Engineering Mechanics (Hibbele

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### Fundamental Principles - **Newton's First Law:** A particle remains at rest or continues to move with constant velocity if the resultant force on it is zero. - **Newton's Second Law:** $\vec{F} = m\vec{a}$ (If the resultant force acting on a particle is not zero, the particle accelerates in the direction of the resultant force, and the magnitude of the acceleration is proportional to the resultant force.) - **Newton's Third Law:** For every action, there is an equal and opposite reaction. - **Newton's Law of Gravitational Attraction:** $F = G \frac{m_1 m_2}{r^2}$ where $G = 6.673 \times 10^{-11} \text{ m}^3/(\text{kg} \cdot \text{s}^2)$ - **Units:** - SI: length (m), time (s), mass (kg), force (N) - US Customary: length (ft), time (s), force (lb), mass (slug) ### Force Vectors - **Scalar:** Quantity with magnitude (e.g., mass, length, time). - **Vector:** Quantity with magnitude and direction (e.g., force, velocity, acceleration). - **Vector Addition (Parallelogram Law):** Resultant vector is the diagonal of the parallelogram formed by two vectors. - **Vector Addition (Triangle Rule):** Head-to-tail connection. - **Cartesian Vectors:** - $\vec{F} = F_x \hat{i} + F_y \hat{j} + F_z \hat{k}$ - Magnitude: $|\vec{F}| = \sqrt{F_x^2 + F_y^2 + F_z^2}$ - Direction Cosines: $\cos \alpha = \frac{F_x}{F}$, $\cos \beta = \frac{F_y}{F}$, $\cos \gamma = \frac{F_z}{F}$ - Unit Vector: $\vec{u}_F = \frac{\vec{F}}{|\vec{F}|} = \cos \alpha \hat{i} + \cos \beta \hat{j} + \cos \gamma \hat{k}$ - **Dot Product:** $\vec{A} \cdot \vec{B} = |\vec{A}||\vec{B}|\cos\theta = A_x B_x + A_y B_y + A_z B_z$ - Used to find the angle between two vectors or the projection of one vector onto another. - Projection of $\vec{A}$ onto $\vec{B}$: $\text{Proj}_B \vec{A} = (\vec{A} \cdot \vec{u}_B) \vec{u}_B$ ### Equilibrium of a Particle - **Free-Body Diagram (FBD):** Essential for solving equilibrium problems. Isolates the particle and shows all external forces acting on it. - **Equations of Equilibrium:** - 2D: $\sum F_x = 0$, $\sum F_y = 0$ - 3D: $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$ - **Procedure for Analysis:** 1. Draw FBD. 2. Establish coordinate system. 3. Apply equations of equilibrium. 4. Solve for unknowns. ### Force System Resultants - **Moment of a Force (Scalar):** $M_O = Fd$ (where $d$ is perpendicular distance from $O$ to line of action of $F$). - **Moment of a Force (Vector):** $\vec{M}_O = \vec{r} \times \vec{F}$ - $\vec{r}$ is position vector from $O$ to any point on the line of action of $\vec{F}$. - Cross product: $\vec{A} \times \vec{B} = (A_y B_z - A_z B_y)\hat{i} - (A_x B_z - A_z B_x)\hat{j} + (A_x B_y - A_y B_x)\hat{k}$ - **Varignon's Theorem:** The moment of a force about a point is equal to the sum of the moments of its components about the same point. - **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 $M = Fd$. - **Resultant of a Force System:** - **Resultant Force:** $\vec{F}_R = \sum \vec{F}$ - **Resultant Moment:** $\vec{M}_{R_O} = \sum \vec{M}_O + \sum \vec{M}_{\text{couples}}$ - For 2D, a force system can be reduced to a single resultant force or a resultant force and a resultant couple moment. - For 3D, a force system can be reduced to a resultant force and a resultant couple moment (wrench). ### Equilibrium of a Rigid Body - **Free-Body Diagram (FBD):** Show all external forces and couple moments. Identify known and unknown forces. - **Equations of Equilibrium:** - **2D:** $\sum F_x = 0$, $\sum F_y = 0$, $\sum M_O = 0$ (at any point $O$) - **3D:** $\sum \vec{F} = 0 \implies \sum F_x = 0, \sum F_y = 0, \sum F_z = 0$ - $\sum \vec{M}_O = 0 \implies \sum M_x = 0, \sum M_y = 0, \sum M_z = 0$ (at any point $O$) - **Supports and Reactions:** - **Roller:** One force, perpendicular to surface. - **Pin/Hinge:** Two force components (x, y). - **Fixed Support:** Two force components (x, y) and one couple moment. - **Ball-and-socket:** Three force components (x, y, z). - **Fixed/Cantilever (3D):** Three force components and three couple moments. ### Structural Analysis - **Trusses:** Members are slender and connected at their ends by frictionless pins. Forces are axial (tension or compression). - **Method of Joints:** Apply particle equilibrium ($\sum F_x=0, \sum F_y=0$) at each joint. Start with joints having few unknowns. - **Method of Sections:** Cut the truss to expose desired members. Apply rigid body equilibrium ($\sum F_x=0, \sum F_y=0, \sum M=0$) to either section. - **Zero-Force Members:** Identify members that carry no load under specific loading conditions (e.g., two non-collinear members connected at an unloaded joint). - **Frames and Machines:** Contain at least one multi-force member. - Disassemble the structure into its members. - Draw FBD for each member and/or connection. - Apply rigid body equilibrium to each part. - Remember action-reaction pairs between connected members. ### Center of Gravity and Centroid - **Center of Gravity (CG):** Point where the entire weight of a body appears to act. - $\bar{x} = \frac{\sum \tilde{x}W}{\sum W}$, $\bar{y} = \frac{\sum \tilde{y}W}{\sum W}$, $\bar{z} = \frac{\sum \tilde{z}W}{\sum W}$ - **Centroid:** Geometric center of an area or volume. - **Area:** $\bar{x} = \frac{\sum \tilde{x}A}{\sum A}$, $\bar{y} = \frac{\sum \tilde{y}A}{\sum A}$ - **Line:** $\bar{x} = \frac{\sum \tilde{x}L}{\sum L}$, $\bar{y} = \frac{\sum \tilde{y}L}{\sum L}$ - **Composite Bodies:** Divide into simpler shapes, find CG/centroid of each, then use summation formulas. - **Pappus-Guldinus Theorems:** - **Surface Area:** $A = \theta \bar{r} L$ (for revolution about an axis) - **Volume:** $V = \theta \bar{r} A$ (for revolution about an axis) - $\theta$ is the angle of revolution in radians. $\bar{r}$ is the perpendicular distance from the centroid of the generating plane curve/area to the axis of revolution. ### Moments of Inertia - **Area Moment of Inertia (Second Moment of Area):** Measures a body's resistance to bending. - $I_x = \int y^2 dA$, $I_y = \int x^2 dA$ - **Polar Moment of Inertia:** $J_O = I_x + I_y = \int r^2 dA$ - **Parallel-Axis Theorem:** $I = \bar{I} + Ad^2$ - $\bar{I}$ is moment of inertia about centroidal axis. - $A$ is area. $d$ is perpendicular distance between parallel axes. - **Mass Moment of Inertia:** Measures a body's resistance to angular acceleration. - $I = \int r^2 dm$ - Parallel-Axis Theorem (Mass): $I = \bar{I} + md^2$ ### Kinematics of a Particle - **Rectilinear Motion (1D):** - Velocity: $v = \frac{ds}{dt}$ - Acceleration: $a = \frac{dv}{dt} = v \frac{dv}{ds}$ - **Constant Acceleration:** - $v = v_0 + a_c t$ - $s = s_0 + v_0 t + \frac{1}{2} a_c t^2$ - $v^2 = v_0^2 + 2 a_c (s - s_0)$ - **Curvilinear Motion (2D/3D):** - Position: $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ - Velocity: $\vec{v} = \frac{d\vec{r}}{dt} = v_x\hat{i} + v_y\hat{j} + v_z\hat{k}$ - Acceleration: $\vec{a} = \frac{d\vec{v}}{dt} = a_x\hat{i} + a_y\hat{j} + a_z\hat{k}$ - **Projectile Motion:** Constant $a_y = -g$, $a_x = 0$. - **Normal and Tangential Components:** $a_t = \dot{v}$, $a_n = \frac{v^2}{\rho}$ (where $\rho$ is radius of curvature) - **Cylindrical Components:** $\vec{v} = \dot{r}\vec{u}_r + r\dot{\theta}\vec{u}_{\theta} + \dot{z}\vec{u}_z$ - $\vec{a} = (\ddot{r} - r\dot{\theta}^2)\vec{u}_r + (r\ddot{\theta} + 2\dot{r}\dot{\theta})\vec{u}_{\theta} + \ddot{z}\vec{u}_z$ ### 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: $\sum F_r = m a_r$, $\sum F_{\theta} = m a_{\theta}$, $\sum F_z = m a_z$ - **Work and Energy:** - **Work of a Force:** $U_{1-2} = \int_{s_1}^{s_2} \vec{F} \cdot d\vec{r}$ - Constant force: $U_{1-2} = (F \cos\theta) \Delta s$ - Weight: $U_g = -W(y_2 - y_1)$ - Spring: $U_s = -\frac{1}{2}k(s_2^2 - s_1^2)$ - **Principle of Work and Energy:** $T_1 + U_{1-2} = T_2$ - Kinetic Energy: $T = \frac{1}{2}mv^2$ - **Conservation of Energy:** $T_1 + V_1 = T_2 + V_2$ (when only conservative forces do work) - Potential Energy: $V_g = Wy$ (gravitational), $V_e = \frac{1}{2}ks^2$ (elastic) - **Impulse and Momentum:** - **Linear Impulse:** $\vec{I} = \int_{t_1}^{t_2} \vec{F} dt$ - **Linear Momentum:** $\vec{L} = m\vec{v}$ - **Principle of Linear Impulse and Momentum:** $m\vec{v}_1 + \sum \int \vec{F} dt = m\vec{v}_2$ - **Conservation of Linear Momentum:** $\sum m\vec{v}_1 = \sum m\vec{v}_2$ (when sum of external impulses is zero) - **Impact:** - **Coefficient of Restitution:** $e = \frac{(v_B)_2 - (v_A)_2}{(v_A)_1 - (v_B)_1}$ (for central impact) - $e=1$ (elastic), $e=0$ (plastic) ### Kinematics of a Rigid Body - **Types of Motion:** - **Translation:** All points have the same velocity and acceleration. - Rectilinear: Straight line path. - Curvilinear: Curved path. - **Rotation About a Fixed Axis:** - Angular Velocity: $\omega = \frac{d\theta}{dt}$ - Angular Acceleration: $\alpha = \frac{d\omega}{dt} = \omega \frac{d\omega}{d\theta}$ - Relationship to linear: $v = \omega r$, $a_t = \alpha r$, $a_n = \omega^2 r$ - **General Plane Motion:** Combination of translation and rotation. - **Relative Velocity:** $\vec{v}_B = \vec{v}_A + \vec{\omega} \times \vec{r}_{B/A}$ - **Instantaneous Center (IC) of Zero Velocity:** Point where velocity is instantaneously zero. Can be used to find velocities of other points. - **Relative Acceleration:** $\vec{a}_B = \vec{a}_A + \vec{\alpha} \times \vec{r}_{B/A} - \omega^2 \vec{r}_{B/A}$ - **Fixed-Axis Rotation with Constant Angular Acceleration:** - $\omega = \omega_0 + \alpha_c t$ - $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha_c t^2$ - $\omega^2 = \omega_0^2 + 2 \alpha_c (\theta - \theta_0)$ ### Kinetics of a Rigid Body - **Equations of Motion:** - **Translation:** $\sum \vec{F} = m\vec{a}_G$ (where $G$ is center of mass) - **Rotation about a Fixed Axis:** - $\sum F_n = m(a_G)_n = m\omega^2 r_G$ - $\sum F_t = m(a_G)_t = m\alpha r_G$ - $\sum M_O = I_O \alpha$ (where $O$ is the fixed axis of rotation) - **General Plane Motion:** - $\sum F_x = m(a_G)_x$ - $\sum F_y = m(a_G)_y$ - $\sum M_G = I_G \alpha$ (moment about center of mass $G$) - **Work and Energy (Rigid Body):** - **Kinetic Energy:** $T = \frac{1}{2} m v_G^2 + \frac{1}{2} I_G \omega^2$ - For rotation about fixed axis $O$: $T = \frac{1}{2} I_O \omega^2$ - **Principle of Work and Energy:** $T_1 + U_{1-2} = T_2$ - **Impulse and Momentum (Rigid Body):** - **Linear Impulse and Momentum:** $m(\vec{v}_G)_1 + \sum \int \vec{F} dt = m(\vec{v}_G)_2$ - **Angular Impulse and Momentum:** $(H_G)_1 + \sum \int M_G dt = (H_G)_2$ - Angular Momentum: $H_G = I_G \omega$ (about center of mass) - Angular Momentum about fixed point $O$: $H_O = I_O \omega$ - **Conservation of Momentum:** If $\sum \int \vec{F} dt = 0$, then $m(\vec{v}_G)_1 = m(\vec{v}_G)_2$. - If $\sum \int M_G dt = 0$, then $(H_G)_1 = (H_G)_2$.