Hibbeler

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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, tends to 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 $\vec{F}$ and a magnitude that is directly proportional to $\vec{F}$. If $m$ is the mass of the particle, then $\vec{F} = m\vec{a}$. 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 Vectors Cartesian Vector Form: $\vec{F} = F_x \hat{i} + F_y \hat{j}$ Magnitude: $F = \sqrt{F_x^2 + F_y^2}$ Direction: $\theta = \arctan\left(\frac{F_y}{F_x}\right)$ Components: $F_x = F \cos\theta$, $F_y = F \sin\theta$ Resultant Force: $\vec{F}_R = \sum \vec{F} = (\sum F_x)\hat{i} + (\sum F_y)\hat{j}$ 2.2. 3D Vectors Cartesian Vector Form: $\vec{F} = F_x \hat{i} + F_y \hat{j} + F_z \hat{k}$ Magnitude: $F = \sqrt{F_x^2 + F_y^2 + F_z^2}$ Direction Cosines: $\cos\alpha = \frac{F_x}{F}$, $\cos\beta = \frac{F_y}{F}$, $\cos\gamma = \frac{F_z}{F}$ $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$ Unit Vector: $\hat{u}_F = \frac{\vec{F}}{F} = \cos\alpha \hat{i} + \cos\beta \hat{j} + \cos\gamma \hat{k}$ Position Vector: $\vec{r} = (x_B - x_A)\hat{i} + (y_B - y_A)\hat{j} + (z_B - z_A)\hat{k}$ Force directed along a line: $\vec{F} = F \hat{u} = F \left(\frac{\vec{r}}{r}\right)$ 2.3. Dot Product $\vec{A} \cdot \vec{B} = AB \cos\theta$ $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z$ Angle between two vectors: $\theta = \arccos\left(\frac{\vec{A} \cdot \vec{B}}{AB}\right)$ Projection of $\vec{A}$ onto $\vec{B}$: $A_B = \vec{A} \cdot \hat{u}_B$ Vector component of $\vec{A}$ parallel to $\vec{B}$: $\vec{A}_\parallel = (\vec{A} \cdot \hat{u}_B) \hat{u}_B$ Vector component of $\vec{A}$ perpendicular to $\vec{B}$: $\vec{A}_\perp = \vec{A} - \vec{A}_\parallel$ 3. Equilibrium of a Particle Free-Body Diagram (FBD): Essential for problem-solving. Shows all external forces acting on the particle. Equilibrium Equations: 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 (Rigid Bodies) 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): $\vec{M}_O = \vec{r} \times \vec{F}$ $\vec{r}$ is position vector from point $O$ to any point on the line of action of $\vec{F}$. Determinant form: $\vec{M}_O = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ r_x & r_y & r_z \\ F_x & F_y & F_z \end{vmatrix}$ 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. $\vec{M}_O = \vec{r} \times (\vec{F}_1 + \vec{F}_2) = \vec{r} \times \vec{F}_1 + \vec{r} \times \vec{F}_2$. 4.2. Moment About an Axis $M_a = \hat{u}_a \cdot (\vec{r} \times \vec{F})$ Determinant form: $M_a = \begin{vmatrix} u_{ax} & u_{ay} & u_{az} \\ r_x & r_y & r_z \\ F_x & F_y & F_z \end{vmatrix}$ 4.3. Couple Moment $\vec{M} = \vec{r} \times \vec{F}$ (where $\vec{F}$ and $-\vec{F}$ form the couple) Magnitude: $M = Fd$, where $d$ is the perpendicular distance between the forces. 4.4. Equivalent Systems Resultant Force: $\vec{F}_R = \sum \vec{F}$ Resultant Moment: $(\vec{M}_R)_O = \sum \vec{M}_O + \sum (\vec{r} \times \vec{F})$ 5. Equilibrium of a Rigid Body Free-Body Diagram (FBD): Crucial for identifying all forces and moments. Equations of Equilibrium: $\sum \vec{F} = 0 \implies \sum F_x = 0, \sum F_y = 0, \sum F_z = 0$ $\sum \vec{M}_O = 0 \implies \sum M_x = 0, \sum M_y = 0, \sum M_z = 0$ (Moment can be taken about any point $O$) Supports and Reactions: Support Type 2D Reactions 3D Reactions Roller 1 force $\perp$ surface 1 force $\perp$ surface Smooth Pin/Hinge 2 force components 3 force components, 3 moments (if fixed) Fixed Support 2 force components, 1 moment 3 force components, 3 moments Cable/Rope 1 force along cable 1 force along cable 6. Trusses, Frames, and Machines 6.1. Trusses Composed of two-force members (only axial force). Method of Joints: Apply $\sum F_x = 0, \sum F_y = 0$ at each joint. Method of Sections: Cut through members, apply $\sum F_x = 0, \sum F_y = 0, \sum M = 0$ to a section. Zero-Force Members: If only two non-collinear members meet at a joint with no external load, both are zero-force members. If three members meet at a joint, two are collinear, and there's no external load, the third member (not collinear) is a zero-force member. 6.2. Frames and Machines Contain multi-force members. Disassemble the structure into its component parts. Apply equilibrium equations to each part. Newton's Third Law is critical for internal forces. 7. Internal Forces Normal Force (N): Perpendicular to the section. Shear Force (V): Tangent to the section. Bending Moment (M): Causes bending. Sign Convention (for positive values): N: Tension (pulling away from section) V: Down on right face, up on left face M: Causes compression in top fibers (concave up) Relations between Load, Shear, and Moment: $\frac{dV}{dx} = w(x)$ (distributed load) $\frac{dM}{dx} = V(x)$ $\Delta V = \int w(x) dx$ $\Delta M = \int V(x) dx$ 8. Friction Static Friction: $F_s \le \mu_s N$, where $\mu_s$ is coefficient of static friction. Max static friction: $(F_s)_{\max} = \mu_s N$. Kinetic Friction: $F_k = \mu_k N$, where $\mu_k$ is coefficient of kinetic friction. ($\mu_k Angle of Friction: $\tan\phi_s = \mu_s$, $\tan\phi_k = \mu_k$. 9. Center of Gravity and Centroid Centroid of Area: $\bar{x} = \frac{\int x dA}{\int dA} = \frac{\sum \tilde{x} A}{\sum A}$ $\bar{y} = \frac{\int y dA}{\int dA} = \frac{\sum \tilde{y} A}{\sum A}$ Centroid of Volume: $\bar{x} = \frac{\int x dV}{\int dV} = \frac{\sum \tilde{x} V}{\sum V}$ $\bar{y} = \frac{\int y dV}{\int dV} = \frac{\sum \tilde{y} V}{\sum V}$ $\bar{z} = \frac{\int z dV}{\int dV} = \frac{\sum \tilde{z} V}{\sum V}$ Common Centroids: Rectangle: $\bar{x} = b/2, \bar{y} = h/2$ Triangle: $\bar{x} = b/3, \bar{y} = h/3$ (from right angle) Semicircle: $\bar{y} = 4r/(3\pi)$ (from diameter) 10. Moment 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: $I_x = \bar{I}_{x'} + Ad_y^2$ $I_y = \bar{I}_{y'} + Ad_x^2$ $J_O = \bar{J}_C + Ad^2$ Product of Inertia: $I_{xy} = \int xy dA$ 11. Virtual Work $\delta U = \sum F \cos\theta \delta s + \sum M \delta\theta = 0$ For a system in equilibrium, the virtual work done by all external forces and couples acting on the system is zero for any virtual displacement consistent with constraints. Used to find equilibrium positions or force reactions without disassembling structures. 12. Kinematics of a Particle 12.1. Rectilinear Motion $v = \frac{ds}{dt}$ $a = \frac{dv}{dt} = \frac{d^2s}{dt^2}$ $a ds = v dv$ Constant Acceleration: $v = v_0 + a_c t$ $s = s_0 + v_0 t + \frac{1}{2} a_c t^2$ $v^2 = v_0^2 + 2 a_c (s - s_0)$ 12.2. Curvilinear Motion 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 $v \frac{dv}{ds}$ $a_n = \frac{v^2}{\rho}$, where $\rho$ is radius of curvature. Cylindrical Components: $\vec{r} = r\hat{u}_r + z\hat{k}$ $\vec{v} = \dot{r}\hat{u}_r + r\dot{\theta}\hat{u}_\theta + \dot{z}\hat{k}$ $\vec{a} = (\ddot{r} - r\dot{\theta}^2)\hat{u}_r + (r\ddot{\theta} + 2\dot{r}\dot{\theta})\hat{u}_\theta + \ddot{z}\hat{k}$ 13. Kinetics of a Particle 13.1. 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: $\sum F_t = ma_t$, $\sum F_n = ma_n$ Cylindrical: $\sum F_r = ma_r$, $\sum F_\theta = ma_\theta$, $\sum F_z = ma_z$ 13.2. Work and Energy Work of a Force: $U_{1-2} = \int_1^2 \vec{F} \cdot d\vec{r}$ Work of Weight: $U_g = -W \Delta y$ Work of Spring: $U_s = -(\frac{1}{2} k s_2^2 - \frac{1}{2} k s_1^2)$ Principle of Work and Energy: $T_1 + U_{1-2} = T_2$ Kinetic Energy: $T = \frac{1}{2} mv^2$ Conservation of Energy: $T_1 + V_1 = T_2 + V_2$ (for conservative forces) Potential Energy: $V = V_g + V_e$ Gravitational: $V_g = Wy$ Elastic: $V_e = \frac{1}{2} ks^2$ 13.3. Impulse and Momentum Linear Momentum: $\vec{L} = m\vec{v}$ Linear Impulse: $\text{Imp} = \int \vec{F} dt$ 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: $\sum m\vec{v}_1 = \sum m\vec{v}_2$ (when sum of external impulses is zero) Impact: Coefficient of Restitution: $e = \frac{(v_B')_n - (v_A')_n}{(v_A)_n - (v_B)_n}$ $e=1$ (elastic), $e=0$ (plastic) 14. Planar Kinematics of a Rigid Body 14.1. Rotation About a Fixed Axis $\alpha = \frac{d\omega}{dt}$, $\omega = \frac{d\theta}{dt}$ $\alpha d\theta = \omega d\omega$ $v = \omega r$ $a_t = \alpha r$, $a_n = \omega^2 r = v^2/r$ 14.2. Absolute Motion Analysis Relate position coordinates of points on the body using geometry. Differentiate with respect to time to get velocity and acceleration. 14.3. Relative Motion Analysis Velocity: $\vec{v}_B = \vec{v}_A + \vec{v}_{B/A}$ For rigid body: $\vec{v}_{B/A} = \vec{\omega} \times \vec{r}_{B/A}$ Acceleration: $\vec{a}_B = \vec{a}_A + \vec{a}_{B/A}$ For rigid body: $\vec{a}_{B/A} = \vec{\alpha} \times \vec{r}_{B/A} - \omega^2 \vec{r}_{B/A}$ Instantaneous Center (IC) of Zero Velocity: For pure rotation, $\vec{v} = \vec{\omega} \times \vec{r}_{IC}$. IC lies on a line perpendicular to $\vec{v}$. 15. Planar Kinetics of a Rigid Body 15.1. Equations of Motion $\sum \vec{F} = m\vec{a}_G$ $\sum M_G = I_G \alpha$ (Moment about center of mass G) $\sum M_P = I_P \alpha$ (Moment about fixed point P) For rolling without slipping: $a_G = \alpha r$ 15.2. Work and Energy Kinetic Energy (Planar): $T = \frac{1}{2} mv_G^2 + \frac{1}{2} I_G \omega^2$ Principle of Work and Energy: $T_1 + U_{1-2} = T_2$ Conservation of Energy: $T_1 + V_1 = T_2 + V_2$ 15.3. Impulse and Momentum Linear Momentum: $m(\vec{v}_G)_1 + \sum \int \vec{F} dt = m(\vec{v}_G)_2$ Angular Momentum: $H_G = I_G \omega$ Principle of Angular Impulse and Momentum: $(H_G)_1 + \sum \int M_G dt = (H_G)_2$ For a fixed axis P: $(H_P)_1 + \sum \int M_P dt = (H_P)_2$ Conservation of Angular Momentum: $\sum (H)_1 = \sum (H)_2$ (if sum of moments about point is zero)