Hibbeler

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I. Statics - Force Vectors 1.1 Scalars & Vectors Scalar: Magnitude only (e.g., mass, volume, length). Vector: Magnitude and direction (e.g., force, velocity, acceleration). 1.2 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: $A_x = A \cos \theta$, $A_y = A \sin \theta$ 1.3 Cartesian Vectors (3D) Unit Vectors: $\hat{i}, \hat{j}, \hat{k}$ along $x, y, z$ axes. Vector: $\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$ Magnitude: $|\vec{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}$ Direction Cosines: $\cos \alpha = A_x / A$, $\cos \beta = A_y / A$, $\cos \gamma = A_z / A$ $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$ Unit Vector: $\hat{u}_A = \vec{A} / A = \cos \alpha \hat{i} + \cos \beta \hat{j} + \cos \gamma \hat{k}$ 1.4 Dot Product $\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta$ $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z$ Angle between vectors: $\theta = \cos^{-1} \left( \frac{\vec{A} \cdot \vec{B}}{|\vec{A}| |\vec{B}|} \right)$ Projection of $\vec{A}$ onto $\vec{B}$: $A_B = \vec{A} \cdot \hat{u}_B$ II. Statics - Equilibrium of a Particle 2.1 Free-Body Diagram (FBD) Isolate the particle. Show all external forces acting on it (known and unknown). Do not show internal forces or forces exerted by the particle. 2.2 Equations of Equilibrium 2D: $\sum F_x = 0$, $\sum F_y = 0$ 3D: $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$ III. Statics - Rigid Body Equilibrium 3.1 Moment of a Force Scalar (2D): $M_O = Fd$ (force $\times$ perpendicular distance). Use right-hand rule for sign. Vector (3D): $\vec{M}_O = \vec{r} \times \vec{F}$ $\vec{M}_O = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ r_x & r_y & r_z \\ F_x & F_y & F_z \end{vmatrix}$ Moment about an axis: $M_{u} = \hat{u} \cdot (\vec{r} \times \vec{F})$ 3.2 Principle of Moments (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: $\vec{M}_O = \vec{r} \times (\vec{F}_1 + \vec{F}_2) = (\vec{r} \times \vec{F}_1) + (\vec{r} \times \vec{F}_2)$ 3.3 Couple Moment Two parallel forces of same magnitude, opposite direction, separated by distance $d$. $M = Fd$ (magnitude), direction perpendicular to plane of forces. Vector: $\vec{M} = \vec{r} \times \vec{F}$ (where $\vec{r}$ connects points on $F$ and $-F$). 3.4 Equivalent Systems Resultant Force: $\vec{F}_R = \sum \vec{F}$ Resultant Moment: $\vec{M}_R = \sum \vec{M}_O + \sum (\vec{r} \times \vec{F})$ 3.5 Equations of Equilibrium for a Rigid Body $\sum \vec{F} = 0$ (Sum of forces is zero) $\sum \vec{M}_O = 0$ (Sum of moments about any point $O$ is zero) 2D: $\sum F_x = 0$, $\sum F_y = 0$, $\sum M_O = 0$ (3 equations) 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 equations) 3.6 Supports and Reactions Support Type 2D Reactions 3D Reactions Roller 1 force $\perp$ surface 1 force $\perp$ surface Pin (Hinge) 2 forces ($F_x, F_y$) 3 forces ($F_x, F_y, F_z$) Fixed (Cantilever) 2 forces, 1 moment 3 forces, 3 moments Cable/Rope 1 tension force along cable 1 tension force along cable IV. Statics - Trusses, Frames, Machines 4.1 Trusses (Pinned Joints) Method of Joints: Apply particle equilibrium ($\sum F_x=0, \sum F_y=0$) to each joint. Assume tension (pulling away from joint). Method of Sections: Cut through members (max 3 unknowns) and apply rigid body equilibrium ($\sum F_x=0, \sum F_y=0, \sum M=0$) to either section. Zero-Force Members: Two non-collinear members meeting at an unloaded joint $\rightarrow$ both are zero-force. Three members, two collinear, meeting at an unloaded joint $\rightarrow$ third is zero-force. 4.2 Frames & Machines Contain multi-force members. Dismember the structure and draw FBD for each member/pin. Internal forces at connections are equal and opposite. V. Statics - Friction 5.1 Dry Friction Static Friction: $F_s \le \mu_s N$ $F_s$ is the friction force, opposing impending motion. $\mu_s$ is the coefficient of static friction. $N$ is the normal force. Maximum static friction: $(F_s)_{max} = \mu_s N$. Occurs just before motion. Kinetic Friction: $F_k = \mu_k N$ $F_k$ is the friction force when motion occurs. $\mu_k$ is the coefficient of kinetic friction. $\mu_k Angle of Static Friction: $\phi_s = \arctan(\mu_s)$. Angle between $R$ (resultant of $F_s$ and $N$) and $N$. 5.2 Wedges & Screws Analyze by drawing FBDs for each component and applying equilibrium equations. Friction forces oppose relative motion or tendency of motion. VI. Statics - Centroid & Moment of Inertia 6.1 Centroid of Area $\bar{x} = \frac{\int x dA}{\int dA} = \frac{\sum \bar{x}_i A_i}{\sum A_i}$ $\bar{y} = \frac{\int y dA}{\int dA} = \frac{\sum \bar{y}_i A_i}{\sum A_i}$ 6.2 Moment of Inertia (Second Moment of Area) $I_x = \int y^2 dA$ $I_y = \int x^2 dA$ $I_{xy} = \int xy dA$ (Product of Inertia) 6.3 Parallel-Axis Theorem $I_x = I_{x'} + Ad_y^2$ $I_y = I_{y'} + Ad_x^2$ $I_{xy} = I_{x'y'} + Ad_x d_y$ Where $I_{x'}, I_{y'}, I_{x'y'}$ are moments of inertia about centroidal axes. VII. Kinematics of a Particle 7.1 Rectilinear Motion Velocity: $v = ds/dt$ Acceleration: $a = dv/dt = d^2s/dt^2$ Relating $v, a, s$: $a ds = v dv$ Constant Acceleration: $v = v_0 + a_c t$ $s = s_0 + v_0 t + \frac{1}{2} a_c t^2$ $v^2 = v_0^2 + 2 a_c (s - s_0)$ 7.2 Curvilinear Motion Position: $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ Velocity: $\vec{v} = \dot{x}\hat{i} + \dot{y}\hat{j} + \dot{z}\hat{k}$ Acceleration: $\vec{a} = \ddot{x}\hat{i} + \ddot{y}\hat{j} + \ddot{z}\hat{k}$ 7.3 Projectile Motion $a_x = 0$, $a_y = -g$ $v_x = (v_0)_x$ $v_y = (v_0)_y - gt$ $x = (v_0)_x t$ $y = (v_0)_y t - \frac{1}{2} g t^2$ $v_y^2 = (v_0)_y^2 - 2g y$ 7.4 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 $a_t = v dv/ds$ $a_n = v^2 / \rho$ (where $\rho$ is radius of curvature) 7.5 Cylindrical Components $\vec{r} = r \hat{u}_r + z \hat{k}$ $\vec{v} = \dot{r} \hat{u}_r + r \dot{\theta} \hat{u}_{\theta} + \dot{z} \hat{k}$ $\vec{a} = (\ddot{r} - r \dot{\theta}^2) \hat{u}_r + (r \ddot{\theta} + 2 \dot{r} \dot{\theta}) \hat{u}_{\theta} + \ddot{z} \hat{k}$ VIII. Kinetics of a Particle 8.1 Newton's Second Law $\sum \vec{F} = m \vec{a}$ Rectangular: $\sum F_x = m a_x$, $\sum F_y = m a_y$, $\sum F_z = m a_z$ Normal/Tangential: $\sum F_t = m a_t$, $\sum F_n = m a_n$, $\sum F_b = 0$ Cylindrical: $\sum F_r = m a_r$, $\sum F_{\theta} = m a_{\theta}$, $\sum F_z = m a_z$ 8.2 Work and Energy Work of a Force: $U_{1-2} = \int_1^2 \vec{F} \cdot d\vec{r}$ Spring Work: $U_s = -\frac{1}{2} k (s_2^2 - s_1^2)$ Kinetic Energy: $T = \frac{1}{2} m v^2$ Principle of Work and Energy: $T_1 + U_{1-2} = T_2$ Conservative Forces: Gravity, spring force. Potential Energy: $V_g = W y$ (gravity), $V_e = \frac{1}{2} k s^2$ (elastic) Conservation of Energy: $T_1 + V_1 = T_2 + V_2$ (if only conservative forces do work) 8.3 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: $\sum (m \vec{v})_1 = \sum (m \vec{v})_2$ (if sum of external impulses is zero) Impact: Coefficient of Restitution: $e = \frac{(v_B)_2 - (v_A)_2}{(v_A)_1 - (v_B)_1}$ $e=1$ (elastic), $e=0$ (plastic) IX. Kinematics of a Rigid Body 9.1 Types of Motion Translation: All points have same velocity and acceleration. Rectilinear: Straight path. Curvilinear: Curved path. Rotation about a Fixed Axis: Angular velocity: $\omega = d\theta/dt$ Angular acceleration: $\alpha = d\omega/dt$ $v = \omega r$ $a_t = \alpha r$, $a_n = \omega^2 r = v^2/r$ General Plane Motion: Translation + Rotation. 9.2 Relative Motion Analysis (Velocity) $\vec{v}_B = \vec{v}_A + \vec{v}_{B/A}$ $\vec{v}_{B/A} = \vec{\omega} \times \vec{r}_{B/A}$ (for rotation about A) 9.3 Relative Motion Analysis (Acceleration) $\vec{a}_B = \vec{a}_A + \vec{a}_{B/A}$ $\vec{a}_{B/A} = \vec{\alpha} \times \vec{r}_{B/A} - \omega^2 \vec{r}_{B/A}$ 9.4 Instantaneous Center of Zero Velocity (IC) Point on a body undergoing general plane motion that has zero velocity at that instant. Can be used to find velocities: $v = \omega r_{IC}$ X. Kinetics of a Rigid Body 10.1 Mass Moment of Inertia $I = \int r^2 dm$ Parallel-Axis Theorem: $I = I_G + md^2$ (where $I_G$ is about centroid G) 10.2 Equations of Motion (Plane Motion) $\sum F_x = m (\bar{a}_x)$ $\sum F_y = m (\bar{a}_y)$ $\sum M_G = I_G \alpha$ (Sum of moments about mass center G) Alternatively, $\sum M_P = \sum (\vec{M}_k)_P$ (Sum of moments about any point P, where $\sum (\vec{M}_k)_P$ includes $m(\vec{a}_G)_x, m(\vec{a}_G)_y$ and $I_G \alpha$ terms) 10.3 Work and Energy (Rigid Body) Kinetic Energy (Plane Motion): $T = \frac{1}{2} m \bar{v}^2 + \frac{1}{2} I_G \omega^2$ Principle of Work and Energy: $T_1 + U_{1-2} = T_2$ Conservation of Energy: $T_1 + V_1 = T_2 + V_2$ 10.4 Impulse and Momentum (Rigid Body) Linear Momentum: $m \vec{\bar{v}}$ Angular Momentum about G: $H_G = I_G \omega$ Principle of Linear Impulse and Momentum: $m (\vec{\bar{v}})_1 + \sum \int_{t_1}^{t_2} \vec{F} dt = m (\vec{\bar{v}})_2$ Principle of Angular Impulse and Momentum: $(H_G)_1 + \sum \int_{t_1}^{t_2} M_G dt = (H_G)_2$ Conservation of Momentum: If $\sum \int \vec{F} dt = 0$, then $m (\vec{\bar{v}})_1 = m (\vec{\bar{v}})_2$. If $\sum \int M_G dt = 0$, then $(H_G)_1 = (H_G)_2$.