Hibbeler Mechanics Cheatsheet

Cheatsheet Content

### Vectors and Forces #### Vector Operations - **Vector Addition:** Head-to-tail method, or component addition: $\vec{R} = \vec{A} + \vec{B} = (A_x+B_x)\hat{i} + (A_y+B_y)\hat{j} + (A_z+B_z)\hat{k}$ - **Dot Product:** $\vec{A} \cdot \vec{B} = |\vec{A}||\vec{B}|\cos\theta = A_x B_x + A_y B_y + A_z B_z$ (scalar result, used for projection) - **Cross Product:** $\vec{A} \times \vec{B} = (A_y B_z - A_z B_y)\hat{i} - (A_x B_z - A_z B_x)\hat{j} + (A_x B_y - A_y B_x)\hat{k}$ (vector result, perpendicular to both $\vec{A}$ and $\vec{B}$, magnitude $|\vec{A}||\vec{B}|\sin\theta$) #### Force Systems - **Resultant Force:** $\vec{F}_R = \sum \vec{F}$ - **Equilibrium:** $\sum \vec{F} = 0$ (static equilibrium condition) - **Components:** $F_x = F \cos\theta$, $F_y = F \sin\theta$ - **Unit Vector:** $\hat{u}_A = \frac{\vec{A}}{|\vec{A}|}$ ### Equilibrium of a Particle - **2D Equilibrium:** $\sum F_x = 0$, $\sum F_y = 0$ - **3D Equilibrium:** $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$ - **Free-Body Diagram (FBD):** Essential for visualizing all forces acting on a particle. ### Equilibrium of a Rigid Body - **Conditions for 2D Equilibrium:** - $\sum F_x = 0$ - $\sum F_y = 0$ - $\sum M_O = 0$ (sum of moments about any point O must be zero) - **Conditions for 3D Equilibrium:** - $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$ - $\sum M_x = 0$, $\sum M_y = 0$, $\sum M_z = 0$ - **Moment of a Force:** - **Scalar (2D):** $M_O = Fd$ (force F times perpendicular distance d from O to line of action of F) - **Vector (3D):** $\vec{M}_O = \vec{r} \times \vec{F}$ (where $\vec{r}$ is position vector from O to any point on line of action of F) - **Couple Moment:** A pair of equal, opposite, and non-collinear forces. $M = Fd$ (magnitude), direction given by right-hand rule. ### Trusses, Frames, and Machines #### Trusses - **Assumptions:** All members are two-force members (only axial force), forces applied at joints. - **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, apply rigid body equilibrium ($\sum F_x=0, \sum F_y=0, \sum M=0$) to either section. #### Frames and Machines - Components are multi-force members. Disassemble the structure into its component parts and draw FBDs for each part. Apply rigid body equilibrium to each part. ### Center of Gravity and Centroid - **Center of Gravity (CG):** Point where entire weight of body acts. - $\bar{x} = \frac{\sum W_i x_i}{\sum W_i}$, $\bar{y} = \frac{\sum W_i y_i}{\sum W_i}$, $\bar{z} = \frac{\sum W_i z_i}{\sum W_i}$ - **Centroid (C):** Geometric center (for homogeneous materials, C=CG). - **Area:** $\bar{x} = \frac{\sum A_i x_i}{\sum A_i}$, $\bar{y} = \frac{\sum A_i y_i}{\sum A_i}$ - **Volume:** $\bar{x} = \frac{\sum V_i x_i}{\sum V_i}$, $\bar{y} = \frac{\sum V_i y_i}{\sum V_i}$, $\bar{z} = \frac{\sum V_i z_i}{\sum V_i}$ - **Pappus-Guldinus Theorems:** For surface area and volume of revolution. ### 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:** $I = I_c + Ad^2$ (where $I_c$ is moment of inertia about centroidal axis, A is area, d is distance between parallel axes). - **Mass Moment of Inertia:** - $I = \int r^2 dm$ - For a slender rod about its center: $I = \frac{1}{12} mL^2$ - For a cylinder about its central axis: $I = \frac{1}{2} mR^2$ - Parallel-Axis Theorem (Mass): $I = I_G + md^2$ (where $I_G$ is about mass center, m is mass, d is distance) ### 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 Motion:** - $v = \frac{ds}{dt}$ - $a = \frac{dv}{dt} = \frac{d^2s}{dt^2}$ - $a ds = v dv$ #### Curvilinear Motion - **Rectangular Components:** - $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ - $\vec{v} = \dot{x}\hat{i} + \dot{y}\hat{j} + \dot{z}\hat{k}$ - $\vec{a} = \ddot{x}\hat{i} + \ddot{y}\hat{j} + \ddot{z}\hat{k}$ - **Normal and Tangential Components:** - $v = \dot{s}$ - $a_t = \dot{v} = v \frac{dv}{ds}$ (tangential acceleration, changes speed) - $a_n = \frac{v^2}{\rho}$ (normal acceleration, changes direction, $\rho$ is radius of curvature) - $a = \sqrt{a_t^2 + a_n^2}$ - **Cylindrical Components:** - $\vec{r} = r\hat{u}_r + z\hat{k}$ - $\vec{v} = \dot{r}\hat{u}_r + r\dot{\theta}\hat{u}_{\theta} + \dot{z}\hat{k}$ - $\vec{a} = (\ddot{r} - r\dot{\theta}^2)\hat{u}_r + (r\ddot{\theta} + 2\dot{r}\dot{\theta})\hat{u}_{\theta} + \ddot{z}\hat{k}$ ### Kinetics of a Particle #### Newton's Second Law - $\sum \vec{F} = m\vec{a}$ - **Rectangular:** $\sum F_x = ma_x$, $\sum F_y = ma_y$, $\sum F_z = ma_z$ - **Normal-Tangential:** $\sum F_t = ma_t$, $\sum F_n = ma_n$ - **Cylindrical:** $\sum F_r = ma_r$, $\sum F_{\theta} = ma_{\theta}$, $\sum F_z = ma_z$ #### Work and Energy - **Work of a Force:** $U_{1-2} = \int_1^2 \vec{F} \cdot d\vec{r}$ - Constant Force: $U_{1-2} = F_c \cos\theta \Delta s$ - Weight: $U_{1-2} = -W \Delta y$ - 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 $T = \frac{1}{2}mv^2$ is kinetic energy) - **Conservation of Energy (Conservative Forces Only):** $T_1 + V_1 = T_2 + V_2$ - **Potential Energy:** - Gravitational: $V_g = Wy$ - Elastic (Spring): $V_e = \frac{1}{2} k s^2$ #### Impulse and Momentum - **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 (No External Impulses):** $\sum m\vec{v}_1 = \sum m\vec{v}_2$ - **Impact:** - **Coefficient of Restitution:** $e = \frac{(v_B')_n - (v_A')_n}{(v_A)_n - (v_B)_n}$ (applies along line of impact) - For perfectly elastic impact, $e=1$. For perfectly plastic, $e=0$. - **Angular Momentum (about a 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 (No External Moments):** $(\vec{H}_O)_1 = (\vec{H}_O)_2$ ### Kinematics of a Rigid Body #### Types of Motion - **Translation:** - Rectilinear: Path is a straight line. - Curvilinear: Path is a curved line. - **Rotation about a Fixed Axis:** - $\alpha = \frac{d\omega}{dt}$ - $\omega = \frac{d\theta}{dt}$ - $v = r\omega$ - $a_t = r\alpha$ - $a_n = r\omega^2 = \frac{v^2}{r}$ - **Constant Angular Acceleration:** - $\omega = \omega_0 + \alpha_c t$ - $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha_c t^2$ - $\omega^2 = \omega_0^2 + 2 \alpha_c (\theta - \theta_0)$ - **General Plane Motion (Translation + 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):** Point where velocity is instantaneously zero. Locating IC simplifies velocity analysis. ### Kinetics of a Rigid Body #### Equations of Motion (Plane Motion) - $\sum F_x = m(\bar{a}_G)_x$ - $\sum F_y = m(\bar{a}_G)_y$ - $\sum M_G = \bar{I}\alpha$ (moment about center of mass G) - OR $\sum M_P = \sum (\mathcal{M}_k)_P$ (moment about any point P, where $\sum (\mathcal{M}_k)_P$ includes moments of $m\bar{a}_x, m\bar{a}_y$, and $\bar{I}\alpha$) #### Work and Energy - **Kinetic Energy (Plane Motion):** $T = \frac{1}{2} m\bar{v}^2 + \frac{1}{2} \bar{I}\omega^2$ - **Principle of Work and Energy:** $T_1 + U_{1-2} = T_2$ #### Impulse and Momentum - **Linear Momentum:** $m\bar{v}_1 + \sum \int_{t_1}^{t_2} \vec{F} dt = m\bar{v}_2$ - **Angular Momentum (about G):** $\bar{H}_G = \bar{I}\omega$ - **Principle of Angular Impulse and Momentum (about G):** $\bar{I}\omega_1 + \sum \int_{t_1}^{t_2} M_G dt = \bar{I}\omega_2$ - **Conservation of Momentum:** - Linear: $\sum m\bar{v}_1 = \sum m\bar{v}_2$ (if $\sum F = 0$) - Angular: $\sum (\bar{H}_G)_1 = \sum (\bar{H}_G)_2$ (if $\sum M_G = 0$)