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 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. Newton's Law of Gravitational Attraction: $F = G \frac{m_1 m_2}{r^2}$, 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. Vector Operations Scalar Multiplication: $B\vec{A} = (BA_x)\hat{i} + (BA_y)\hat{j} + (BA_z)\hat{k}$ Vector Addition (Parallelogram Law): $\vec{R} = \vec{A} + \vec{B}$ Vector Subtraction: $\vec{R}' = \vec{A} - \vec{B} = \vec{A} + (-\vec{B})$ 2.2. Cartesian Vectors Unit Vector: $\vec{u}_A = \frac{\vec{A}}{|\vec{A}|} = \frac{A_x}{A}\hat{i} + \frac{A_y}{A}\hat{j} + \frac{A_z}{A}\hat{k}$ Position Vector: $\vec{r} = (x_2-x_1)\hat{i} + (y_2-y_1)\hat{j} + (z_2-z_1)\hat{k}$ Force Vector along a Line: $\vec{F} = F \vec{u} = F \left( \frac{\vec{r}}{|\vec{r}|} \right)$ 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: $\theta = \cos^{-1}\left( \frac{\vec{A} \cdot \vec{B}}{|\vec{A}| |\vec{B}|} \right)$ Projection of $\vec{A}$ onto $\vec{B}$: $A_B = \vec{A} \cdot \vec{u}_B = |\vec{A}| \cos\theta$ 2.4. Cross Product $\vec{C} = \vec{A} \times \vec{B} = (A_yB_z - A_zB_y)\hat{i} - (A_xB_z - A_zB_x)\hat{j} + (A_xB_y - A_yB_x)\hat{k}$ $|\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}| \sin\theta$ 3. Equilibrium of a Particle Condition for Equilibrium: $\sum \vec{F} = 0$ Cartesian Components: $\sum F_x = 0$ $\sum F_y = 0$ $\sum F_z = 0$ Free-Body Diagram (FBD): Essential for solving equilibrium problems. Show all external forces acting on the particle. 4. Force System Resultants 4.1. Moment of a Force Scalar Formulation (2D): $M_O = Fd$ (magnitude), direction by right-hand rule. Vector Formulation: $\vec{M}_O = \vec{r} \times \vec{F}$ $\vec{r}$ is position vector from point $O$ to any point on line of action of $\vec{F}$. Varignon's Theorem: The moment of a force about a point is equal to the sum of the moments of its components about the 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 Couple A couple consists of two parallel forces that are equal in magnitude, opposite in direction, and separated by a perpendicular distance $d$. Moment of a Couple: $M = Fd$ (scalar), $\vec{M} = \vec{r} \times \vec{F}$ (vector) 4.3. Resultant of a Force and Couple System Resultant Force: $\vec{F}_R = \sum \vec{F}$ Resultant Moment: $\vec{M}_R = \sum \vec{M}_O + \sum \vec{M}_{couple}$ 5. Equilibrium of a Rigid Body Conditions for Equilibrium: $\sum \vec{F} = 0 \implies \sum F_x = 0, \sum F_y = 0, \sum F_z = 0$ $\sum \vec{M}_O = 0 \implies \sum M_x = 0, \sum M_y = 0, \sum M_z = 0$ Support 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 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 ($M_x, M_y, M_z$) 6. Trusses, Frames, and Machines 6.1. Trusses Assumptions: Members are connected by pins, 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 to expose internal forces. Apply rigid body equilibrium ($\sum F_x=0, \sum F_y=0, \sum M=0$) to the section. Zero-Force Members: If only two non-collinear members connect at a joint and no external load or reaction acts at the joint, both members are zero-force members. If three members connect at a joint, two of which are collinear, and no external load or reaction acts at the joint, the third member is a zero-force member. 6.2. Frames and Machines Can have multi-force members (members with forces applied at more than two points). Break down into individual components and apply equilibrium equations to each part. Internal forces between connected parts are equal and opposite (Newton's 3rd Law). 7. Internal Forces Axial Force ($N$): Acts perpendicular to the section. Shear Force ($V$): Acts in the plane of the section. Bending Moment ($M$): Tends to bend the member. Sign Convention (for beams): Axial: Tension positive. Shear: Upward on left face positive. Moment: Causing compression in top fiber (sagging) positive. Relations between Load, Shear, and Moment: $\frac{dV}{dx} = -w(x)$ (Negative of distributed load) $\frac{dM}{dx} = V(x)$ 8. Friction Static Friction: $F_s \le \mu_s N$ (Maximum static friction: $F_{s,max} = \mu_s N$) Kinetic Friction: $F_k = \mu_k N$ (when sliding occurs) $\mu_s > \mu_k$ Angle of Static Friction: $\phi_s = \tan^{-1}(\mu_s)$ Angle of Kinetic Friction: $\phi_k = \tan^{-1}(\mu_k)$ Wedges and Screws: Analyze using FBDs and equilibrium equations, considering friction forces. 9. Center of Gravity and Centroid Center of Gravity (CG): Point where the entire weight of a body can be considered to act. $\bar{x} = \frac{\int x dW}{\int dW}$, $\bar{y} = \frac{\int y dW}{\int dW}$, $\bar{z} = \frac{\int z dW}{\int dW}$ Centroid: Geometric center of an area or volume. Area: $\bar{x} = \frac{\int x dA}{A}$, $\bar{y} = \frac{\int y dA}{A}$ Composite Bodies: $\bar{X} = \frac{\sum \bar{x}_i A_i}{\sum A_i}$, $\bar{Y} = \frac{\sum \bar{y}_i A_i}{\sum A_i}$ Theorems of Pappus and Guldinus: Area of Revolution: $A = \theta \bar{r} L$ (for a line rotated) Volume of Revolution: $V = \theta \bar{r} A$ (for an area rotated) 10. Moment of Inertia Moment of Inertia of Area: $I_x = \int y^2 dA$ $I_y = \int x^2 dA$ Polar Moment of Inertia: $J_O = I_x + I_y = \int r^2 dA$ Parallel-Axis Theorem: $I_x = \bar{I}_{x'} + Ad_y^2$ $I_y = \bar{I}_{y'} + Ad_x^2$ $J_O = \bar{J}_C + Ad^2$ Radius of Gyration: $k = \sqrt{\frac{I}{A}}$ Moment of Inertia of Mass: $I = \int r^2 dm$ Parallel-Axis Theorem (mass): $I = \bar{I} + md^2$