Hibbeler Engineering Mechanics

Cheatsheet Content

1. General Principles Newton's First Law: A particle originally at rest, or moving in a straight line with constant velocity, will 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 $\vec{F}$ and a magnitude that is directly proportional to $\vec{F}$. If $\vec{F}$ is applied to a particle of mass $m$, this law may be expressed as $\vec{F} = m\vec{a}$. 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 $G = 6.673 \times 10^{-11} \text{ m}^3/(\text{kg} \cdot \text{s}^2)$. Weight: $W = mg$, where $g = 9.81 \text{ m/s}^2$ or $32.2 \text{ ft/s}^2$. 2. Force Vectors 2.1. Scalar & Vector Quantities Scalar: Magnitude only (e.g., mass, length, time). Vector: Magnitude and direction (e.g., force, velocity, acceleration). 2.2. Vector Operations Vector Addition (Parallelogram Law): $\vec{R} = \vec{A} + \vec{B}$. Resultant is diagonal of parallelogram formed by $\vec{A}$ and $\vec{B}$. Vector Subtraction: $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$. Resolution of a Vector: $\vec{F} = F_x \hat{i} + F_y \hat{j}$ (2D), $\vec{F} = F_x \hat{i} + F_y \hat{j} + F_z \hat{k}$ (3D). Magnitude: $F = \sqrt{F_x^2 + F_y^2 + F_z^2}$. Direction Cosines: $\cos \alpha = F_x/F$, $\cos \beta = F_y/F$, $\cos \gamma = F_z/F$. Note: $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$. Unit Vector: $\vec{u}_F = \frac{\vec{F}}{F} = \cos \alpha \hat{i} + \cos \beta \hat{j} + \cos \gamma \hat{k}$. 2.3. Dot Product $\vec{A} \cdot \vec{B} = AB \cos \theta$. $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z$. Angle between two vectors: $\theta = \cos^{-1}\left(\frac{\vec{A} \cdot \vec{B}}{AB}\right)$. Projection of $\vec{A}$ onto $\vec{B}$: $A_B = \vec{A} \cdot \vec{u}_B$. Vector component: $\vec{A}_B = (\vec{A} \cdot \vec{u}_B)\vec{u}_B$. 3. Equilibrium of a Particle Free-Body Diagram (FBD): Essential for solving equilibrium problems. Isolate the particle and show all external forces acting on it. Equations of Equilibrium: 2D: $\sum F_x = 0$, $\sum F_y = 0$. 3D: $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$. 4. Force System Resultants 4.1. Moment of a Force (Scalar Formulation) $M_O = Fd$, where $d$ is the perpendicular distance from $O$ to the line of action of $F$. Right-Hand Rule: Curl fingers in direction of rotation, thumb points in direction of moment vector. 4.2. Moment of a Force (Vector Formulation) $\vec{M}_O = \vec{r} \times \vec{F}$, where $\vec{r}$ is a position vector from $O$ to any point on the line of action of $\vec{F}$. Determinant form: $$ \vec{M}_O = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ r_x & r_y & r_z \\ F_x & F_y & F_z \end{vmatrix} $$ 4.3. Moment of a Force about a Specified Axis $M_L = \vec{u}_L \cdot (\vec{r} \times \vec{F})$, where $\vec{u}_L$ is the unit vector along the axis $L$. 4.4. Couple Moment $\vec{M} = \vec{r} \times \vec{F}$ (for two equal, opposite, and non-collinear forces). Magnitude $M = Fd$. 4.5. Resultants 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 5.1. Types of Supports & Reactions Support Type Reactions Roller Force $\perp$ surface Pin/Hinge Two force components Fixed Support Two force components, one moment Cable/Rope Tension along cable 5.2. Equations of Equilibrium 2D (Coplanar): $\sum F_x = 0$, $\sum F_y = 0$, $\sum M_O = 0$. (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) 5.3. Two-Force Members If a rigid body is subjected to only two forces, they must be equal, opposite, and collinear. 5.4. Three-Force Members If a rigid body is subjected to only three forces, they must be concurrent or parallel. 6. Trusses, Frames, and Machines 6.1. Trusses Assumptions: Members are joined by pins, loads applied at joints, members are two-force members. 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 section. Zero-Force Members: If only two non-collinear members connect at a joint and no external load/reaction, both are zero-force. If three members connect at a joint, two are collinear, and no external load/reaction, the third is zero-force. 6.2. Frames and Machines Components are generally multi-force members. Disassemble the structure into its component parts. Apply equilibrium equations to each part, considering action-reaction forces at connections. 7. Internal Forces Normal Force ($N$): Perpendicular to cross-section. Shear Force ($V$): Tangent to cross-section. Bending Moment ($M$): Moment about the centroidal axis of cross-section. Sign Convention: Axial: Tension positive. Shear: Up on right face, down on left face positive. Moment: Causing compression at top fiber (concave down) positive. Relations: $\frac{dV}{dx} = -w(x)$ (distributed load) $\frac{dM}{dx} = V(x)$ 8. Friction Static Friction: $F_s \le \mu_s N$. Max static friction $F_{s,max} = \mu_s N$. Kinetic Friction: $F_k = \mu_k N$. Note: $\mu_k Angle of Friction: $\tan \phi_s = \mu_s$. 9. Center of Gravity and Centroid Discrete Particles: $\bar{r} = \frac{\sum \tilde{r} W}{\sum W}$ or $\bar{x} = \frac{\sum \tilde{x} W}{\sum W}$, etc. Areas: $\bar{x} = \frac{\int \tilde{x} dA}{\int dA}$, $\bar{y} = \frac{\int \tilde{y} dA}{\int dA}$. For composite areas: $\bar{x} = \frac{\sum \tilde{x} A}{\sum A}$, etc. Volumes: $\bar{x} = \frac{\int \tilde{x} dV}{\int dV}$, etc. For composite volumes: $\bar{x} = \frac{\sum \tilde{x} V}{\sum V}$, etc. 10. Moments of Inertia 10.1. 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: $I = \bar{I} + Ad^2$, where $\bar{I}$ is moment of inertia about centroidal axis, $A$ is area, $d$ is distance between parallel axes. 10.2. Mass Moment of Inertia $I = \int r^2 dm$. Parallel-Axis Theorem: $I = \bar{I} + md^2$. 11. Kinematics of a Particle 11.1. Rectilinear Motion $v = \frac{ds}{dt}$ $a = \frac{dv}{dt} = \frac{d^2s}{dt^2}$ $a \, ds = v \, dv$ Constant Acceleration: $v = v_0 + a_c t$ $s = s_0 + v_0 t + \frac{1}{2} a_c t^2$ $v^2 = v_0^2 + 2 a_c (s - s_0)$ 11.2. 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: $\vec{v} = v \vec{u}_t$ $\vec{a} = a_t \vec{u}_t + a_n \vec{u}_n$, where $a_t = \dot{v}$ and $a_n = \frac{v^2}{\rho}$. ($\rho$ is radius of curvature) Cylindrical Components: $\vec{r} = r\vec{u}_r + z\vec{u}_z$ $\vec{v} = \dot{r}\vec{u}_r + r\dot{\theta}\vec{u}_\theta + \dot{z}\vec{u}_z$ $\vec{a} = (\ddot{r} - r\dot{\theta}^2)\vec{u}_r + (r\ddot{\theta} + 2\dot{r}\dot{\theta})\vec{u}_\theta + \ddot{z}\vec{u}_z$ 12. Kinetics of a Particle 12.1. 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: $\sum F_t = m a_t$, $\sum F_n = m a_n$. Cylindrical: $\sum F_r = m a_r$, $\sum F_\theta = m a_\theta$, $\sum F_z = m a_z$. 12.2. Work and Energy Work of a Force: $U_{1-2} = \int_{s_1}^{s_2} F \cos \theta \, ds$. Work of a Weight: $U_{1-2} = -W \Delta y$. Work of a Spring: $U_{1-2} = \frac{1}{2} k (s_1^2 - s_2^2)$. Principle of Work and Energy: $T_1 + U_{1-2} = T_2$. Kinetic energy $T = \frac{1}{2} m v^2$. Conservative Forces: Gravity, spring force. Non-conservative: Friction, drag. Conservation of Energy: $T_1 + V_1 = T_2 + V_2$. Gravitational Potential Energy: $V_g = Wy$. Elastic Potential Energy: $V_e = \frac{1}{2} k s^2$. 12.3. Impulse and Momentum Linear Impulse: $\int_{t_1}^{t_2} \vec{F} dt$. Linear Momentum: $\vec{L} = m\vec{v}$. Principle of Linear Impulse and Momentum: $m\vec{v}_1 + \sum \int_{t_1}^{t_2} \vec{F} dt = m\vec{v}_2$. Conservation of Linear Momentum: If $\sum \int \vec{F} dt = 0$, then $m\vec{v}_1 = m\vec{v}_2$. Impact: Coefficient of Restitution: $e = \frac{(v_B)_2 - (v_A)_2}{(v_A)_1 - (v_B)_1}$. $e=1$ (elastic), $e=0$ (plastic). Angular Impulse: $\int_{t_1}^{t_2} \vec{M}_O dt$. Angular Momentum: $(\vec{H}_O)_1 = \vec{r}_1 \times m\vec{v}_1$. 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: If $\sum \int \vec{M}_O dt = 0$, then $(\vec{H}_O)_1 = (\vec{H}_O)_2$. 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: $\omega = \frac{d\theta}{dt}$, $\alpha = \frac{d\omega}{dt}$. $\alpha \, d\theta = \omega \, d\omega$. $v = r\omega$, $a_t = r\alpha$, $a_n = r\omega^2 = v^2/r$. General Plane Motion (Relative Velocity): $\vec{v}_B = \vec{v}_A + \vec{\omega} \times \vec{r}_{B/A}$. General Plane Motion (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): For finding velocities in general plane motion. $\vec{v}_{IC} = 0$. 14. Planar Kinetics of a Rigid Body Equations of Motion: $\sum F_x = m(\bar{a}_x)$ $\sum F_y = m(\bar{a}_y)$ $\sum M_G = \bar{I}\alpha$ (Sum of moments about mass center $G$) Alternatively, $\sum M_P = \sum (\mathcal{M}_k)_P$ (Sum of moments about any point $P$) Work and Energy: $T_1 + V_1 + U_{1-2}^{\text{nc}} = T_2 + V_2$. Kinetic Energy: $T = \frac{1}{2} m \bar{v}^2 + \frac{1}{2} \bar{I} \omega^2$. For rotation about fixed axis O: $T = \frac{1}{2} I_O \omega^2$. Impulse and Momentum: Linear: $m(\vec{v}_G)_1 + \sum \int \vec{F} dt = m(\vec{v}_G)_2$. Angular: $(\vec{H}_G)_1 + \sum \int \vec{M}_G dt = (\vec{H}_G)_2$. (About mass center $G$) For rotation about fixed axis O: $(I_O \omega)_1 + \sum \int M_O dt = (I_O \omega)_2$. 15. Vibrations 15.1. Undamped Free Vibration Equation: $m\ddot{x} + kx = 0$. Natural Frequency: $\omega_n = \sqrt{k/m}$ (rad/s). Period: $\tau = 2\pi/\omega_n$. Frequency: $f = 1/\tau = \omega_n/(2\pi)$. Solution: $x(t) = A \sin(\omega_n t) + B \cos(\omega_n t)$. 15.2. Damped Free Vibration Equation: $m\ddot{x} + c\dot{x} + kx = 0$. Damping Ratio: $\zeta = \frac{c}{c_c} = \frac{c}{2m\omega_n}$. Natural Frequency (damped): $\omega_d = \omega_n \sqrt{1 - \zeta^2}$. Solutions depend on $\zeta$: $\zeta $\zeta = 1$ (critically damped): Fastest return to equilibrium without oscillation. $\zeta > 1$ (overdamped): Slow return to equilibrium without oscillation.