Hibbeler

Cheatsheet Content

1. General Principles Newton's First Law: A particle originally at rest, or moving in a straight line with constant velocity, tends to 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 the force and a magnitude that is directly proportional to the force. $ \vec{F} = m\vec{a} $ Newton's Third Law: The mutual forces of action and reaction between two particles are equal, opposite, and collinear. Gravitational Law: $F = G \frac{m_1 m_2}{r^2}$ where $G = 66.73 \times 10^{-12} \text{ m}^3/(\text{kg} \cdot \text{s}^2)$. Weight: $W = mg$ where $g = 9.81 \text{ m/s}^2$ or $32.2 \text{ ft/s}^2$. 2. Force Vectors 2.1. 2D Vector Components $F_x = F \cos \theta$, $F_y = F \sin \theta$ $F = \sqrt{F_x^2 + F_y^2}$ $\theta = \arctan(F_y/F_x)$ Unit Vector: $\vec{u} = \vec{F}/F = \cos\theta \hat{i} + \sin\theta \hat{j}$ 2.2. 3D Vector Components $\vec{F} = F_x \hat{i} + F_y \hat{j} + F_z \hat{k}$ $F = \sqrt{F_x^2 + F_y^2 + F_z^2}$ Direction Cosines: $\cos \alpha = F_x/F$, $\cos \beta = F_y/F$, $\cos \gamma = F_z/F$ $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$ Unit Vector: $\vec{u} = \frac{F_x}{F}\hat{i} + \frac{F_y}{F}\hat{j} + \frac{F_z}{F}\hat{k} = \cos\alpha \hat{i} + \cos\beta \hat{j} + \cos\gamma \hat{k}$ 2.3. 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: $\theta = \arccos\left(\frac{\vec{A} \cdot \vec{B}}{|\vec{A}||\vec{B}|}\right)$ Projection: $A_B = \vec{A} \cdot \vec{u}_B$ 3. Equilibrium of a Particle Condition: $\sum \vec{F} = \vec{0}$ In 2D: $\sum F_x = 0$, $\sum F_y = 0$ In 3D: $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$ Procedure: Draw Free-Body Diagram (FBD). Establish coordinate system. Apply equations of equilibrium. 4. Force System Resultants 4.1. Moment of a Force (Scalar) $M_O = Fd$ (where $d$ is perpendicular distance) Positive for counter-clockwise rotation. 4.2. Moment of a Force (Vector) $\vec{M}_O = \vec{r} \times \vec{F}$ $\vec{M}_O = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ r_x & r_y & r_z \\ F_x & F_y & F_z \end{vmatrix}$ 4.3. Moment about an Axis $M_{axis} = \vec{u}_{axis} \cdot (\vec{r} \times \vec{F})$ $M_{axis} = \begin{vmatrix} u_x & u_y & u_z \\ r_x & r_y & r_z \\ F_x & F_y & F_z \end{vmatrix}$ 4.4. Couple Moment $\vec{M} = \vec{r} \times \vec{F}$ (where $\vec{r}$ is vector from line of action of $-\vec{F}$ to $\vec{F}$) Magnitude $M = Fd$ 4.5. Resultant of a Force System Resultant Force: $\vec{F}_R = \sum \vec{F}$ Resultant Moment: $(\vec{M}_R)_O = \sum (\vec{M})_O + \sum (\vec{r} \times \vec{F})$ 5. Equilibrium of a Rigid Body Conditions: $\sum \vec{F} = \vec{0}$ and $\sum \vec{M}_O = \vec{0}$ In 2D (3 eqns): $\sum F_x = 0$, $\sum F_y = 0$, $\sum M_O = 0$ In 3D (6 eqns): $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$, $\sum M_x = 0$, $\sum M_y = 0$, $\sum M_z = 0$ Support Reactions: Type 2D Reaction 3D Reaction Roller 1 Force (normal) 1 Force (normal) Pin/Hinge 2 Forces ($F_x, F_y$) 3 Forces ($F_x, F_y, F_z$) Fixed Support 2 Forces, 1 Moment 3 Forces, 3 Moments Ball-and-Socket - 3 Forces ($F_x, F_y, F_z$) 6. Trusses, Frames, and Machines 6.1. Trusses Assumptions: Members are two-force members, loads 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, analyze equilibrium of a section ($\sum F_x=0, \sum F_y=0, \sum M=0$). 6.2. Frames and Machines Contain multi-force members. Disassemble into component parts. Apply rigid body equilibrium to each part. Internal forces are equal and opposite. 7. Internal Forces Axial Force (N): Normal to cross-section. Shear Force (V): Tangent to cross-section. Bending Moment (M): Causes bending. Procedure: Find external reactions. Cut the member at desired point. Draw FBD of section, assume positive directions for N, V, M. Apply equilibrium equations. Relations: $dV/dx = -w(x)$ (Distributed load) $dM/dx = V(x)$ (Shear force) 8. Friction Static Friction: $F_s \le \mu_s N$ (up to impending motion) Kinetic Friction: $F_k = \mu_k N$ (when motion occurs) $\mu_s > \mu_k$ Angle of Friction: $\phi_s = \arctan(\mu_s)$ Wedges: Analyze as connected rigid bodies. Screws: $M = Wr \tan(\theta_L \pm \phi_s)$ for raising/lowering, where $\theta_L$ is lead angle, $\phi_s$ is friction angle. 9. Centroid and Moment of Inertia 9.1. Centroid of an Area $\bar{x} = \frac{\sum \tilde{x} A}{\sum A}$, $\bar{y} = \frac{\sum \tilde{y} A}{\sum A}$ For composite areas, divide into simple shapes. 9.2. Moment of Inertia of an Area $I_x = \int y^2 dA$, $I_y = \int x^2 dA$ Parallel-Axis Theorem: $I_x = \bar{I}_x + Ad_y^2$, $I_y = \bar{I}_y + Ad_x^2$ Radius of Gyration: $k = \sqrt{I/A}$ Product of Inertia: $I_{xy} = \int xy dA$ 10. Virtual Work Principle of Virtual Work: $\delta U = 0$ for equilibrium. $\delta U = \sum F_i \delta s_i + \sum M_j \delta \theta_j = 0$ Virtual displacements $\delta s_i$ and virtual rotations $\delta \theta_j$ are infinitesimal, consistent with constraints. 11. Kinematics of a Particle 11.1. Rectilinear Motion $v = ds/dt$ $a = dv/dt = v dv/ds$ 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 Motion (Rectangular Components) 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: $\vec{a} = \ddot{x}\hat{i} + \ddot{y}\hat{j} + \ddot{z}\hat{k}$ 11.3. Curvilinear Motion (Normal and Tangential Components) $\vec{v} = v\vec{u}_t$ $\vec{a} = \dot{v}\vec{u}_t + \frac{v^2}{\rho}\vec{u}_n = a_t\vec{u}_t + a_n\vec{u}_n$ $\rho$ is radius of curvature. 11.4. Curvilinear Motion (Cylindrical Components) Position: $\vec{r} = r\vec{u}_r + z\vec{u}_z$ Velocity: $\vec{v} = \dot{r}\vec{u}_r + r\dot{\theta}\vec{u}_{\theta} + \dot{z}\vec{u}_z$ Acceleration: $\vec{a} = (\ddot{r} - r\dot{\theta}^2)\vec{u}_r + (r\ddot{\theta} + 2\dot{r}\dot{\theta})\vec{u}_{\theta} + \ddot{z}\vec{u}_z$ 11.5. Relative Motion $\vec{r}_B = \vec{r}_A + \vec{r}_{B/A}$ $\vec{v}_B = \vec{v}_A + \vec{v}_{B/A}$ $\vec{a}_B = \vec{a}_A + \vec{a}_{B/A}$ 12. Kinetics of a Particle 12.1. Newton's Second Law $\sum \vec{F} = m\vec{a}$ 12.2. Work and Energy Work of a Force: $U_{1-2} = \int_1^2 \vec{F} \cdot d\vec{r}$ Kinetic Energy: $T = \frac{1}{2}mv^2$ Principle of Work and Energy: $T_1 + U_{1-2} = T_2$ Conservative Forces/Potential Energy: Gravity: $V_g = Wy$ Spring: $V_e = \frac{1}{2}ks^2$ Conservation of Energy: $T_1 + V_1 = T_2 + V_2$ (when only conservative forces do work) 12.3. 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: $\sum (m\vec{v})_1 = \sum (m\vec{v})_2$ (when sum of external impulses is zero) Impact (Coefficient of Restitution): $e = \frac{(v_B)_2 - (v_A)_2}{(v_A)_1 - (v_B)_1}$ 13. Planar Kinematics of a Rigid Body Translation: $\vec{v}_B = \vec{v}_A$, $\vec{a}_B = \vec{a}_A$ Rotation about a Fixed Axis: $\omega = d\theta/dt$, $\alpha = d\omega/dt = \omega d\omega/d\theta$ $v = \omega r$, $a_t = \alpha r$, $a_n = \omega^2 r = v^2/r$ Absolute General Planar Motion: $\vec{v}_P = \vec{v}_O + \vec{\omega} \times \vec{r}_{P/O}$ Relative Motion Analysis: $\vec{v}_B = \vec{v}_A + \vec{\omega} \times \vec{r}_{B/A}$ $\vec{a}_B = \vec{a}_A + \vec{\alpha} \times \vec{r}_{B/A} - \omega^2 \vec{r}_{B/A}$ Instantaneous Center of Zero Velocity (IC): Point where velocity is momentarily zero. Used to find velocities. 14. Planar Kinetics of a Rigid Body Equations of Motion: $\sum F_x = m(\bar{a}_x)$ $\sum F_y = m(\bar{a}_y)$ $\sum M_G = \bar{I}\alpha$ (moment about center of mass G) Alternatively, $\sum M_P = \sum (\mathcal{M}_k)_P$ (moment about point P) Mass Moment of Inertia: $I = \int r^2 dm$ Parallel-Axis Theorem: $I = \bar{I} + md^2$ 14.1. Work and Energy Kinetic Energy: $T = \frac{1}{2}m\bar{v}^2 + \frac{1}{2}\bar{I}\omega^2$ Principle of Work and Energy: $T_1 + U_{1-2} = T_2$ 14.2. Impulse and Momentum Linear Momentum: $m(\vec{v}_G)_1 + \sum \int \vec{F} dt = m(\vec{v}_G)_2$ Angular Momentum: $(\vec{H}_G)_1 + \sum \int \vec{M}_G dt = (\vec{H}_G)_2$ $(\vec{H}_G) = \bar{I}\vec{\omega}$ (for planar motion) 15. Vibrations 15.1. Undamped Free Vibration Equation: $m\ddot{x} + kx = 0$ Natural Frequency: $\omega_n = \sqrt{k/m}$ (rad/s) or $f_n = \frac{1}{2\pi}\sqrt{k/m}$ (Hz) Period: $\tau = 2\pi/\omega_n$ Solution: $x(t) = A \sin(\omega_n t) + B \cos(\omega_n t)$ 15.2. Torsional Vibration Equation: $I\ddot{\theta} + K\theta = 0$ (K is torsional stiffness) Natural Frequency: $\omega_n = \sqrt{K/I}$