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, 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. Scalar & Vector Operations Vector Addition (Parallelogram Law): $\vec{R} = \vec{A} + \vec{B}$ Vector Subtraction: $\vec{R}' = \vec{A} - \vec{B} = \vec{A} + (-\vec{B})$ Components: $F_x = F \cos\theta$, $F_y = F \sin\theta$ Magnitude: $F = \sqrt{F_x^2 + F_y^2}$ Direction: $\theta = \arctan\left(\frac{F_y}{F_x}\right)$ 2.2. Cartesian Vectors Unit Vector: $\vec{u}_A = \frac{\vec{A}}{A}$ Vector Representation: $\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$ Magnitude: $A = \sqrt{A_x^2 + A_y^2 + A_z^2}$ Position Vector: $\vec{r} = (x_B - x_A)\hat{i} + (y_B - y_A)\hat{j} + (z_B - z_A)\hat{k}$ Force Vector from Position Vector: $\vec{F} = F \vec{u} = F \left(\frac{\vec{r}}{r}\right)$ 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 Vectors: $\theta = \arccos\left(\frac{\vec{A} \cdot \vec{B}}{AB}\right)$ Projection of $\vec{A}$ onto $\vec{B}$: $A_B = \vec{A} \cdot \vec{u}_B$ 3. Equilibrium of a Particle Free-Body Diagram (FBD): Essential for problem-solving. Isolate the particle and show all external forces. Equations of Equilibrium (2D): $\sum F_x = 0$ $\sum F_y = 0$ Equations of Equilibrium (3D): $\sum F_x = 0$ $\sum F_y = 0$ $\sum F_z = 0$ 4. Force System Resultants 4.1. Cross Product $\vec{C} = \vec{A} \times \vec{B} = (A_y B_z - A_z B_y)\hat{i} - (A_x B_z - A_z B_x)\hat{j} + (A_x B_y - A_y B_x)\hat{k}$ Magnitude: $C = AB \sin\theta$ 4.2. Moment of a Force Scalar (2D): $M_O = Fd$ (positive counter-clockwise) Vector (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. 4.3. Couple Moment $\vec{M} = \vec{r} \times \vec{F}$ or $M = Fd$ 4.4. Resultant of a Force and Couple System $\vec{F}_R = \sum \vec{F}$ $\vec{M}_R = \sum \vec{M}_O + \sum \vec{M}_{couple}$ 5. Equilibrium of a Rigid Body 5.1. Types of Supports and Reactions Support Type 2D Reactions 3D Reactions Roller 1 force $\perp$ surface 1 force $\perp$ surface Pin/Hinge 2 force components 3 force components, 0 moments Fixed Support 2 force components, 1 moment 3 force components, 3 moments Ball-and-Socket N/A 3 force components Journal Bearing N/A 2 force components, 2 moments 5.2. Equations of Equilibrium 2D: $\sum F_x = 0$ $\sum F_y = 0$ $\sum M_O = 0$ (moment about any point O) 3D: $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$ $\sum M_x = 0$, $\sum M_y = 0$, $\sum M_z = 0$ (moments about x, y, z axes) 6. Trusses, Frames, and Machines 6.1. Trusses (Pin-connected members) Assumptions: Forces 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 into sections and apply rigid body equilibrium ($\sum F_x = 0, \sum F_y = 0, \sum M_O = 0$) to a section. Zero-Force Members: If only two non-collinear members connect at a joint and no external load or reaction is applied, both are zero-force members. If three members meet at a joint, two of which are collinear, and no external load or reaction is applied, the third member is a zero-force member. 6.2. Frames and Machines (Multi-force members) Separate the structure into its component parts. Draw FBD for each part and for the entire structure. Apply rigid body equilibrium equations to each part. Internal forces at connections are equal and opposite. 7. Internal Forces Axial Force (N): Perpendicular to the cross-section. Shear Force (V): Parallel to the cross-section. Bending Moment (M): Causes rotation about an axis in the cross-section. Sign Convention: Axial: Tension (+), Compression (-) Shear: Up on right face (+), Down on right face (-) Moment: Causes compression at top (+), Causes tension at top (-) Shear and Moment Diagrams: Graphical representation of V and M along the beam. Relationships: $\frac{dV}{dx} = w(x)$ (Distributed load) $\frac{dM}{dx} = V(x)$ (Shear force) 8. Friction Static Friction: $F_s \le \mu_s N$ (Maximum static friction $F_{s,max} = \mu_s N$) Kinetic Friction: $F_k = \mu_k N$ $\mu_s > \mu_k$ Angle of Static Friction: $\tan\phi_s = \mu_s$ Angle of Repose: Angle at which object on incline starts to slip. Wedges and Screws: Apply equilibrium to the wedge/screw and the block(s) it supports, considering friction forces. 9. Center of Gravity and Centroid Center of Gravity (CG): Point where the entire weight of a body can be considered to act. Centroid: Geometric center of an area or volume. For homogeneous materials, CG and centroid coincide. Centroid of an Area: $\bar{x} = \frac{\int \tilde{x} dA}{\int dA} = \frac{\sum \tilde{x} A}{\sum A}$ $\bar{y} = \frac{\int \tilde{y} dA}{\int dA} = \frac{\sum \tilde{y} A}{\sum A}$ Theorems of Pappus and Guldinus: Area of Surface of Revolution: $A = \theta \bar{r} L$ (where $\theta$ is angle in radians, $\bar{r}$ is centroidal distance, $L$ is length of generating curve) Volume of Body of Revolution: $V = \theta \bar{r} A$ (where $A$ is area of generating curve) 10. Moments of Inertia Moment of Inertia for Area: $I_x = \int y^2 dA$ $I_y = \int x^2 dA$ $J_O = I_x + I_y = \int r^2 dA$ (Polar Moment of Inertia) Parallel-Axis Theorem: $I_x = \bar{I}_{x'} + A d_y^2$ $I_y = \bar{I}_{y'} + A d_x^2$ $J_O = \bar{J}_C + A d^2$ Radius of Gyration: $k = \sqrt{\frac{I}{A}}$ Product of Inertia: $I_{xy} = \int xy dA$ Principal Moments of Inertia: Max/min moments of inertia, occur at principal axes. 11. Kinematics of a Particle 11.1. Rectilinear Motion Velocity: $v = \frac{ds}{dt}$ Acceleration: $a = \frac{dv}{dt} = \frac{d^2s}{dt^2}$ Chain Rule: $a = v \frac{dv}{ds}$ 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: $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ Velocity: $\vec{v} = \frac{d\vec{r}}{dt} = \dot{x}\hat{i} + \dot{y}\hat{j} + \dot{z}\hat{k}$ Acceleration: $\vec{a} = \frac{d\vec{v}}{dt} = \ddot{x}\hat{i} + \ddot{y}\hat{j} + \ddot{z}\hat{k}$ Normal and Tangential Components: $v = \dot{s}$ $a_t = \dot{v} = \frac{dv}{dt}$ $a_n = \frac{v^2}{\rho}$ (where $\rho$ is radius of curvature) $a = \sqrt{a_t^2 + a_n^2}$ Cylindrical Components: $\vec{v} = \dot{r}\hat{u}_r + r\dot{\theta}\hat{u}_\theta + \dot{z}\hat{u}_z$ $\vec{a} = (\ddot{r} - r\dot{\theta}^2)\hat{u}_r + (r\ddot{\theta} + 2\dot{r}\dot{\theta})\hat{u}_\theta + \ddot{z}\hat{u}_z$ 12. Kinetics of a Particle (Force & Acceleration) 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: $\sum F_t = m a_t$ $\sum F_n = m a_n = m \frac{v^2}{\rho}$ Cylindrical Coordinates: $\sum F_r = m a_r = m (\ddot{r} - r\dot{\theta}^2)$ $\sum F_\theta = m a_\theta = m (r\ddot{\theta} + 2\dot{r}\dot{\theta})$ $\sum F_z = m a_z = m \ddot{z}$ 13. Kinetics of a Particle (Work & Energy) Work of a Force: $U_{1-2} = \int_1^2 \vec{F} \cdot d\vec{r}$ Work of Weight: $U_{1-2} = -W \Delta y$ 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 $T = \frac{1}{2} m v^2$ (Kinetic Energy) Conservation of Energy: $T_1 + V_1 = T_2 + V_2$ (for conservative forces) Gravitational Potential Energy: $V_g = Wy$ Elastic Potential Energy: $V_e = \frac{1}{2} k s^2$ 14. Kinetics of a Particle (Impulse & Momentum) Linear Impulse: $\vec{I} = \int \vec{F} dt$ 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: $m_A \vec{v}_{A1} + m_B \vec{v}_{B1} = m_A \vec{v}_{A2} + m_B \vec{v}_{B2}$ (when no external impulses) Impact: Coefficient of Restitution: $e = \frac{(v_B)_2 - (v_A)_2}{(v_A)_1 - (v_B)_1}$ Elastic Impact: $e=1$ Plastic Impact: $e=0$ Angular Momentum: $(H_O)_1 + \sum \int_{t_1}^{t_2} M_O dt = (H_O)_2$ where $H_O = (\vec{r} \times m\vec{v})$ 15. Planar Kinematics of a Rigid Body Translation: All points have same velocity and acceleration. Rotation About a Fixed Axis: $\omega = \frac{d\theta}{dt}$ $\alpha = \frac{d\omega}{dt}$ $v = r\omega$ $a_t = r\alpha$ $a_n = r\omega^2 = \frac{v^2}{r}$ 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): For finding velocities. 16. Planar Kinetics of a Rigid Body Equations of Motion: $\sum F_x = m(\bar{a}_G)_x$ $\sum F_y = m(\bar{a}_G)_y$ $\sum M_G = \bar{I}_G \alpha$ (Moment about center of mass G) $\sum M_P = \sum (\mathcal{M}_k)_P$ (Moment about any point P) Mass Moment of Inertia: $I = \int r^2 dm$ Radius of Gyration: $k = \sqrt{\frac{I}{m}}$ Rolling Motion: For no slip, $v_G = r\omega$, $a_G = r\alpha$.