Hibbeler Mechanics Cheatsheet

Cheatsheet Content

### Fundamental Principles - **Newton's First Law:** A particle remains at rest or moves with constant velocity if the net force acting on it is zero. - **Newton's Second Law:** $\vec{F} = m\vec{a}$. The acceleration of a particle is proportional to the net force acting on it and inversely proportional to its mass. - **Newton's Third Law:** For every action, there is an equal and opposite reaction. - **Newton's Law of Gravitational Attraction:** $F = G \frac{m_1 m_2}{r^2}$, where $G = 6.673 \times 10^{-11} \text{ m}^3/(\text{kg}\cdot\text{s}^2)$. - **Weight:** $W = mg$, where $g = 9.81 \text{ m/s}^2$ or $32.2 \text{ ft/s}^2$. ### Vectors - **Scalar:** Quantity with magnitude only (e.g., mass, length, time). - **Vector:** Quantity with magnitude and direction (e.g., force, velocity, acceleration). - **Vector Addition (Parallelogram Law):** $\vec{R} = \vec{A} + \vec{B}$. - **Vector Subtraction:** $\vec{R}' = \vec{A} - \vec{B} = \vec{A} + (-\vec{B})$. - **Cartesian Vector Representation:** $\vec{F} = F_x \hat{i} + F_y \hat{j} + F_z \hat{k}$ - **Unit Vector:** $\vec{u}_A = \frac{\vec{A}}{|\vec{A}|} = \frac{A_x}{A} \hat{i} + \frac{A_y}{A} \hat{j} + \frac{A_z}{A} \hat{k}$. - **Dot Product:** $\vec{A} \cdot \vec{B} = |\vec{A}||\vec{B}|\cos\theta = A_x B_x + A_y B_y + A_z B_z$. - Used to find the angle between two vectors or the projection of one vector onto another. - **Cross Product:** $\vec{C} = \vec{A} \times \vec{B}$. Magnitude $|\vec{C}| = |\vec{A}||\vec{B}|\sin\theta$. Direction by right-hand rule. - $\vec{A} \times \vec{B} = (A_y B_z - A_z B_y)\hat{i} - (A_x B_z - A_z B_x)\hat{j} + (A_x B_y - A_y B_x)\hat{k}$. - Used to find a vector perpendicular to two other vectors (e.g., moment). ### Particle Equilibrium - **Conditions for Equilibrium:** $\sum \vec{F} = 0$. - 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 solving equilibrium problems. Isolate the particle and show all external forces acting on it. ### Rigid Body Equilibrium - **Conditions for Equilibrium:** - $\sum \vec{F} = 0$ (Sum of forces is zero) - $\sum \vec{M}_O = 0$ (Sum of moments about any point O is zero) - **2D Equilibrium Equations:** - $\sum F_x = 0$ - $\sum F_y = 0$ - $\sum M_O = 0$ (where O is any point in the plane) - **3D Equilibrium Equations:** - $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$ - $\sum M_x = 0$, $\sum M_y = 0$, $\sum M_z = 0$ (moments about x, y, z axes) - **Moment of a Force:** - Scalar: $M_O = Fd$, where $d$ is the perpendicular distance from O to the line of action of F. - Vector: $\vec{M}_O = \vec{r} \times \vec{F}$, where $\vec{r}$ is the position vector from O to any point on the line of action of F. - **Couple Moment:** $\vec{M} = \vec{r} \times \vec{F}$, where $\vec{r}$ is a position vector from the line of action of $-\vec{F}$ to the line of action of $\vec{F}$. Magnitude $M = Fd$. ### Trusses - **Assumptions:** Members are two-force members (only axial force), forces applied at joints via pins. - **Method of Joints:** 1. Draw FBD of entire truss to find external reactions. 2. Draw FBD of each joint. Assume unknown member forces are in tension. 3. Apply $\sum F_x = 0$, $\sum F_y = 0$ at each joint. 4. Work through joints with a maximum of two unknown forces. - **Method of Sections:** 1. Draw FBD of entire truss to find external reactions. 2. Cut the truss through members of interest (max 3 unknowns). 3. Draw FBD of one section. 4. Apply $\sum F_x = 0$, $\sum F_y = 0$, $\sum M_O = 0$ to solve for unknown member forces. - **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 connect at a joint, two are collinear, and no external load/reaction, the third non-collinear member is a zero-force member. ### Frames and Machines - **Frames:** Stationary structures designed to support loads. Contain at least one multi-force member. - **Machines:** Structures containing moving parts, designed to transmit or alter the effect of forces. - **Analysis Procedure:** 1. Draw FBD of the entire structure to find external reactions. 2. Disassemble the structure and draw FBD for each member. 3. Apply Newton's Third Law for internal forces at connections (equal and opposite). 4. Apply equilibrium equations ($\sum F_x = 0, \sum F_y = 0, \sum M_O = 0$) to each member or combination of members to solve for unknowns. ### Center of Gravity & Centroid - **Center of Gravity (CG):** Point where the entire weight of a body can be considered to act. - $\bar{x} = \frac{\sum \tilde{x}W}{\sum W}$, $\bar{y} = \frac{\sum \tilde{y}W}{\sum W}$, $\bar{z} = \frac{\sum \tilde{z}W}{\sum W}$ - **Centroid:** Geometric center of an area or volume. - **Area:** $\bar{x} = \frac{\int \tilde{x} dA}{\int dA}$, $\bar{y} = \frac{\int \tilde{y} dA}{\int dA}$ - For composite areas: $\bar{x} = \frac{\sum \tilde{x}A}{\sum A}$, $\bar{y} = \frac{\sum \tilde{y}A}{\sum A}$ - **Theorems of Pappus and Guldinus:** - **Area of surface of revolution:** $A = \theta \bar{r} L$ (where $\theta$ is angle of revolution, $\bar{r}$ is distance from axis to centroid of curve, $L$ is length of curve). For full revolution, $\theta = 2\pi$. - **Volume of body of revolution:** $V = \theta \bar{r} A$ (where $\theta$ is angle of revolution, $\bar{r}$ is distance from axis to centroid of area, $A$ is area). For full revolution, $\theta = 2\pi$. ### Moments of Inertia - **Area Moment of Inertia:** Measures resistance to bending. - $I_x = \int y^2 dA$, $I_y = \int x^2 dA$ - **Parallel-Axis Theorem:** $I = I_c + Ad^2$, where $I_c$ is moment of inertia about centroidal axis, $A$ is area, $d$ is distance between parallel axes. - **Polar Moment of Inertia:** $J_O = \int r^2 dA = I_x + I_y$. - **Mass Moment of Inertia:** Measures resistance to angular acceleration. - $I = \int r^2 dm$ - **Parallel-Axis Theorem:** $I = I_G + md^2$, where $I_G$ is mass moment of inertia about centroidal axis, $m$ is mass, $d$ is distance between parallel axes. - **Radius of Gyration:** $k = \sqrt{I/m}$ ### Kinematics of a Particle - **Rectilinear Motion (Constant Acceleration):** - $v = v_0 + a_c t$ - $s = s_0 + v_0 t + \frac{1}{2} a_c t^2$ - $v^2 = v_0^2 + 2 a_c (s - s_0)$ - **General Rectilinear Motion:** - $v = \frac{ds}{dt}$ - $a = \frac{dv}{dt} = v \frac{dv}{ds}$ - **Curvilinear Motion (Cartesian):** - $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ - $\vec{v} = \dot{x}\hat{i} + \dot{y}\hat{j} + \dot{z}\hat{k}$ - $\vec{a} = \ddot{x}\hat{i} + \ddot{y}\hat{j} + \ddot{z}\hat{k}$ - **Curvilinear Motion (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 $a_t = v \frac{dv}{ds}$ (tangential acceleration, changes speed) - $a_n = \frac{v^2}{\rho}$ (normal acceleration, changes direction, $\rho$ is radius of curvature) - **Curvilinear Motion (Cylindrical Components):** - $\vec{r} = r \hat{u}_r + z \hat{k}$ - $\vec{v} = \dot{r} \hat{u}_r + r \dot{\theta} \hat{u}_{\theta} + \dot{z} \hat{k}$ - $\vec{a} = (\ddot{r} - r \dot{\theta}^2) \hat{u}_r + (r \ddot{\theta} + 2 \dot{r} \dot{\theta}) \hat{u}_{\theta} + \ddot{z} \hat{k}$ ### Kinetics of a Particle - **Newton's Second Law:** $\sum \vec{F} = m\vec{a}$ - **Rectangular Components:** $\sum F_x = ma_x$, $\sum F_y = ma_y$, $\sum F_z = ma_z$ - **Normal and Tangential Components:** $\sum F_t = ma_t$, $\sum F_n = ma_n$ - **Cylindrical Components:** $\sum F_r = ma_r$, $\sum F_{\theta} = ma_{\theta}$, $\sum F_z = ma_z$ - **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: $U_s = -\frac{1}{2} k (s_2^2 - s_1^2)$ (where $s$ is deformation from unstretched length) - **Principle of Work and Energy:** $T_1 + U_{1-2} = T_2$ - $T = \frac{1}{2} mv^2$ (kinetic energy) - **Conservation of Energy:** $T_1 + V_1 = T_2 + V_2$ (when only conservative forces do work) - **Gravitational Potential Energy:** $V_g = Wy$ - **Elastic Potential Energy:** $V_e = \frac{1}{2} ks^2$ - **Impulse and Momentum:** - **Linear Impulse:** $\vec{I} = \int \vec{F} dt$ - **Linear Momentum:** $\vec{p} = m\vec{v}$ - **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:** $\sum m\vec{v}_1 = \sum m\vec{v}_2$ (when net external impulse is zero) - **Impact:** - **Coefficient of Restitution:** $e = \frac{(v_B)_2 - (v_A)_2}{(v_A)_1 - (v_B)_1}$ (for central impact) - For perfectly elastic impact, $e=1$. For perfectly plastic impact, $e=0$. - **Angular Momentum:** - **Angular Momentum of a Particle about 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:** $(\vec{H}_O)_1 = (\vec{H}_O)_2$ (when net external moment about O is zero) ### Kinematics of a Rigid Body - **Types of Motion:** - **Translation:** All points have the same velocity and acceleration. - Rectilinear: Straight path. - Curvilinear: Curved path. - **Rotation about a Fixed Axis:** All points move in concentric circles. - Angular velocity: $\omega = \frac{d\theta}{dt}$ - Angular acceleration: $\alpha = \frac{d\omega}{dt} = \omega \frac{d\omega}{d\theta}$ - For constant angular acceleration: - $\omega = \omega_0 + \alpha_c t$ - $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha_c t^2$ - $\omega^2 = \omega_0^2 + 2 \alpha_c (\theta - \theta_0)$ - Velocity of point P: $v = \omega r$ - Acceleration of point P: $a_t = \alpha r$, $a_n = \omega^2 r = v^2/r$ - **General Plane Motion:** Translation + Rotation. - **Relative Velocity:** $\vec{v}_B = \vec{v}_A + \vec{v}_{B/A} = \vec{v}_A + (\vec{\omega} \times \vec{r}_{B/A})$ - **Relative Acceleration:** $\vec{a}_B = \vec{a}_A + \vec{a}_{B/A} = \vec{a}_A + (\vec{\alpha} \times \vec{r}_{B/A}) - \omega^2 \vec{r}_{B/A}$ - **Instantaneous Center (IC) of Zero Velocity:** For plane motion, a point about which the body appears to rotate at that instant. - $v = \omega r_{IC}$ ### Kinetics of a Rigid Body - **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 moments about the mass center G) - Or $\sum M_P = I_P \alpha$ (Sum moments about a fixed point P) - **Work and Energy (Plane Motion):** - **Kinetic Energy:** $T = \frac{1}{2} m \bar{v}^2 + \frac{1}{2} I_G \omega^2$ - **Principle of Work and Energy:** $T_1 + U_{1-2} = T_2$ - **Conservation of Energy:** $T_1 + V_1 = T_2 + V_2$ - **Impulse and Momentum (Plane Motion):** - **Linear Impulse and Momentum:** $m(\vec{v}_G)_1 + \sum \int \vec{F} dt = m(\vec{v}_G)_2$ - **Angular Impulse and Momentum:** $(H_G)_1 + \sum \int M_G dt = (H_G)_2$ - $(H_G)_1 = I_G \omega_1$ - **Conservation of Angular Momentum:** $(H_G)_1 = (H_G)_2$ (if $\sum \int M_G dt = 0$)