Hibbeler Mechanics Cheatsheet

Cheatsheet Content

### Vectors and Forces - **Scalar:** Quantity with magnitude only (mass, length, time). - **Vector:** Quantity with magnitude and direction (force, velocity, acceleration). - **Vector Representation:** - Cartesian: $\vec{F} = F_x \hat{i} + F_y \hat{j} + F_z \hat{k}$ - Magnitude: $|\vec{F}| = \sqrt{F_x^2 + F_y^2 + F_z^2}$ - Unit Vector: $\hat{u}_F = \frac{\vec{F}}{|\vec{F}|}$ - **Dot 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. - **Cross Product:** $\vec{A} \times \vec{B} = (A_y B_z - A_z B_y)\hat{i} + (A_z B_x - A_x B_z)\hat{j} + (A_x B_y - A_y B_x)\hat{k}$ - Magnitude: $|\vec{A} \times \vec{B}| = |\vec{A}||\vec{B}|\sin\theta$ - Direction: Right-hand rule. Used to find moment of a force. ### Equilibrium of a Particle - **Definition:** A particle is in equilibrium if the resultant force acting on it is zero. - **Newton's First Law:** $\sum \vec{F} = 0$ - **Equations of Equilibrium:** - 2D: $\sum F_x = 0$, $\sum F_y = 0$ - 3D: $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$ - **Free-Body Diagram (FBD):** Essential for solving equilibrium problems. Isolate the particle and show all external forces acting on it. ### Moment of a Force - **Definition:** The turning effect of a force about a point or axis. - **Scalar Formulation (2D):** $M_O = Fd$ (where $d$ is perpendicular distance from $O$ to force line of action). - Sign convention (e.g., counter-clockwise positive). - **Vector Formulation (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 $\vec{F}$. - **Principle of Transmissibility:** A force can be moved along its line of action without changing its external effect on a rigid body. - **Varignon's Theorem:** The moment of a resultant force about a point is equal to the sum of the moments of its components about the same point. - $\vec{M}_O = \vec{r} \times (\vec{F}_1 + \vec{F}_2) = (\vec{r} \times \vec{F}_1) + (\vec{r} \times \vec{F}_2)$ ### Couples - **Definition:** Two parallel forces of equal magnitude and opposite direction, separated by a perpendicular distance $d$. - **Moment of a Couple:** $M = Fd$ - A couple produces only rotation, no translation. - The moment of a couple is a free vector (its effect is the same regardless of where it is applied). ### Equivalent Systems - **Resultant Force:** $\vec{R} = \sum \vec{F}$ - **Resultant Moment:** $\vec{M}_R = \sum \vec{M}_O + \sum (\vec{r} \times \vec{F})$ - **Reduction to a Force-Couple System:** Any system of forces and couples acting on a rigid body can be reduced to an equivalent resultant force $\vec{R}$ acting at a specified point $O$ and a resultant couple moment $\vec{M}_R$. ### Equilibrium of a Rigid Body - **Definition:** A rigid body is in equilibrium if the resultant force and resultant couple moment acting on it are both zero. - **Equations of Equilibrium:** - 2D: $\sum F_x = 0$, $\sum F_y = 0$, $\sum M_O = 0$ - 3D: $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$, $\sum M_x = 0$, $\sum M_y = 0$, $\sum M_z = 0$ - **Support Reactions:** | Type of Support | 2D Reactions | 3D Reactions | |-----------------|--------------|--------------| | Roller | $F \perp$ sur. | $F \perp$ sur. | | Pin | $F_x, F_y$ | $F_x, F_y, F_z, M_x, M_y$ (no $M_z$) | | Fixed | $F_x, F_y, M$| $F_x, F_y, F_z, M_x, M_y, M_z$ | - **Two-Force Member:** A member subjected to forces only at two points. The forces must be equal, opposite, and collinear. - **Three-Force Member:** A member subjected to forces only at three points. The lines of action of the three forces must be concurrent or parallel. ### Trusses - **Assumptions:** - Members are joined by smooth pins. - All loads are applied at the joints. - Members are two-force members (axial force only). - **Method of Joints:** 1. Draw FBD of entire truss to find support reactions. 2. Draw FBD for each joint. 3. Apply $\sum F_x = 0$ and $\sum F_y = 0$ to find forces in members. 4. Start with joints having few unknown forces (max 2 unknowns). - **Method of Sections:** 1. Draw FBD of entire truss to find support reactions. 2. Cut the truss through the members whose forces are to be determined (max 3 unknowns). 3. Draw FBD of one section. 4. Apply $\sum F_x = 0$, $\sum F_y = 0$, $\sum M = 0$ to solve for forces. - **Zero-Force Members:** - If only two non-collinear members meet at an unloaded joint, both are zero-force members. - If three members meet at an unloaded joint, and two are collinear, the third is a zero-force member. ### Frames and Machines - **Frames:** Structures designed to support loads, usually stationary and fully constrained. - **Machines:** Structures designed to transmit and modify forces, usually containing moving parts. - **Analysis Procedure:** 1. Draw FBD of the entire frame/machine (if possible) to find external reactions. 2. Disassemble the frame/machine into its component parts. 3. Draw FBD for each component, showing all internal and external forces. Remember Newton's Third Law (action-reaction pairs are equal and opposite). 4. Apply equilibrium equations ($\sum F_x = 0, \sum F_y = 0, \sum M = 0$) to each component to solve for unknown forces. ### Internal Forces - **Procedure (Method of Sections):** 1. Cut the member at the desired location. 2. Draw FBD of either segment. 3. Apply equilibrium equations to find: - **Normal Force ($N$):** Perpendicular to the cut section. - **Shear Force ($V$):** Parallel to the cut section. - **Bending Moment ($M$):** Moment about the centroid of the cut section. - **Sign Convention:** - **Normal Force:** Tension is positive. - **Shear Force:** Positive if it tends to rotate the segment clockwise. - **Bending Moment:** Positive if it causes compression in the top fibers and tension in the bottom fibers (or smiley face). ### Friction - **Static Friction ($F_s$):** Opposes impending motion. - $F_s \le \mu_s N$ (where $\mu_s$ is coefficient of static friction, $N$ is normal force). - Maximum static friction: $(F_s)_{max} = \mu_s N$. - **Kinetic Friction ($F_k$):** Opposes relative motion when sliding occurs. - $F_k = \mu_k N$ (where $\mu_k$ is coefficient of kinetic friction). - Generally, $\mu_k ### Center of Gravity and Centroid - **Center of Gravity (CG):** Point where the entire weight of a body appears to act. - $\bar{x} = \frac{\sum W_i x_i}{\sum W_i}$, $\bar{y} = \frac{\sum W_i y_i}{\sum W_i}$, $\bar{z} = \frac{\sum W_i z_i}{\sum W_i}$ - **Centroid:** Geometric center of an area or volume. - For Area: $\bar{x} = \frac{\int x dA}{\int dA}$, $\bar{y} = \frac{\int y dA}{\int dA}$ - For Composite Areas/Volumes: $\bar{x} = \frac{\sum \bar{x}_i A_i}{\sum A_i}$ (similar for $y, z$ and volumes). - **Theorems of Pappus and Guldinus:** - **Area of surface of revolution:** $A = \theta \bar{r} L$ (where $L$ is length of curve, $\bar{r}$ is centroidal distance, $\theta$ is angle of revolution in radians). - **Volume of body of revolution:** $V = \theta \bar{r} A$ (where $A$ is area of region, $\bar{r}$ is centroidal distance, $\theta$ is angle of revolution in radians). ### Moments of Inertia - **Area Moment of Inertia:** Measures resistance to bending/buckling. - $I_x = \int y^2 dA$, $I_y = \int x^2 dA$ - **Polar Moment of Inertia:** $J_O = I_x + I_y = \int r^2 dA$ - **Parallel-Axis Theorem:** $I = I_c + Ad^2$ - $I_c$ is moment of inertia about centroidal axis. - $A$ is area. - $d$ is perpendicular distance between parallel axes. - **Radius of Gyration:** $k = \sqrt{\frac{I}{A}}$ - Represents the distance from an axis at which the entire area could be concentrated to yield the same moment of inertia. - **Mass Moment of Inertia:** Measures resistance to angular acceleration. - $I = \int r^2 dm$ - For composite bodies: $I = \sum (I_c + md^2)$