Hibbeler

Cheatsheet Content

1. Fundamental Concepts Newton's Laws of Motion: Particle remains at rest or moves with constant velocity if no net force acts on it. $F = ma$ (Net force = mass $\times$ acceleration) Action-reaction: Forces between two particles are equal and opposite. Units: SI: mass (kg), length (m), time (s), force (N). $1 \text{ N} = 1 \text{ kg} \cdot \text{m/s}^2$ US Customary: force (lb), length (ft), time (s), mass (slug). $1 \text{ slug} = 1 \text{ lb} \cdot \text{s}^2/\text{ft}$ Scalars & Vectors: Scalar: Magnitude only (e.g., mass, time, energy). Vector: Magnitude and direction (e.g., force, velocity, acceleration). 2. Vector Operations Vector Addition (Parallelogram Law): $\vec{R} = \vec{A} + \vec{B}$ Vector Subtraction: $\vec{R}' = \vec{A} - \vec{B} = \vec{A} + (-\vec{B})$ Resolution into Components: 2D: $\vec{F} = F_x \hat{i} + F_y \hat{j}$ where $F_x = F \cos \theta$, $F_y = F \sin \theta$ 3D: $\vec{F} = F_x \hat{i} + F_y \hat{j} + F_z \hat{k}$ Direction Cosines: $\cos \alpha = F_x/F$, $\cos \beta = F_y/F$, $\cos \gamma = F_z/F$ $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$ Dot Product (Scalar 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 angle between vectors or projection of one vector onto another. Projection of $\vec{A}$ onto $\vec{B}$: $A_B = \vec{A} \cdot \hat{u}_B = \vec{A} \cdot (\vec{B}/|\vec{B}|)$ Cross Product (Vector Product): $\vec{C} = \vec{A} \times \vec{B}$ Magnitude: $|\vec{C}| = |\vec{A}| |\vec{B}| \sin \theta$ Direction: Right-hand rule, $\vec{C}$ is perpendicular to plane of $\vec{A}$ and $\vec{B}$. Cartesian Form: $\vec{A} \times \vec{B} = (A_y B_z - A_z B_y)\hat{i} - (A_x B_z - A_z B_x)\hat{j} + (A_x B_y - A_y B_x)\hat{k}$ Used for calculating moments. 3. Equilibrium of a Particle Condition: Net force is zero. $\sum \vec{F} = 0$ 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 identifying all forces acting on the particle. 4. Force System Resultants Moment of a Force about a Point O: $\vec{M}_O = \vec{r} \times \vec{F}$ Scalar (2D): $M_O = Fd$ (d = perpendicular distance from O to line of action of F). Vector (3D): Use cross product. Moment of a Force about an Axis: $M_{axis} = \hat{u}_{axis} \cdot (\vec{r} \times \vec{F})$ Couple Moment: $\vec{M} = \vec{r} \times \vec{F}$ (for two equal, opposite, non-collinear forces) Magnitude: $M = Fd$ (d = perpendicular distance between forces). Resultant of a Force System: $\vec{F}_R = \sum \vec{F}$ $\vec{M}_{R_O} = \sum (\vec{r} \times \vec{F}) + \sum \vec{M}_{couples}$ A force system can be reduced to a single resultant force $\vec{F}_R$ and a resultant couple moment $\vec{M}_{R_O}$ at an arbitrary point O. 5. Equilibrium of a Rigid Body Conditions: $\sum \vec{F} = 0$ (Sum of forces is zero) $\sum \vec{M}_O = 0$ (Sum of moments about any point O is zero) 2D Equations: $\sum F_x = 0$, $\sum F_y = 0$, $\sum M_O = 0$ 3D Equations: $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$, $\sum M_x = 0$, $\sum M_y = 0$, $\sum M_z = 0$ Support Reactions: Type of Support Reactions (2D) Roller 1 force $\perp$ surface Pin/Hinge 2 force components ($F_x, F_y$) Fixed Support 2 force components ($F_x, F_y$) + 1 moment ($M_z$) Cable/Rope 1 tensile force along cable 6. Trusses & Frames Trusses: Members are two-force members (tension or compression). Forces act only at joints. Method of Joints: Apply particle equilibrium ($\sum F_x=0, \sum F_y=0$) at each joint. Method of Sections: Cut the truss, apply rigid body equilibrium ($\sum F_x=0, \sum F_y=0, \sum M=0$) to a section. Zero-Force Members: Two non-collinear members connected at an unloaded joint $\implies$ both are zero-force. Three members, two collinear, one non-collinear at an unloaded joint $\implies$ non-collinear is zero-force. Frames & Machines: Members are generally multi-force members. Disassemble the structure into its component parts. Draw FBD for each component and for the overall structure. Apply rigid body equilibrium to each part. 7. Centroid & Center of Gravity Centroid of Area: $(\bar{x}, \bar{y}, \bar{z})$ $\bar{x} = \frac{\int x dA}{\int dA}$, $\bar{y} = \frac{\int y dA}{\int dA}$, $\bar{z} = \frac{\int z dA}{\int dA}$ Composite Bodies: $\bar{X} = \frac{\sum \bar{x}_i A_i}{\sum A_i}$, etc. Center of Gravity: $(\bar{x}, \bar{y}, \bar{z})$ (for mass/weight) $\bar{x} = \frac{\int x dW}{\int dW}$, $\bar{y} = \frac{\int y dW}{\int dW}$, $\bar{z} = \frac{\int z dW}{\int dW}$ For homogeneous material, centroid of volume/area/length = center of gravity. Theorems of Pappus-Guldinus: Area of surface of revolution: $A = \theta \bar{r} L$ ($\theta$ in radians) Volume of body of revolution: $V = \theta \bar{r} A$ ($\theta$ in radians) 8. 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: $I_x = \bar{I}_x + A d_y^2$ ($d_y$ is distance from centroidal x-axis to parallel x-axis) $I_y = \bar{I}_y + A d_x^2$ $J_O = \bar{J}_C + A d^2$ Radius of Gyration: $k = \sqrt{I/A}$ Product of Inertia: $I_{xy} = \int xy dA$ Parallel-Axis Theorem: $I_{xy} = \bar{I}_{xy} + A \bar{x} \bar{y}$ Principal Moments of Inertia: $\tan 2\theta_p = \frac{-I_{xy}}{(I_x - I_y)/2}$ $I_{max/min} = \frac{I_x + I_y}{2} \pm \sqrt{\left(\frac{I_x - I_y}{2}\right)^2 + I_{xy}^2}$ 9. Friction Static Friction: $F_s \le \mu_s N$ (N is normal force) Maximum static friction: $F_{s,max} = \mu_s N$ (at impending motion) Kinetic Friction: $F_k = \mu_k N$ (when motion occurs) $\mu_k Angle of Friction: $\phi_s = \tan^{-1}(\mu_s)$ Wedges: Often involve friction on multiple surfaces. Draw FBDs for each component. Belts: $T_2 = T_1 e^{\mu_s \beta}$ ($\beta$ in radians) 10. Virtual Work Principle of Virtual Work: $\delta U = 0$ for equilibrium. $\delta U = \sum F \delta s + \sum M \delta \theta = 0$ Virtual displacement $\delta s$ and virtual rotation $\delta \theta$ are infinitesimal, imaginary displacements consistent with constraints. Conservative Forces: Work done is independent of path. (e.g., gravity, spring force) Potential Energy: $V = V_g + V_e$ Gravitational: $V_g = W y$ Elastic (Spring): $V_e = \frac{1}{2} k s^2$ Stability of Equilibrium: Stable: $V$ is minimum. $\frac{d^2V}{ds^2} > 0$ Unstable: $V$ is maximum. $\frac{d^2V}{ds^2} Neutral: $V$ is constant. $\frac{d^2V}{ds^2} = 0$ 11. Kinematics of a Particle Rectilinear Motion: Velocity: $v = ds/dt$ Acceleration: $a = dv/dt = v dv/ds$ Constant Acceleration: $v = v_0 + a_c t$ $s = s_0 + v_0 t + \frac{1}{2} a_c t^2$ $v^2 = v_0^2 + 2 a_c (s - s_0)$ Curvilinear Motion: Position: $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ Velocity: $\vec{v} = d\vec{r}/dt = v_x\hat{i} + v_y\hat{j} + v_z\hat{k}$ Acceleration: $\vec{a} = d\vec{v}/dt = a_x\hat{i} + a_y\hat{j} + a_z\hat{k}$ Projectile Motion: (constant $a_y = -g$, $a_x = 0$) $v_x = (v_0)_x$ $x = x_0 + (v_0)_x t$ $v_y = (v_0)_y - gt$ $y = y_0 + (v_0)_y t - \frac{1}{2} gt^2$ Normal and Tangential Components: $\vec{v} = v \hat{u}_t$ $\vec{a} = a_t \hat{u}_t + a_n \hat{u}_n$ $a_t = dv/dt$ (change in speed) $a_n = v^2/\rho$ (change in direction, $\rho$ = radius of curvature) Cylindrical Components: $\vec{r} = r \hat{u}_r$ $\vec{v} = \dot{r} \hat{u}_r + r \dot{\theta} \hat{u}_{\theta}$ $\vec{a} = (\ddot{r} - r \dot{\theta}^2) \hat{u}_r + (r \ddot{\theta} + 2 \dot{r} \dot{\theta}) \hat{u}_{\theta}$ 12. Kinetics of a Particle Newton's Second Law: $\sum \vec{F} = m\vec{a}$ Equations of Motion: 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: $\sum F_r = m a_r$, $\sum F_{\theta} = m a_{\theta}$, $\sum F_z = m a_z$ Work and Energy: Work of a Force: $U_{1-2} = \int_1^2 \vec{F} \cdot d\vec{r}$ Kinetic Energy: $T = \frac{1}{2} m v^2$ Principle of Work and Energy: $T_1 + \sum U_{1-2} = T_2$ Conservation of Energy (for conservative forces): $T_1 + V_1 = T_2 + V_2$ Impulse and Momentum: Linear Momentum: $\vec{L} = m\vec{v}$ Linear Impulse: $\text{Imp}_{1-2} = \int_1^2 \vec{F} dt$ Principle of Linear Impulse-Momentum: $m\vec{v}_1 + \sum \int_1^2 \vec{F} dt = m\vec{v}_2$ Conservation of Linear Momentum (if $\sum \vec{F}_{ext} = 0$): $m_A \vec{v}_{A1} + m_B \vec{v}_{B1} = m_A \vec{v}_{A2} + m_B \vec{v}_{B2}$ Impact: Coefficient of Restitution: $e = \frac{(v_B)_2 - (v_A)_2}{(v_A)_1 - (v_B)_1}$ (along line of impact) Elastic: $e=1$, Plastic: $e=0$ Angular Momentum about Point O: $(\vec{H}_O)_1 + \sum \int_1^2 \vec{M}_O dt = (\vec{H}_O)_2$ For a particle: $\vec{H}_O = \vec{r} \times m\vec{v}$ 13. Kinematics of a Rigid Body Types of Motion: Translation: All particles have same velocity and acceleration. Rotation about a Fixed Axis: $\omega = d\theta/dt$, $\alpha = d\omega/dt = \omega d\omega/d\theta$ $v = \omega r$ $a_t = \alpha r$, $a_n = \omega^2 r$ General Plane Motion: Translation + Rotation. $\vec{v}_B = \vec{v}_A + \vec{\omega} \times \vec{r}_{B/A}$ $\vec{a}_B = \vec{a}_A + \vec{\alpha} \times \vec{r}_{B/A} - \omega^2 \vec{r}_{B/A}$ Instantaneous Center (IC) of Zero Velocity: Point where $\vec{v}=0$ for general plane motion. Used to find velocities. 14. Kinetics of a Rigid Body Equations of Motion: $\sum \vec{F} = m\vec{a}_G$ $\sum \vec{M}_G = I_G \vec{\alpha}$ (moment about center of mass G) For rotation about a fixed axis O: $\sum \vec{M}_O = I_O \vec{\alpha}$ Mass Moment of Inertia: $I = \int r^2 dm$ Parallel-Axis Theorem: $I = I_G + m d^2$ Work and Energy: Kinetic Energy: $T = \frac{1}{2} m v_G^2 + \frac{1}{2} I_G \omega^2$ For fixed axis rotation O: $T = \frac{1}{2} I_O \omega^2$ Principle of Work and Energy: $T_1 + \sum U_{1-2} = T_2$ Impulse and Momentum: Linear: $m(\vec{v}_G)_1 + \sum \int \vec{F} dt = m(\vec{v}_G)_2$ Angular (about G): $(I_G \vec{\omega})_1 + \sum \int \vec{M}_G dt = (I_G \vec{\omega})_2$ Angular (about fixed point O): $(I_O \vec{\omega})_1 + \sum \int \vec{M}_O dt = (I_O \vec{\omega})_2$ Conservation of Momentum: If net impulse is zero, momentum is conserved.