}$ 11. Kinetics of a Rigid Body (Plane Motion) 11.1. Mass Moment of Inertia $I = \int r^2 dm$ Parallel-Axis Theorem: $I_O = \bar{I} + m d^2$, where $\bar{I}$ is about centroidal axis, $d$ is distance. 11.2. Equations of Motion $\sum \vec{F} = m\vec{a}_G$ $\sum M_G = \bar{I}\vec{\alpha}$ (moment about center of mass G) Alternatively, $\sum M_P = \sum (\mathcal{M}_k)_P$ (moment about any point P, where $(\mathcal{M}_k)_P$ are kinetic moments). For fixed-axis rotation about O: $\sum M_O = I_O \alpha$ 11.3. Work and Energy Kinetic Energy (Plane Motion): $T = \frac{1}{2}m v_G^2 + \frac{1}{2}\bar{I} \omega^2$ Kinetic Energy (Fixed Axis Rotation O): $T = \frac{1}{2}I_O \omega^2$ Principle of Work and Energy: $T_1 + \sum U_{1-2} = T_2$ 11.4. Impulse and Momentum Linear Momentum: $m(\vec{v}_G)_1 + \sum \int_{t_1}^{t_2} \vec{F} dt = m(\vec{v}_G)_2$ Angular Momentum (about G): $(\vec{H}_G)_1 + \sum \int_{t_1}^{t_2} \vec{M}_G dt = (\vec{H}_G)_2$, where $H_G = \bar{I}\omega$. Angular Momentum (about fixed point O): $(\vec{H}_O)_1 + \sum \int_{t_1}^{t_2} \vec{M}_O dt = (\vec{H}_O)_2$, where $H_O = I_O\omega$. Conservation of Energy/Momentum: Apply principles when appropriate sums are zero.