Hibbeler Mechanics Cheatsheet

Cheatsheet Content

### Vectors and Forces - **Scalar:** Quantity with magnitude only (e.g., mass, time, length). - **Vector:** Quantity with both magnitude and direction (e.g., 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:** $\vec{C} = \vec{A} \times \vec{B}$ - Magnitude: $|\vec{C}| = |\vec{A}||\vec{B}|\sin\theta$ - Direction: Right-hand rule. - Cartesian: $(A_y B_z - A_z B_y)\hat{i} - (A_x B_z - A_z B_x)\hat{j} + (A_x B_y - A_y B_x)\hat{k}$ - **Force Systems:** - **Concurrent:** Lines of action intersect at a single point. - **Coplanar:** All forces lie in the same plane. - **Resultant Force:** $\vec{F}_R = \sum \vec{F}$ ### 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. - Show all external forces acting on it (known and unknown). - Label coordinate system. ### Moment and Couples - **Moment of a Force (Torque):** 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). - Vector (3D): $\vec{M}_O = \vec{r} \times \vec{F}$ (where $\vec{r}$ is position vector from $O$ to any point on 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. - $\vec{M}_O = \vec{r} \times \vec{F}_R = \vec{r} \times (\vec{F}_1 + \vec{F}_2 + ...) = (\vec{r} \times \vec{F}_1) + (\vec{r} \times \vec{F}_2) + ...$ - **Couple:** Two parallel forces of the same magnitude but opposite direction, separated by a perpendicular distance $d$. - Resultant force is zero. - Produces a pure rotational effect (moment). - Magnitude of couple moment: $M = Fd$. - Vector: $\vec{M} = \vec{r} \times \vec{F}$ (where $\vec{r}$ connects any point on one force to any point on the other). ### Equilibrium of a Rigid Body - **Definition:** A rigid body is in equilibrium if it is not accelerating (both translation and rotation). - **Equations of Equilibrium:** - $\sum \vec{F} = 0 \Rightarrow \sum F_x = 0, \sum F_y = 0, \sum F_z = 0$ (Translational Equilibrium) - $\sum \vec{M}_O = 0 \Rightarrow \sum M_x = 0, \sum M_y = 0, \sum M_z = 0$ (Rotational Equilibrium) - **Support Reactions:** Understand common support types and their associated reactions: - **Roller:** One force (perpendicular to surface). - **Pin/Hinge:** Two force components (x and y). - **Fixed Support:** Three force components (x, y) and one moment (z). - **Smooth Surface:** One force (normal to surface). - **Rough Surface:** One normal force, one friction force. - **Two-Force Member:** A member subjected to forces at only two points; forces must be equal, opposite, and collinear. - **Three-Force Member:** A member subjected to forces at only three points; lines of action must be concurrent or parallel. ### Trusses, Frames, and Machines - **Truss:** Structure composed of slender members joined at their ends. - Members are typically two-force members (axial force only). - Assumed pin-connected, loads applied at joints. - **Methods of Analysis:** - **Method of Joints:** Apply particle equilibrium ($\sum F_x=0, \sum F_y=0$) at each joint. - **Method of Sections:** Cut through members, apply rigid body equilibrium ($\sum F_x=0, \sum F_y=0, \sum M=0$) to a section. - **Frames and Machines:** Structures with at least one multi-force member. - Can contain two-force members, but also members with forces along their length or moments. - Components are typically analyzed by disassembling the structure and drawing FBDs for each part. - Apply rigid body equilibrium to each component. ### Centroids and Moments of Inertia - **Centroid (Geometric Center):** - For area A: $\bar{x} = \frac{\int x dA}{\int dA}$, $\bar{y} = \frac{\int y dA}{\int dA}$ - For composite areas: $\bar{x} = \frac{\sum \bar{x}_i A_i}{\sum A_i}$, $\bar{y} = \frac{\sum \bar{y}_i A_i}{\sum A_i}$ - **Moment of Inertia (Second Moment of Area):** Measures an area's resistance to bending or buckling. - $I_x = \int y^2 dA$, $I_y = \int x^2 dA$ - **Parallel-Axis Theorem:** $I = I_c + Ad^2$ - $I_c$: Moment of inertia about the centroidal axis. - $A$: Area. - $d$: Perpendicular distance between the parallel axes. - **Radius of Gyration:** $k = \sqrt{I/A}$ - **Product of Inertia:** $I_{xy} = \int xy dA$ ### Friction - **Static Friction ($F_s$):** Opposes impending motion. $0 \le F_s \le \mu_s N$ - **Kinetic Friction ($F_k$):** Opposes motion. $F_k = \mu_k N$ - **Coefficient of Static Friction ($\mu_s$):** Ratio of max static friction to normal force. - **Coefficient of Kinetic Friction ($\mu_k$):** Ratio of kinetic friction to normal force ($\mu_k ### Kinematics of a Particle (Rectilinear Motion) - **Position:** $s(t)$ - **Velocity:** $v = \frac{ds}{dt}$ - **Acceleration:** $a = \frac{dv}{dt} = \frac{d^2s}{dt^2}$ - **Constant Acceleration Formulas:** - $v = v_0 + a_c t$ - $s = s_0 + v_0 t + \frac{1}{2} a_c t^2$ - $v^2 = v_0^2 + 2 a_c (s - s_0)$ - **Dependent Motion:** Relate motion of one particle to another using constraint equations for cable lengths. ### Kinematics of a Particle (Curvilinear Motion) - **Rectangular Components (x, y, z):** - Position: $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ - Velocity: $\vec{v} = \dot{x}\hat{i} + \dot{y}\hat{j} + \dot{z}\hat{k}$ - Acceleration: $\vec{a} = \ddot{x}\hat{i} + \ddot{y}\hat{j} + \ddot{z}\hat{k}$ - **Normal and Tangential Components (n, t):** - Velocity: $\vec{v} = v \hat{u}_t$ - Acceleration: $\vec{a} = a_t \hat{u}_t + a_n \hat{u}_n$ - Tangential: $a_t = \dot{v}$ (change in speed) - Normal: $a_n = \frac{v^2}{\rho}$ (change in direction, $\rho$ is radius of curvature) - **Cylindrical Components (r, $\theta$, z):** - Position: $\vec{r} = r\hat{u}_r + z\hat{k}$ - Velocity: $\vec{v} = \dot{r}\hat{u}_r + r\dot{\theta}\hat{u}_{\theta} + \dot{z}\hat{k}$ - Acceleration: $\vec{a} = (\ddot{r} - r\dot{\theta}^2)\hat{u}_r + (r\ddot{\theta} + 2\dot{r}\dot{\theta})\hat{u}_{\theta} + \ddot{z}\hat{k}$ ### Kinetics of a Particle (Newton's Second Law) - **Newton's Second Law:** $\sum \vec{F} = m\vec{a}$ - **Rectangular:** $\sum F_x = ma_x$, $\sum F_y = ma_y$, $\sum F_z = ma_z$ - **Normal-Tangential:** $\sum F_t = ma_t$, $\sum F_n = ma_n = m\frac{v^2}{\rho}$ - **Cylindrical:** $\sum F_r = ma_r$, $\sum F_{\theta} = ma_{\theta}$, $\sum F_z = ma_z$ - **Equation of Motion:** Apply Newton's Second Law to FBDs. - **Centripetal Force:** Force causing normal acceleration, $F_n = m\frac{v^2}{\rho}$. ### Work and Energy of a Particle - **Work of a Force:** $U = \int \vec{F} \cdot d\vec{r}$ - For constant force: $U = (F\cos\theta)s$ - Work of 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:** $P = \frac{dU}{dt} = \vec{F} \cdot \vec{v}$ - **Conservative Forces:** Work done is independent of path (gravity, spring). - **Gravitational Potential Energy:** $V_g = W y$ - **Elastic Potential Energy:** $V_e = \frac{1}{2} k s^2$ - **Conservation of Energy:** $T_1 + V_1 = T_2 + V_2$ (for conservative systems) ### Impulse and Momentum of a Particle - **Linear Momentum:** $\vec{L} = m\vec{v}$ - **Linear Impulse:** $\vec{I} = \int_{t_1}^{t_2} \vec{F} dt$ - **Principle of Linear Impulse and Momentum:** $m\vec{v}_1 + \sum \int_{t_1}^{t_2} \vec{F} dt = m\vec{v}_2$ - Or $\vec{L}_1 + \sum \vec{I}_{1-2} = \vec{L}_2$ - **Conservation of Linear Momentum:** $\sum m\vec{v}_1 = \sum m\vec{v}_2$ (when no external impulse or $\sum \vec{F} = 0$) - **Impact (Collisions):** - **Coefficient of Restitution (e):** $e = \frac{(v_B)_2 - (v_A)_2}{(v_A)_1 - (v_B)_1}$ - $e=1$: Perfectly elastic (conservation of kinetic energy). - $e=0$: Perfectly plastic (bodies stick together). - **Central Impact:** Initial and final velocities are along the line of impact. - **Oblique Impact:** Velocities are not along the line of impact. Momentum conserved perpendicular to line of impact. ### Kinematics of a Rigid Body - **Translation:** All points move along parallel paths. - Rectilinear: Straight paths. - Curvilinear: Curved paths. - **Rotation about a Fixed Axis:** - Angular Position: $\theta$ - Angular Velocity: $\omega = \frac{d\theta}{dt}$ - Angular Acceleration: $\alpha = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}$ - Constant Angular Acceleration: - $\omega = \omega_0 + \alpha_c t$ - $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha_c t^2$ - $\omega^2 = \omega_0^2 + 2\alpha_c (\theta - \theta_0)$ - Velocity of point P: $v = \omega r$ - Acceleration of point P: $a_t = \alpha r$, $a_n = \omega^2 r$ - **General Plane Motion:** Translation + Rotation. - **Relative Velocity:** $\vec{v}_B = \vec{v}_A + \vec{v}_{B/A} = \vec{v}_A + \vec{\omega} \times \vec{r}_{B/A}$ - **Relative Acceleration:** $\vec{a}_B = \vec{a}_A + \vec{a}_{B/A} = \vec{a}_A + \vec{\alpha} \times \vec{r}_{B/A} - \omega^2 \vec{r}_{B/A}$ - **Instantaneous Center (IC) of Zero Velocity:** Point on the body (or extension) that has zero velocity at a given instant. - Used to find velocities of other points: $v = \omega r_{IC}$. ### Mass Moment of Inertia - **Definition:** Measure of a body's resistance to angular acceleration. - For a rigid body: $I = \int r^2 dm$ - **Parallel-Axis Theorem:** $I = I_G + md^2$ - $I_G$: Mass moment of inertia about the centroidal axis. - $m$: Mass. - $d$: Perpendicular distance between parallel axes. - **Radius of Gyration:** $k = \sqrt{I/m}$ ### Kinetics of a Rigid Body - **Equations of Motion (Plane Motion):** - $\sum F_x = m(\bar{a}_x)$ - $\sum F_y = m(\bar{a}_y)$ - $\sum M_G = I_G \alpha$ (Moment about center of mass G) - Alternatively, $\sum M_P = I_P \alpha$ if P is a fixed axis of rotation. - **Work and Energy (Rigid Body):** - **Translational Kinetic Energy:** $T = \frac{1}{2}m v_G^2$ - **Rotational Kinetic Energy:** $T = \frac{1}{2}I_G \omega^2$ - **Total Kinetic Energy (General Plane Motion):** $T = \frac{1}{2}m v_G^2 + \frac{1}{2}I_G \omega^2$ - **Principle of Work and Energy:** $T_1 + \sum U_{1-2} = T_2$ - **Impulse and Momentum (Rigid Body):** - **Linear Impulse and Momentum:** $m(\vec{v}_G)_1 + \sum \int \vec{F} dt = m(\vec{v}_G)_2$ - **Angular Impulse and Momentum:** $(H_G)_1 + \sum \int M_G dt = (H_G)_2$ - Where angular momentum $H_G = I_G \omega$ - **Conservation of Angular Momentum:** $\sum (H_G)_1 = \sum (H_G)_2$ (if $\sum M_G = 0$)