Hibbeler Mechanics (2-Page)

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### Fundamental Principles - **Newton's First Law:** A particle remains at rest or continues to move with constant velocity if there is no unbalanced force acting on it. - **Newton's Second Law:** The acceleration of a particle is proportional to the resultant force acting on it and is in the direction of this force. $\sum \vec{F} = m\vec{a}$ - **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 $G = 6.673 \times 10^{-11} \text{ m}^3/(\text{kg} \cdot \text{s}^2)$ - **Units:** - SI: meter (m), kilogram (kg), second (s), Newton (N) - US Customary: foot (ft), slug (slug), second (s), pound (lb) - $1 \text{ slug} = 1 \text{ lb} \cdot \text{s}^2 / \text{ft}$ - Weight: $W = mg$ ### Force Vectors - **Scalar:** Quantity with magnitude (e.g., mass, length, time). - **Vector:** Quantity with magnitude and direction (e.g., force, velocity). - **Vector Addition (Parallelogram Law):** Resultant vector is the diagonal of the parallelogram formed by the two vectors. - **Vector Components:** - Cartesian: $\vec{F} = F_x \hat{i} + F_y \hat{j} + F_z \hat{k}$ - Magnitude: $F = \sqrt{F_x^2 + F_y^2 + F_z^2}$ - Direction Cosines: $\cos\alpha = F_x/F$, $\cos\beta = F_y/F$, $\cos\gamma = F_z/F$ - Unit Vector: $\vec{u} = \frac{\vec{F}}{F} = \cos\alpha \hat{i} + \cos\beta \hat{j} + \cos\gamma \hat{k}$ - **Dot Product:** $\vec{A} \cdot \vec{B} = AB \cos\theta = A_x B_x + A_y B_y + A_z B_z$ - Useful for finding angle between vectors or projection of one vector onto another. - Projection of $\vec{A}$ onto $\vec{B}$: $A_B = \vec{A} \cdot \vec{u}_B$ ### Equilibrium of a Particle - **Conditions for Equilibrium:** $\sum \vec{F} = \vec{0}$ - In 2D: $\sum F_x = 0$, $\sum F_y = 0$ - In 3D: $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$ - **Free-Body Diagram (FBD):** Essential for visualizing all forces acting on a particle. ### Force System Resultants - **Moment of a Force (about a point O):** $\vec{M}_O = \vec{r} \times \vec{F}$ - Magnitude: $M_O = rF\sin\theta = Fd$ (where $d$ is perpendicular distance) - Right-hand rule for direction. - Cartesian Components: $\vec{M}_O = (r_y F_z - r_z F_y)\hat{i} + (r_z F_x - r_x F_z)\hat{j} + (r_x F_y - r_y F_x)\hat{k}$ - **Moment of a Force (about an axis a):** $M_a = \vec{u}_a \cdot (\vec{r} \times \vec{F})$ - **Couple Moment:** $\vec{M} = \vec{r} \times \vec{F}$ (for two equal, opposite, and non-collinear forces) - Magnitude: $M = Fd$ - **Simplifying a Force and Couple System:** - Replace all forces with a single resultant force $\vec{F}_R = \sum \vec{F}$ acting at a point O. - Replace all moments with a single resultant couple moment $\vec{M}_{R_O} = \sum \vec{M}_O + \sum \vec{M}_{couple}$ ### Equilibrium of a Rigid Body - **Conditions for Equilibrium:** - $\sum \vec{F} = \vec{0}$ (Sum of forces is zero) - $\sum \vec{M}_O = \vec{0}$ (Sum of moments about any point O is zero) - **Support Reactions:** - **Pin/Hinge:** Two force components (e.g., $R_x, R_y$) - **Roller:** One force component (normal to surface) - **Fixed Support:** Two force components and one moment (e.g., $R_x, R_y, M_z$) - **Rope/Cable:** Tension (always away from body) - **Smooth Surface:** Normal force (perpendicular to surface) - **Two-Force Member:** If a rigid body is subjected to only two forces, these forces must be collinear, equal in magnitude, and opposite in direction. - **Three-Force Member:** If a rigid body is subjected to only three forces, these forces must be either concurrent or parallel. ### Trusses - **Assumptions:** - Members are joined by pins (frictionless). - Loads are applied only at joints. - Members are two-force members. - **Method of Joints:** - Draw FBD of each joint. - Apply $\sum F_x = 0$ and $\sum F_y = 0$ at each joint. - Start at a joint with at most two unknown member forces. - **Method of Sections:** - Cut the truss into two sections through members with unknown forces. - Draw FBD of one section. - Apply $\sum F_x = 0$, $\sum F_y = 0$, and $\sum M = 0$. - Can solve for up to three unknown forces. ### Frames and Machines - **Frames:** Structures designed to support loads, usually stationary and fully constrained. Contain multi-force members. - **Machines:** Structures designed to transmit and modify forces, usually containing moving parts. - **Analysis Procedure:** 1. Draw FBD of the entire structure to find external reactions. 2. Disassemble the structure into its individual members. 3. Draw FBD for each member, showing all forces (action-reaction pairs are equal and opposite). 4. Apply equilibrium equations ($\sum F_x = 0$, $\sum F_y = 0$, $\sum M = 0$) to each member or combination of members. ### Center of Gravity and Centroid - **Center of Gravity (CG):** Point where the entire weight of a body can be considered to act. - $W\bar{x} = \sum W_i x_i$, $W\bar{y} = \sum W_i y_i$, $W\bar{z} = \sum W_i z_i$ - **Centroid:** Geometric center of an area or volume. - **Area:** $\bar{x} = \frac{\sum \tilde{x}_i A_i}{\sum A_i}$, $\bar{y} = \frac{\sum \tilde{y}_i A_i}{\sum A_i}$ - **Volume:** $\bar{x} = \frac{\sum \tilde{x}_i V_i}{\sum V_i}$, $\bar{y} = \frac{\sum \tilde{y}_i V_i}{\sum V_i}$, $\bar{z} = \frac{\sum \tilde{z}_i V_i}{\sum V_i}$ - **Theorems of Pappus and Guldinus:** - **Area of surface of revolution:** $A = \theta \bar{r} L$ (for arc length L rotated by angle $\theta$) - **Volume of body of revolution:** $V = \theta \bar{r} A$ (for area A rotated by angle $\theta$) ### Moments of Inertia - **Area Moment of Inertia:** - $I_x = \int y^2 dA$ - $I_y = \int x^2 dA$ - Polar Moment of Inertia: $J_O = I_x + I_y = \int r^2 dA$ - **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. - **Radius of Gyration:** $k = \sqrt{I/A}$ ### Friction - **Static Friction:** $F_s \le \mu_s N$ - $F_s$ is static friction force. - $\mu_s$ is coefficient of static friction. - $N$ is normal force. - Maximum static friction: $F_{s,max} = \mu_s N$ (impending motion). - **Kinetic Friction:** $F_k = \mu_k N$ - $F_k$ is kinetic friction force. - $\mu_k$ is coefficient of kinetic friction ($\mu_k ### Work and Energy - **Work of a Force:** $U = \int \vec{F} \cdot d\vec{r}$ - Constant force: $U = (F\cos\theta)s$ - Weight: $U_g = -W\Delta y$ - Spring: $U_s = -\frac{1}{2}k(s_2^2 - s_1^2)$ - **Principle of Work and Energy:** $T_1 + \sum U_{1-2} = T_2$ - Kinetic Energy: $T = \frac{1}{2}mv^2$ - **Conservative Forces:** Work done is independent of path (e.g., gravity, elastic spring force). - **Potential Energy:** - Gravitational: $V_g = Wy$ - Elastic: $V_e = \frac{1}{2}ks^2$ - **Conservation of Energy:** $T_1 + V_1 = T_2 + V_2$ (for conservative systems) ### Impulse and Momentum - **Linear Impulse:** $\vec{I} = \int \vec{F} dt$ - **Linear Momentum:** $\vec{L} = m\vec{v}$ - **Principle of Linear Impulse and Momentum:** $m\vec{v}_1 + \sum \int_{t_1}^{t_2} \vec{F} dt = m\vec{v}_2$ - $\vec{L}_1 + \sum \vec{I}_{1-2} = \vec{L}_2$ - **Conservation of Linear Momentum:** $\sum (m\vec{v})_1 = \sum (m\vec{v})_2$ (when no external impulse). - **Impact:** - **Coefficient of Restitution:** $e = \frac{(v_B)_2 - (v_A)_2}{(v_A)_1 - (v_B)_1}$ - $e=1$ (perfectly elastic), $e=0$ (perfectly plastic).