Engineering Mechanics (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 $\vec{F}$ and a magnitude that is directly proportional to $\vec{F}$. If $\vec{F}$ is applied to a particle of mass $m$, 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 $G = 66.73(10^{-12}) \text{ m}^3/(\text{kg} \cdot \text{s}^2)$. Weight: $W = mg$, where $g = 9.81 \text{ m/s}^2$ or $32.2 \text{ ft/s}^2$. 2. Force Vectors 2.1 Scalars and Vectors Scalar: Magnitude only (e.g., mass, volume, length). Vector: Magnitude and direction (e.g., force, velocity, acceleration). 2.2 Vector Operations Vector Addition (Parallelogram Law): Resultant $\vec{R} = \vec{A} + \vec{B}$. Vector Subtraction: $\vec{R}' = \vec{A} - \vec{B} = \vec{A} + (-\vec{B})$. Dot Product: $\vec{A} \cdot \vec{B} = |\vec{A}||\vec{B}| \cos \theta = A_x B_x + A_y B_y + A_z B_z$. Projection of $\vec{A}$ onto $\vec{B}$: $A_B = \vec{A} \cdot \frac{\vec{B}}{|\vec{B}|}$. Cross Product: $\vec{C} = \vec{A} \times \vec{B} = (A_y B_z - A_z B_y)\hat{i} - (A_x B_z - A_z B_x)\hat{j} + (A_x B_y - A_y B_x)\hat{k}$. Magnitude: $|\vec{C}| = |\vec{A}||\vec{B}| \sin \theta$. Direction: Right-hand rule. 2.3 Cartesian Vectors Unit Vector: $\hat{u}_A = \frac{\vec{A}}{|\vec{A}|} = \frac{A_x}{A}\hat{i} + \frac{A_y}{A}\hat{j} + \frac{A_z}{A}\hat{k}$. Position Vector: $\vec{r} = (x_2-x_1)\hat{i} + (y_2-y_1)\hat{j} + (z_2-z_1)\hat{k}$. Force Vector along a Line: $\vec{F} = F \hat{u} = F \left( \frac{\vec{r}}{|\vec{r}|} \right)$. 3. 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: $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$. 4. Force System Resultants 4.1 Moment of a Force Scalar Formulation (2D): $M_O = Fd$ (force times perpendicular distance). Counter-clockwise moments are positive. Vector Formulation: $\vec{M}_O = \vec{r} \times \vec{F}$. $\vec{r}$ is position vector from point O to any point on line of action of $\vec{F}$. 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. $\vec{M}_O = \vec{r} \times (\vec{F}_1 + \vec{F}_2) = (\vec{r} \times \vec{F}_1) + (\vec{r} \times \vec{F}_2)$. Moment about an Axis: $M_a = \hat{u}_a \cdot (\vec{r} \times \vec{F})$. 4.2 Couple Moment A couple consists of two parallel forces that are equal in magnitude, opposite in direction, and separated by a perpendicular distance $d$. Couple Moment: $M = Fd$ (scalar), or $\vec{M} = \vec{r} \times \vec{F}$ (vector). The moment is independent of the point about which it is computed. 4.3 Resultants of a Force System Resultant Force: $\vec{F}_R = \sum \vec{F}$. Resultant Couple Moment: $\vec{M}_R = \sum \vec{M} + \sum (\vec{r} \times \vec{F})$. A force system can be reduced to a single resultant force $\vec{F}_R$ acting at a specific point, and a resultant couple moment $\vec{M}_R$. If $\vec{F}_R = 0$, the system reduces to a pure couple moment. If $\vec{M}_R = 0$, the system reduces to a single resultant force. 5. Equilibrium of a Rigid Body Free-Body Diagram (FBD): Crucial for rigid body analysis. Shows all external forces and moments. Equations of Equilibrium: 2D: $\sum F_x = 0$, $\sum F_y = 0$, $\sum M_O = 0$. 3D: $\sum \vec{F} = 0 \implies \sum F_x = 0, \sum F_y = 0, \sum F_z = 0$. $\sum \vec{M}_O = 0 \implies \sum M_x = 0, \sum M_y = 0, \sum M_z = 0$. Supports and Reactions: Type of Support 2D Reactions 3D Reactions Cable/Rope 1 force (tension) 1 force (tension) Roller 1 normal force 1 normal force Smooth Pin/Hinge 2 force components 3 force components Fixed Support 2 force components, 1 moment 3 force components, 3 moments 6. Trusses, Frames, and Machines 6.1 Trusses Composed of slender members joined at their endpoints. Assumed to be pin-connected, and loads applied only at joints. Members are two-force members (tension or compression). Method of Joints: Apply particle equilibrium ($\sum F_x = 0, \sum F_y = 0$) at each joint. Method of Sections: Cut through members to be analyzed, isolate a section, and apply rigid body equilibrium ($\sum F_x = 0, \sum F_y = 0, \sum M = 0$). 6.2 Frames and Machines Structures with at least one multi-force member. Disassemble the structure into its component members. Draw FBD for each member. Apply rigid body equilibrium equations to each member. 7. Internal Forces Internal forces (normal force $N$, shear force $V$, bending moment $M$) are found by making an imaginary cut through the member and drawing an FBD of one of the sections. Sign Convention (for positive internal forces): Normal Force: Tension is positive. Shear Force: Upward on the right face of the cut, or downward on the left face. Bending Moment: Causes compression in the top fibers and tension in the bottom fibers (smiles). Relations between Load, Shear, and Moment: $\frac{dV}{dx} = w(x)$ (slope of shear diagram = distributed load intensity) $\frac{dM}{dx} = V(x)$ (slope of moment diagram = shear force) $\Delta V = \int w(x) dx$ (change in shear = area under load diagram) $\Delta M = \int V(x) dx$ (change in moment = area under shear diagram) 8. Friction Static Friction: $F_s \le \mu_s N$, where $\mu_s$ is the coefficient of static friction. Maximum static friction occurs when motion is impending ($F_s = (\mu_s)_{max} N$). Kinetic Friction: $F_k = \mu_k N$, where $\mu_k$ is the coefficient of kinetic friction ($\mu_k Angle of Static Friction: $\phi_s = \tan^{-1}(\mu_s)$. Angle of Repose: The maximum angle of inclination of an inclined plane at which an object will not slide down due to static friction. 9. Centroids and Moments of Inertia 9.1 Centroid of Area Centroid Coordinates: $ \bar{x} = \frac{\int x dA}{\int dA} = \frac{\sum \bar{x}A}{\sum A} $ $ \bar{y} = \frac{\int y dA}{\int dA} = \frac{\sum \bar{y}A}{\sum A} $ $ \bar{z} = \frac{\int z dA}{\int dA} = \frac{\sum \bar{z}A}{\sum A} $ Pappus-Guldinus Theorems: Area of Surface of Revolution: $A = \theta \bar{r} L$, where $L$ is arc length. Volume of Body of Revolution: $V = \theta \bar{r} A$, where $A$ is area. 9.2 Moments of Inertia Area Moment of Inertia: $ I_x = \int y^2 dA $ $ I_y = \int x^2 dA $ Polar Moment of Inertia: $J_O = \int r^2 dA = I_x + I_y$. 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 $ where $\bar{I}$ is moment of inertia about centroidal axis, $A$ is area, $d$ is distance between parallel axes. Radius of Gyration: $k = \sqrt{I/A}$. 10. Virtual Work Principle of Virtual Work: For a body to be in equilibrium, the virtual work done by all external forces and couples acting on the body must be zero for any virtual displacement consistent with the system's constraints. $ \delta U = \sum F_i \delta s_i + \sum M_j \delta \theta_j = 0 $ Useful for determining equilibrium position or unknown forces without disassembling a structure. 11. Kinematics of a Particle 11.1 Rectilinear Kinematics Velocity: $v = \frac{ds}{dt}$. Acceleration: $a = \frac{dv}{dt} = \frac{d^2s}{dt^2}$. Differential Relation: $a ds = v dv$. Constant Acceleration: $v = v_0 + a_c t$ $s = s_0 + v_0 t + \frac{1}{2} a_c t^2$ $v^2 = v_0^2 + 2 a_c (s - s_0)$ 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: $v = \dot{s}$ $a_t = \dot{v} = \frac{dv}{dt}$ (tangent to path) $a_n = \frac{v^2}{\rho}$ (normal to path, towards center of curvature $\rho$) $a = \sqrt{a_t^2 + a_n^2}$ Cylindrical Components: $\vec{r} = r\hat{u}_r + z\hat{k}$ $\vec{v} = \dot{r}\hat{u}_r + r\dot{\theta}\hat{u}_\theta + \dot{z}\hat{k}$ $\vec{a} = (\ddot{r} - r\dot{\theta}^2)\hat{u}_r + (r\ddot{\theta} + 2\dot{r}\dot{\theta})\hat{u}_\theta + \ddot{z}\hat{k}$ 12. Kinetics of a Particle 12.1 Newton's Second Law $\sum \vec{F} = m\vec{a}$. Rectangular Coordinates: $\sum F_x = ma_x$, $\sum F_y = ma_y$, $\sum F_z = ma_z$. Normal and Tangential Coordinates: $\sum F_t = ma_t$, $\sum F_n = ma_n = m\frac{v^2}{\rho}$, $\sum F_b = 0$. Cylindrical Coordinates: $\sum F_r = m a_r$, $\sum F_\theta = m a_\theta$, $\sum F_z = m a_z$. 12.2 Work and Energy Work of a Force: $U_{1-2} = \int_{s_1}^{s_2} \vec{F} \cdot d\vec{r}$. Constant force: $U_{1-2} = (F \cos \theta) \Delta s$. Spring force: $U_{1-2} = -\frac{1}{2} k (s_2^2 - s_1^2)$. Weight: $U_{1-2} = -W \Delta y$. Principle of Work and Energy: $T_1 + U_{1-2} = T_2$. (Kinetic energy $T = \frac{1}{2}mv^2$). Conservation of Energy (Conservative Forces Only): $T_1 + V_1 = T_2 + V_2$. Gravitational Potential Energy: $V_g = Wy$. Elastic Potential Energy: $V_e = \frac{1}{2}ks^2$. 12.3 Impulse and 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 (when $\sum \vec{F} dt = 0$): $m\vec{v}_1 = m\vec{v}_2$. Impact: Coefficient of Restitution: $e = \frac{(v_B)_2 - (v_A)_2}{(v_A)_1 - (v_B)_1}$. For perfectly elastic impact, $e=1$. For perfectly plastic impact, $e=0$. 12.4 Angular Momentum Angular Momentum about point O: $\vec{H}_O = \vec{r} \times m\vec{v}$. Principle of Angular Impulse and Momentum: $(\vec{H}_O)_1 + \sum \int_{t_1}^{t_2} \vec{M}_O dt = (\vec{H}_O)_2$. Conservation of Angular Momentum (when $\sum \vec{M}_O dt = 0$): $(\vec{H}_O)_1 = (\vec{H}_O)_2$. 13. Planar Kinematics of a Rigid Body Translation: All particles have the same velocity and acceleration. Rotation about a Fixed Axis: Angular velocity: $\omega = \frac{d\theta}{dt}$. Angular acceleration: $\alpha = \frac{d\omega}{dt}$. $v = r\omega$, $a_t = r\alpha$, $a_n = r\omega^2 = v^2/r$. General Plane Motion: Combination of translation and rotation. Relative Velocity: $\vec{v}_B = \vec{v}_A + \vec{\omega} \times \vec{r}_{B/A}$. Relative Acceleration: $\vec{a}_B = \vec{a}_A + \vec{\alpha} \times \vec{r}_{B/A} - \omega^2 \vec{r}_{B/A}$. Instantaneous Center of Zero Velocity (IC): A point on or off the body about which the body appears to be rotating instantaneously. 14. Planar Kinetics of a Rigid Body Equation of Motion: $ \sum F_x = m(\bar{a}_x) $ $ \sum F_y = m(\bar{a}_y) $ $ \sum M_G = \bar{I}\alpha $ where $\bar{a}_x, \bar{a}_y$ are acceleration components of the mass center G, and $\bar{I}$ is the mass moment of inertia about G. Mass Moment of Inertia: $\bar{I} = \int r^2 dm$. Parallel-Axis Theorem: $I_O = \bar{I} + md^2$. Work and Energy: $T_1 + U_{1-2} = T_2$. Kinetic Energy: $T = \frac{1}{2}m\bar{v}^2 + \frac{1}{2}\bar{I}\omega^2$. Impulse and Momentum: Linear: $m(\vec{v}_G)_1 + \sum \int \vec{F} dt = m(\vec{v}_G)_2$. Angular: $(\vec{H}_G)_1 + \sum \int \vec{M}_G dt = (\vec{H}_G)_2$, where $\vec{H}_G = \bar{I}\vec{\omega}$.