Engineering Mechanics (Hibbeler)

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 $m$, then $\vec{F} = m\vec{a}$. Newton's Third Law: The mutual forces of action and reaction between two particles are equal, opposite, and collinear. Gravitational Law (Newton): $F = G \frac{m_1 m_2}{r^2}$, where $G = 66.73 \times 10^{-12} \text{ m}^3/(\text{kg} \cdot \text{s}^2)$. Weight: $W = mg$. On Earth, $g = 9.81 \text{ m/s}^2$ 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). Represented by $\vec{F}$ or $\mathbf{F}$. 2.2. Vector Operations Addition (Parallelogram Law): Resultant $\vec{R} = \vec{A} + \vec{B}$. Tail-to-head method. Subtraction: $\vec{R}' = \vec{A} - \vec{B} = \vec{A} + (-\vec{B})$. Components: $\vec{F} = F_x \hat{i} + F_y \hat{j}$ (2D), $\vec{F} = F_x \hat{i} + F_y \hat{j} + F_z \hat{k}$ (3D). Magnitude: $F = \sqrt{F_x^2 + F_y^2}$ (2D), $F = \sqrt{F_x^2 + F_y^2 + F_z^2}$ (3D). Direction Cosines (3D): $\cos \alpha = F_x/F$, $\cos \beta = F_y/F$, $\cos \gamma = F_z/F$. Note: $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$. Unit Vector: $\vec{u}_F = \vec{F}/F = \frac{F_x}{F}\hat{i} + \frac{F_y}{F}\hat{j} + \frac{F_z}{F}\hat{k}$. 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 of $\vec{A}$ parallel to $\vec{B}$: $\vec{A}_B = (\vec{A} \cdot \vec{u}_B) \vec{u}_B$. 3. Equilibrium of a Particle Condition for Equilibrium: $\sum \vec{F} = \vec{0}$. 2D Equations: $\sum F_x = 0$, $\sum F_y = 0$. 3D Equations: $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$. Free-Body Diagram (FBD): Essential for solving equilibrium problems. Isolate the particle and show all external forces acting on it. 4. Force System Resultants 4.1. Moment of a Force (Torque) Scalar (2D): $M_O = Fd$, where $d$ is the perpendicular distance from $O$ to the line of action of $F$. Convention: CCW (+), CW (-). Vector (3D): $\vec{M}_O = \vec{r} \times \vec{F}$. $\vec{r}$ is position vector from point $O$ to any point on the line of action of $\vec{F}$. Determinant form: $\vec{M}_O = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ r_x & r_y & r_z \\ F_x & F_y & F_z \end{vmatrix}$. Varignon's Theorem: The moment of a force about a point is equal to the sum of the moments of its components about the same point. $\vec{M}_O = \vec{r} \times (\vec{F}_1 + \vec{F}_2) = \vec{r} \times \vec{F}_1 + \vec{r} \times \vec{F}_2$. 4.2. Moment of a Force about an Axis $M_{axis} = \vec{u}_{axis} \cdot (\vec{r} \times \vec{F})$. $\vec{u}_{axis}$ is the unit vector along the axis. $\vec{r}$ is position vector from any point on the axis to any point on the line of action of $\vec{F}$. 4.3. Couple Moment Two parallel forces, equal in magnitude, opposite in direction, separated by a perpendicular distance $d$. $M_c = Fd$ (scalar, 2D). $\vec{M}_c = \vec{r} \times \vec{F}$ (vector, 3D), where $\vec{r}$ connects any point on the line of action of $-\vec{F}$ to any point on the line of action of $\vec{F}$. A couple moment is a free vector (its effect is independent of its point of application). 4.4. Reduction of a Simple Distributed Loading Resultant Force: $F_R = \int w(x) dx = \text{Area under the loading curve}$. Location of Resultant: $\bar{x} = \frac{\int x w(x) dx}{\int w(x) dx}$. For common shapes (rectangle, triangle), this is the centroid of the area. 5. Equilibrium of a Rigid Body Conditions for Equilibrium: $\sum \vec{F} = \vec{0}$ (Translational Equilibrium) $\sum \vec{M}_O = \vec{0}$ (Rotational Equilibrium, about any point $O$) 2D Equations: $\sum F_x = 0$ $\sum F_y = 0$ $\sum M_O = 0$ (scalar moment equation about any point $O$) 3D Equations: $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$ $\sum M_x = 0$, $\sum M_y = 0$, $\sum M_z = 0$ (vector moment equations about any point $O$) Supports and Reactions: Type of Support 2D Reactions 3D Reactions Roller 1 Force ($\perp$ surface) 1 Force ($\perp$ surface) 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 Cable/Rope 1 Force (tension, along cable) 1 Force (tension, along cable) 6. Trusses, Frames, and Machines 6.1. Trusses Members are two-force members (only axial force, tension or compression). Forces applied only at joints. Method of Joints: Apply particle equilibrium ($\sum F_x=0, \sum F_y=0$) at each joint. Start at a joint with at most two unknown member forces. Method of Sections: Cut through the truss, exposing at most three unknown member forces. Apply rigid body equilibrium ($\sum F_x=0, \sum F_y=0, \sum M_O=0$) to the section. 6.2. Frames and Machines Contain multi-force members (subjected to more than two forces). Disassemble the structure into its component members. Draw FBD for each member and/or pin. Apply rigid body equilibrium equations to each part. Action-reaction forces between connected members are equal and opposite. 7. Internal Forces Axial Force (N): Perpendicular to cross-section. Shear Force (V): Tangential to cross-section. Bending Moment (M): Causes bending. Procedure: Find external reactions. Cut the member at the point of interest. Draw FBD of either section. Apply equilibrium equations ($\sum F_x=0, \sum F_y=0, \sum M=0$) to find N, V, M. Sign Convention: Axial: Tension (+), Compression (-). Shear: Up on right face (+), Down on right face (-). Moment: Causes compression in top fibres (+), Causes tension in top fibres (-). Relationships: $dV/dx = w(x)$ (slope of shear diagram = distributed load intensity) $dM/dx = V(x)$ (slope of moment diagram = shear force) 8. Friction Static Friction ($F_s$): Acts to prevent motion. $0 \le F_s \le \mu_s N$. Kinetic Friction ($F_k$): Acts when motion occurs. $F_k = \mu_k N$. $\mu_s$ (coefficient of static friction) is generally greater than $\mu_k$ (coefficient of kinetic friction). Angle of Static Friction: $\phi_s = \tan^{-1}(\mu_s)$. Angle of Repose: The maximum angle of inclination of a surface for an object to remain at rest. 9. Centroids and Moment of Inertia 9.1. Centroid of an Area $\bar{x} = \frac{\int \tilde{x} dA}{\int dA}$, $\bar{y} = \frac{\int \tilde{y} dA}{\int dA}$. For composite areas: $\bar{x} = \frac{\sum \tilde{x}_i A_i}{\sum A_i}$, $\bar{y} = \frac{\sum \tilde{y}_i A_i}{\sum A_i}$. 9.2. Moment of Inertia (Area) $I_x = \int y^2 dA$, $I_y = \int x^2 dA$. Parallel-Axis Theorem: $I_x = \bar{I}_x + A d_y^2$. $I_y = \bar{I}_y + A d_x^2$. $\bar{I}_x, \bar{I}_y$ are moments of inertia about centroidal axes. $d_x, d_y$ are distances from centroidal axes to parallel reference axes. Radius of Gyration: $k = \sqrt{I/A}$. 10. Kinematics of a Particle 10.1. Rectilinear Motion Velocity: $v = ds/dt$. Acceleration: $a = dv/dt = d^2s/dt^2$. Relationship: $a \, ds = v \, dv$. Constant Acceleration ($a_c$): $v = v_0 + a_c t$. $s = s_0 + v_0 t + \frac{1}{2} a_c t^2$. $v^2 = v_0^2 + 2 a_c (s - s_0)$. 10.2. Curvilinear Motion Position Vector: $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$. Velocity: $\vec{v} = d\vec{r}/dt = v_x\hat{i} + v_y\hat{j} + v_z\hat{k}$. Acceleration: $\vec{a} = d\vec{v}/dt = a_x\hat{i} + a_y\hat{j} + a_z\hat{k}$. Rectangular Components: $v_x = \dot{x}$, $a_x = \ddot{x}$, etc. Normal and Tangential Components: $v = \dot{s}$ (always tangent to path). $a_t = \dot{v} = dV/dt$ (tangential acceleration). $a_n = v^2/\rho$ (normal acceleration, always towards center of curvature). $\vec{a} = a_t \hat{u}_t + a_n \hat{u}_n$. $\rho$ is the radius of curvature: $\rho = \frac{[1+(dy/dx)^2]^{3/2}}{|d^2y/dx^2|}$. Cylindrical Components (Polar Coordinates): $\vec{r} = r\hat{u}_r$. $\vec{v} = \dot{r}\hat{u}_r + r\dot{\theta}\hat{u}_\theta$. $\vec{a} = (\ddot{r} - r\dot{\theta}^2)\hat{u}_r + (r\ddot{\theta} + 2\dot{r}\dot{\theta})\hat{u}_\theta$. 11. Kinetics of a Particle 11.1. Newton's Second Law $\sum \vec{F} = m\vec{a}$. Rectangular Components: $\sum F_x = m a_x$, $\sum F_y = m a_y$, $\sum F_z = m a_z$. Normal & Tangential Components: $\sum F_t = m a_t$, $\sum F_n = m a_n = m(v^2/\rho)$. Cylindrical Components: $\sum F_r = m a_r$, $\sum F_\theta = m a_\theta$. 11.2. Work and Energy Work of a Force: $U_{1-2} = \int_{s_1}^{s_2} \vec{F} \cdot d\vec{r}$. Constant force: $U_{1-2} = (F \cos \theta) \Delta s$. Weight: $U_g = -W \Delta y$. Spring: $U_s = -\frac{1}{2}k(s_2^2 - s_1^2)$. Kinetic Energy: $T = \frac{1}{2}mv^2$. Principle of Work and Energy: $T_1 + \sum U_{1-2} = T_2$. Power: $P = dU/dt = \vec{F} \cdot \vec{v}$. Efficiency: $\epsilon = (P_{out}/P_{in}) \times 100\%$. Conservative Forces: Work is independent of path (gravity, spring). 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). 11.3. Impulse and Momentum Linear Momentum: $\vec{L} = m\vec{v}$. Linear Impulse: $\text{Imp}_{1-2} = \int_{t_1}^{t_2} \vec{F} dt$. 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 $\sum \vec{F}_{ext} = \vec{0}$, then $\sum (m\vec{v})_{before} = \sum (m\vec{v})_{after}$. Coefficient of Restitution ($e$): For impact, $e = \frac{(v_B')_n - (v_A')_n}{(v_A)_n - (v_B)_n}$. Elastic impact: $e=1$. Plastic impact: $e=0$. 12. Kinematics of a Rigid Body 12.1. Rotation About a Fixed Axis Angular Velocity: $\omega = d\theta/dt$. Angular Acceleration: $\alpha = d\omega/dt = d^2\theta/dt^2$. Relationship: $\alpha \, d\theta = \omega \, d\omega$. Constant Angular Acceleration ($\alpha_c$): $\omega = \omega_0 + \alpha_c t$. $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha_c t^2$. $\omega^2 = \omega_0^2 + 2 \alpha_c (\theta - \theta_0)$. Velocity of a point P: $\vec{v}_P = \vec{\omega} \times \vec{r}_P$. Magnitude $v_P = \omega r_P$. Acceleration of a point P: $\vec{a}_P = \vec{\alpha} \times \vec{r}_P + \vec{\omega} \times (\vec{\omega} \times \vec{r}_P)$. Tangential: $a_t = \alpha r$. Normal: $a_n = \omega^2 r = v^2/r$. 12.2. General Plane Motion Combination of translation and rotation. Relative Velocity: $\vec{v}_B = \vec{v}_A + \vec{v}_{B/A} = \vec{v}_A + \vec{\omega} \times \vec{r}_{B/A}$. Instantaneous Center (IC) of Zero Velocity: Locate a point on the body that has zero velocity. All other points rotate about the IC. $v = r_{IC} \omega$. Relative Acceleration: $\vec{a}_B = \vec{a}_A + \vec{a}_{B/A} = \vec{a}_A + \vec{\alpha} \times \vec{r}_{B/A} - \omega^2 \vec{r}_{B/A}$. 13. Kinetics of a Rigid Body (Plane Motion) 13.1. Equations of Motion $\sum \vec{F} = m\vec{a}_G$ (translational motion of mass center G). $\sum M_G = I_G \alpha$ (rotational motion about mass center G). $\sum M_O = I_O \alpha$ (if O is a fixed axis of rotation or IC of zero acceleration). $I_G$: mass moment of inertia about G. $I_O$: mass moment of inertia about O. $I_O = I_G + md^2$ (Parallel-Axis Theorem for mass). 13.2. Work and Energy Kinetic Energy: $T = \frac{1}{2} m v_G^2 + \frac{1}{2} I_G \omega^2$. Principle of Work and Energy: $T_1 + \sum U_{1-2} = T_2$. 13.3. Impulse and Momentum Linear Momentum: $m(\vec{v}_G)_1 + \sum \int_{t_1}^{t_2} \vec{F} dt = m(\vec{v}_G)_2$. Angular Momentum about G: $(H_G)_1 + \sum \int_{t_1}^{t_2} M_G dt = (H_G)_2$. $H_G = I_G \omega$. Angular Momentum about O (fixed axis): $(H_O)_1 + \sum \int_{t_1}^{t_2} M_O dt = (H_O)_2$. $H_O = I_O \omega$. 14. Vibrations 14.1. Undamped Free Vibration Equation of Motion: $m\ddot{x} + kx = 0$. Natural Frequency: $\omega_n = \sqrt{k/m}$ (rad/s). Period: $\tau = 2\pi/\omega_n$. Frequency: $f = 1/\tau = \omega_n/(2\pi)$ (Hz). Solution: $x(t) = A \sin(\omega_n t) + B \cos(\omega_n t)$ or $x(t) = C \sin(\omega_n t + \phi)$. 14.2. Damped Free Vibration Equation of Motion: $m\ddot{x} + c\dot{x} + kx = 0$. Damping Ratio: $\zeta = c/(2m\omega_n) = c/(2\sqrt{km})$. Overdamped ($\zeta > 1$): $x(t) = A e^{s_1 t} + B e^{s_2 t}$. Critically Damped ($\zeta = 1$): $x(t) = (A + Bt) e^{-\omega_n t}$. Underdamped ($\zeta $x(t) = e^{-\zeta\omega_n t} (A \sin(\omega_d t) + B \cos(\omega_d t))$. Damped Natural Frequency: $\omega_d = \omega_n \sqrt{1 - \zeta^2}$.