Hibbeler

Cheatsheet Content

1. General Principles Newton's Laws of Motion: 1st Law: A particle originally at rest, or moving in a straight line with constant velocity, tends to remain in this state provided it is not subjected to an unbalanced force. 2nd Law: A particle acted upon by an unbalanced force $\mathbf{F}$ experiences an acceleration $\mathbf{a}$ that has the same direction as $\mathbf{F}$ and a magnitude that is directly proportional to $\mathbf{F}$. If $\mathbf{F}$ is applied to a particle of mass $m$, then $\mathbf{F} = m\mathbf{a}$. 3rd Law: The mutual forces of action and reaction between two particles are equal, opposite, and collinear. Newton's Law of Gravitational Attraction: $F = G \frac{m_1 m_2}{r^2}$ $G = 66.73 \times 10^{-12} \text{ m}^3 / (\text{kg} \cdot \text{s}^2)$ (Universal gravitational constant) Weight: $W = mg$, where $g = 9.81 \text{ m/s}^2$ or $32.2 \text{ ft/s}^2$. Units: SI: mass (kg), length (m), time (s), force (N) US Customary: mass (slug), length (ft), time (s), force (lb) 2. Force Vectors Scalars vs. Vectors: Scalar: Magnitude only (mass, volume, length). Vector: Magnitude and direction (force, velocity, position). Vector Operations: Vector Addition (Parallelogram Law): $\mathbf{R} = \mathbf{A} + \mathbf{B}$ Vector Subtraction: $\mathbf{R'} = \mathbf{A} - \mathbf{B} = \mathbf{A} + (-\mathbf{B})$ Components: $F_x = F \cos \theta$, $F_y = F \sin \theta$ Magnitude: $F = \sqrt{F_x^2 + F_y^2}$ Direction: $\theta = \arctan(F_y/F_x)$ Cartesian Vectors (3D): $\mathbf{F} = F_x \mathbf{i} + F_y \mathbf{j} + F_z \mathbf{k}$ Magnitude: $F = \sqrt{F_x^2 + F_y^2 + F_z^2}$ Direction Cosines: $\cos \alpha = F_x/F$, $\cos \beta = F_y/F$, $\cos \gamma = F_z/F$ Unit Vector: $\mathbf{u}_F = \mathbf{F}/F = \cos \alpha \mathbf{i} + \cos \beta \mathbf{j} + \cos \gamma \mathbf{k}$ $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$ Dot Product: $\mathbf{A} \cdot \mathbf{B} = AB \cos \theta = A_x B_x + A_y B_y + A_z B_z$ Used to find angle between vectors or projection of one vector onto another. Projection of $\mathbf{A}$ onto $\mathbf{B}$: $A_B = \mathbf{A} \cdot \mathbf{u}_B$ Cross Product: $\mathbf{C} = \mathbf{A} \times \mathbf{B}$ Magnitude: $C = AB \sin \theta$ (direction by right-hand rule) Determinant Form: $$ \mathbf{A} \times \mathbf{B} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix} $$ Used to find moment of a force. 3. Equilibrium of a Particle Free-Body Diagram (FBD): Essential for solving equilibrium problems. Isolates the particle and shows all external forces acting on it. Equations of Equilibrium: 2D: $\sum F_x = 0$, $\sum F_y = 0$ 3D: $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$ Spring Force: $F_s = ks$ $k$: spring stiffness (N/m or lb/ft) $s$: deformation (length stretched or compressed) 4. Force System Resultants Moment of a Force: Measures the tendency of a force to rotate a body about a point or axis. Scalar (2D): $M_O = Fd$ (where $d$ is perpendicular distance from $O$ to force line of action). Use sign convention (e.g., CCW positive). Vector (3D): $\mathbf{M}_O = \mathbf{r} \times \mathbf{F}$ (where $\mathbf{r}$ is position vector from $O$ to any point on line of action of $\mathbf{F}$). 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. $\mathbf{M}_O = \mathbf{r} \times (\mathbf{F}_1 + \mathbf{F}_2) = \mathbf{r} \times \mathbf{F}_1 + \mathbf{r} \times \mathbf{F}_2$. Moment of a Force about an Axis: $M_{u} = \mathbf{u} \cdot (\mathbf{r} \times \mathbf{F})$ (where $\mathbf{u}$ is unit vector along the axis). Couple Moment: Two parallel forces with the same magnitude, opposite direction, and separated by a perpendicular distance $d$. $M = Fd$. Always a free vector. Resultant of a Force System: Equivalent force $\mathbf{F}_R = \sum \mathbf{F}$. Equivalent moment $\mathbf{M}_R = \sum \mathbf{M} + \sum (\mathbf{r} \times \mathbf{F})$. 5. Equilibrium of a Rigid Body Free-Body Diagram (FBD): Crucial. Show all external forces and moments. Equations of Equilibrium: 2D: $\sum F_x = 0$, $\sum F_y = 0$, $\sum M_O = 0$ 3D: $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$, $\sum M_x = 0$, $\sum M_y = 0$, $\sum M_z = 0$ Supports and Reactions: Type of Support 2D Reactions 3D Reactions Cable/Rope 1 force (tension) 1 force (tension) Roller 1 force $\perp$ surface 1 force $\perp$ surface Smooth Pin/Hinge 2 forces ($F_x, F_y$) 3 forces ($F_x, F_y, F_z$) Fixed Support 2 forces ($F_x, F_y$) & 1 moment ($M_z$) 3 forces ($F_x, F_y, F_z$) & 3 moments ($M_x, M_y, M_z$) Two-Force Members: If a body is subjected to only two forces, these forces must be collinear, equal in magnitude, and opposite in direction. Three-Force Members: If a body is subjected to only three forces, they must be concurrent or parallel. 6. Trusses, Frames, and Machines Trusses: Members are two-force members (only axial force). Method of Joints: Apply particle equilibrium ($\sum F_x = 0, \sum F_y = 0$) at each joint. Method of Sections: Cut through members and apply rigid body equilibrium ($\sum F_x = 0, \sum F_y = 0, \sum M = 0$) to a section. Zero-Force Members: Identify to simplify analysis. Frames & Machines: Contain multi-force members. Disassemble the structure into its component parts. Draw FBD for each part. Apply rigid body equilibrium to each part. Action-reaction forces between connected parts are equal and opposite. 7. Internal Forces Internal Loadings: Normal force ($N$), Shear force ($V$), Bending moment ($M$). Cut the member at the point of interest. Draw FBD of either segment. Apply equilibrium equations to find $N, V, M$. Shear and Moment Diagrams: $dV/dx = w(x)$ (slope of shear diagram = distributed load intensity) $dM/dx = V(x)$ (slope of moment diagram = shear force) $\Delta V = \int w(x) dx$ (change in shear = area under load diagram) $\Delta M = \int V(x) dx$ (change in moment = area under shear diagram) 8. Friction Static Friction: $F_s \le \mu_s N$ $F_s$: static friction force $\mu_s$: coefficient of static friction $N$: normal force Impending motion: $F_s = \mu_s N$ Kinetic Friction: $F_k = \mu_k N$ $F_k$: kinetic friction force $\mu_k$: coefficient of kinetic friction ($\mu_k Angle of Static Friction: $\tan \phi_s = \mu_s$ Wedges, Screws, Bearings, Belts: Apply friction principles. 9. Center of Gravity and Centroid Center of Gravity (CG): Point where the entire weight of a body can be considered to act. $\bar{x} = \frac{\sum \tilde{x} W}{\sum W}$, $\bar{y} = \frac{\sum \tilde{y} W}{\sum W}$, $\bar{z} = \frac{\sum \tilde{z} W}{\sum W}$ Centroid: Geometric center of an area or volume. Area: $\bar{x} = \frac{\int \tilde{x} dA}{\int dA}$, $\bar{y} = \frac{\int \tilde{y} dA}{\int dA}$ For composite areas/volumes: $\bar{x} = \frac{\sum \tilde{x} A}{\sum A}$, $\bar{y} = \frac{\sum \tilde{y} A}{\sum A}$ Theorems of Pappus-Guldinus: Surface Area: $A = \theta \bar{L} L$ (revolving a line) Volume: $V = \theta \bar{A} A$ (revolving an area) 10. Moments of Inertia Area Moment of Inertia: Measure of an area's resistance to bending. $I_x = \int y^2 dA$, $I_y = \int x^2 dA$ Polar Moment of Inertia: $J_O = \int r^2 dA = I_x + I_y$ Parallel-Axis Theorem: Used to find moment of inertia about an axis parallel to a centroidal axis. $I = \bar{I} + Ad^2$ $\bar{I}$: moment of inertia about centroidal axis $A$: area $d$: perpendicular distance between parallel axes Radius of Gyration: $k = \sqrt{I/A}$ Mass Moment of Inertia: Measure of a body's resistance to angular acceleration. $I = \int r^2 dm$ Parallel-Axis Theorem: $I = \bar{I} + md^2$ 11. Virtual Work Principle of Virtual Work: For a body in equilibrium, the virtual work done by all external forces and couple moments acting on the body is zero for any virtual displacement consistent with the body's constraints. $\delta U = \sum F \cos \theta \delta s + \sum M \delta \theta = 0$ Conservative Forces & Potential Energy: Elastic potential energy: $V_e = \frac{1}{2} k s^2$ Gravitational potential energy: $V_g = W y$ Total potential energy: $V = V_g + V_e$ Equilibrium: $\frac{dV}{ds} = 0$ Stability of Equilibrium: Stable: $\frac{d^2 V}{ds^2} > 0$ (minimum potential energy) Unstable: $\frac{d^2 V}{ds^2} Neutral: $\frac{d^2 V}{ds^2} = 0$ (constant potential energy) 12. 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: Rectangular Components: $\mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}$, $\mathbf{v} = \dot{x}\mathbf{i} + \dot{y}\mathbf{j} + \dot{z}\mathbf{k}$, $\mathbf{a} = \ddot{x}\mathbf{i} + \ddot{y}\mathbf{j} + \ddot{z}\mathbf{k}$ Normal and Tangential Components: $\mathbf{v} = v \mathbf{u}_t$ $\mathbf{a} = a_t \mathbf{u}_t + a_n \mathbf{u}_n$ $a_t = \dot{v}$, $a_n = v^2/\rho$ ($\rho$ is radius of curvature) Cylindrical Components: $\mathbf{r} = r\mathbf{u}_r + z\mathbf{k}$ $\mathbf{v} = \dot{r}\mathbf{u}_r + r\dot{\theta}\mathbf{u}_\theta + \dot{z}\mathbf{k}$ $\mathbf{a} = (\ddot{r} - r\dot{\theta}^2)\mathbf{u}_r + (r\ddot{\theta} + 2\dot{r}\dot{\theta})\mathbf{u}_\theta + \ddot{z}\mathbf{k}$ 13. Kinetics of a Particle Equation of Motion: $\sum \mathbf{F} = m\mathbf{a}$ 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: $\sum F_r = m a_r$, $\sum F_\theta = m a_\theta$, $\sum F_z = m a_z$ Work and Energy: Work of a Force: $U_{1-2} = \int \mathbf{F} \cdot d\mathbf{r}$ Kinetic Energy: $T = \frac{1}{2} m v^2$ Principle of Work and Energy: $T_1 + U_{1-2} = T_2$ Power: $P = \mathbf{F} \cdot \mathbf{v} = Fv \cos \theta$ Efficiency: $\epsilon = P_{out}/P_{in}$ Impulse and Momentum: Linear Momentum: $\mathbf{L} = m\mathbf{v}$ Principle of Linear Impulse and Momentum: $m(\mathbf{v}_1) + \sum \int_{t_1}^{t_2} \mathbf{F} dt = m(\mathbf{v}_2)$ Conservation of Linear Momentum: If $\sum \mathbf{F} = 0$, then $\sum m\mathbf{v}_1 = \sum m\mathbf{v}_2$. Impact: Coefficient of Restitution: $e = \frac{(v_B)_2 - (v_A)_2}{(v_A)_1 - (v_B)_1}$ $e=1$: elastic impact; $e=0$: plastic impact. 14. Planar Kinematics of a Rigid Body Types of Motion: Translation: $\mathbf{v}_A = \mathbf{v}_B$, $\mathbf{a}_A = \mathbf{a}_B$ Rotation about a Fixed Axis: $\omega = d\theta/dt$, $\alpha = d\omega/dt = \omega (d\omega/d\theta)$ $v = \omega r$, $a_t = \alpha r$, $a_n = \omega^2 r = v^2/r$ General Plane Motion: Combination of translation and rotation. Relative Motion Analysis: Velocity: $\mathbf{v}_B = \mathbf{v}_A + \mathbf{v}_{B/A}$ (where $\mathbf{v}_{B/A} = \mathbf{\omega} \times \mathbf{r}_{B/A}$) Acceleration: $\mathbf{a}_B = \mathbf{a}_A + \mathbf{a}_{B/A}$ (where $\mathbf{a}_{B/A} = \mathbf{\alpha} \times \mathbf{r}_{B/A} - \omega^2 \mathbf{r}_{B/A}$) Instantaneous Center of Zero Velocity (IC): For general plane motion, a point exists where velocity is momentarily zero. Useful for velocity analysis. 15. 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, where $\sum (\mathcal{M}_k)_P$ is the sum of moments of kinetic terms) Kinetic Energy (Plane Motion): $T = \frac{1}{2} m \bar{v}^2 + \frac{1}{2} \bar{I} \omega^2$ Work and Energy: $T_1 + U_{1-2} = T_2$ Angular Momentum: $H_G = \bar{I}\omega$ Impulse and Momentum: Linear: $m(\bar{\mathbf{v}}_1) + \sum \int \mathbf{F} dt = m(\bar{\mathbf{v}}_2)$ Angular: $\bar{I}\omega_1 + \sum \int M_G dt = \bar{I}\omega_2$