Hibbeler Mechanics

Cheatsheet Content

### Vectors #### Scalar & Vector Quantities - **Scalar:** Magnitude only (e.g., mass, length, time) - **Vector:** Magnitude and direction (e.g., force, velocity, acceleration) #### Vector Operations - **Addition (Parallelogram Law):** $\vec{R} = \vec{A} + \vec{B}$ - Head-to-tail or parallelogram method - **Subtraction:** $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$ - **Components:** $\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$ - Magnitude: $|\vec{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}$ - Direction Cosines: $\cos\alpha = A_x/|\vec{A}|$, $\cos\beta = A_y/|\vec{A}|$, $\cos\gamma = A_z/|\vec{A}|$ - **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. - If $\vec{A} \cdot \vec{B} = 0$, then $\vec{A} \perp \vec{B}$ (if $\vec{A}, \vec{B} \neq 0$) - **Cross Product:** $\vec{C} = \vec{A} \times \vec{B} = (A_y B_z - A_z B_y)\hat{i} - (A_x B_z - A_z B_x)\hat{j} + (A_x B_y - A_y B_x)\hat{k}$ - Magnitude: $|\vec{A} \times \vec{B}| = |\vec{A}||\vec{B}|\sin\theta$ - Direction: Right-hand rule, perpendicular to plane of $\vec{A}$ and $\vec{B}$. - Used to calculate moment of a force. - If $\vec{A} \times \vec{B} = 0$, then $\vec{A} \parallel \vec{B}$ (if $\vec{A}, \vec{B} \neq 0$) ### Force Systems #### Concurrent Forces - All forces pass through a single point. - **Equilibrium:** $\sum \vec{F} = \vec{0}$ - In 2D: $\sum F_x = 0$, $\sum F_y = 0$ - In 3D: $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$ #### Moment of a Force - **Scalar (2D):** $M_O = Fd$ (Perpendicular distance $d$) - Sign convention: Counter-clockwise (+), Clockwise (-) - **Vector (3D):** $\vec{M}_O = \vec{r} \times \vec{F}$ - $\vec{r}$ is position vector from point O to any point on line of action of $\vec{F}$. - **Varignon's Theorem:** Moment of a resultant force equals sum of moments of its components. - $\vec{M}_O = \vec{r} \times (\vec{F}_1 + \vec{F}_2) = (\vec{r} \times \vec{F}_1) + (\vec{r} \times \vec{F}_2)$ #### Couple Moment - Two parallel forces, equal magnitude, opposite direction, separated by distance $d$. - **Magnitude:** $M = Fd$ - **Vector:** $\vec{M} = \vec{r} \times \vec{F}$ (where $\vec{r}$ connects the line of action of one force to the other) - A couple moment is a free vector (can be moved anywhere without changing its effect). #### Resultants of Force Systems - **Equivalent Force-Couple System:** Any system of forces and moments can be reduced to a single resultant force $\vec{R}$ acting at a point O and a resultant couple moment $\vec{M}_R_O$. - $\vec{R} = \sum \vec{F}$ - $\vec{M}_R_O = \sum \vec{M}_O + \sum (\vec{r} \times \vec{F})$ - **Wrench:** If $\vec{R}$ and $\vec{M}_R$ are collinear, the system reduces to a wrench. - $M_R = (\vec{u} \cdot \vec{M}_R) \vec{u}$ where $\vec{u}$ is unit vector in direction of $\vec{R}$. #### Distributed Loads - **Concentrated Force Equivalence:** Area under the distributed load curve. - **Location:** Centroid of the area under the distributed load curve. - For a rectangular load $w_0$: $F_R = w_0 L$, acts at $L/2$. - For a triangular load $w_0$: $F_R = \frac{1}{2} w_0 L$, acts at $L/3$ from the base. ### Equilibrium of a Rigid Body #### Free-Body Diagrams (FBDs) - Isolate the body. - Show all external forces and moments acting on the body. - Include reactions from supports (see table below). #### Equations of Equilibrium - **2D:** - $\sum F_x = 0$ - $\sum F_y = 0$ - $\sum M_O = 0$ (moment about any arbitrary point O) - **3D:** - $\sum F_x = 0$ - $\sum F_y = 0$ - $\sum F_z = 0$ - $\sum M_x = 0$ - $\sum M_y = 0$ - $\sum M_z = 0$ (or $\sum \vec{M}_O = \vec{0}$) #### Common 2D Supports and Reactions | Type of Support | Reaction | |---|---| | Roller | Force $\perp$ surface | | Pin (Hinge) | Two force components (x, y) | | Fixed Support | Two force components (x, y) & a moment | | Smooth Surface | Force $\perp$ surface | | Rocker | Force $\perp$ surface | | Cable | Force along cable (tension) | #### Common 3D Supports and Reactions | Type of Support | Reaction | |---|---| | Ball & Socket | Three force components (x, y, z) | | Smooth Journal Bearing | Two force components ($\perp$ shaft) | | Thrust Bearing | Two force components ($\perp$ shaft) & one force component (along shaft) | | Fixed Support | Three force components (x, y, z) & three moments (x, y, z) | | Hinge | Two force components ($\perp$ pin) & two moments ($\perp$ pin) | | Cable | Force along cable (tension) | ### Trusses and Frames #### Trusses - Members are slender and connected by pins at their ends. - All forces are axial (tension or compression). - **Assumptions:** 1. Loads applied only at joints. 2. Members are connected by smooth pins. - **Methods of Analysis:** - **Method of Joints:** Apply $\sum F_x = 0$, $\sum F_y = 0$ to each joint. - Start at a joint with 2 unknown member forces. - **Method of Sections:** Cut through members to expose internal forces. - Apply $\sum F_x = 0$, $\sum F_y = 0$, $\sum M_O = 0$ to isolated section. - Choose section to isolate desired forces, often 3 unknowns. - **Zero-Force Members:** 1. Two non-collinear members at an unloaded joint: Both are zero-force. 2. Three members at an unloaded joint, two of which are collinear: The third member is zero-force. #### Frames and Machines - Structures with at least one multi-force member (member with 3+ forces or forces not acting at ends). - **Analysis:** Disassemble into individual members and apply equilibrium equations to each part. - Identify two-force members (forces are axial, equal and opposite). - Act-react pairs: Forces between connected members are equal and opposite. - Pin reactions at connecting points are unknowns. ### Internal Forces #### Shear Force and Bending Moment - **Axial Force (N):** Normal to cross-section (tension/compression). - **Shear Force (V):** Tangential to cross-section. - **Bending Moment (M):** Moment about centroidal axis. #### Sign Convention (for beams) - **Axial:** Tension (+) - **Shear:** Up on right face (+), Down on right face (-) - **Moment:** Compression at top (+), Tension at top (-) (or smiley face) #### Shear and Moment Diagrams - **Relations:** - $\frac{dV}{dx} = -w(x)$ (slope of shear diagram = negative distributed load) - $\frac{dM}{dx} = V(x)$ (slope of moment diagram = shear force) - $\Delta V = -\int w(x) dx$ (change in shear = negative area under load curve) - $\Delta M = \int V(x) dx$ (change in moment = area under shear curve) - **Procedure:** 1. Calculate reactions. 2. Cut beam at arbitrary sections and apply equilibrium to find $V(x)$ and $M(x)$. 3. Plot $V$ and $M$ vs. $x$. 4. Note discontinuities at concentrated forces/moments. ### Friction #### Dry Friction (Coulomb Friction) - **Static Friction ($F_s$):** Opposes impending motion. - $0 \le F_s \le \mu_s N$ - $\mu_s$: coefficient of static friction. - $N$: normal force. - $F_s^{max} = \mu_s N$ (occurs at impending motion). - **Kinetic Friction ($F_k$):** Opposes actual motion. - $F_k = \mu_k N$ - $\mu_k$: coefficient of kinetic friction. ($\mu_k \theta_L$, screw is self-locking) - If $\phi_s ### Center of Gravity & Centroid #### Center of Gravity (CG) - Point where the entire weight of a body can be considered to act. - $W \bar{x} = \sum W_i x_i$ - $W \bar{y} = \sum W_i y_i$ - $W \bar{z} = \sum W_i z_i$ #### Centroid - Geometric center of an area or volume. - **Area:** - $\bar{x} = \frac{\int x dA}{\int dA}$ - $\bar{y} = \frac{\int y dA}{\int dA}$ - **Composite Bodies/Areas:** - $\bar{x} = \frac{\sum \bar{x}_i A_i}{\sum A_i}$ - $\bar{y} = \frac{\sum \bar{y}_i A_i}{\sum A_i}$ - Similarly for volumes and lines. #### Theorems of Pappus and Guldinus - **Area of surface of revolution:** $A = \theta \bar{r} L$ - $\theta$: angle of revolution (radians), $\bar{r}$: distance from axis of revolution to centroid of generating line, $L$: length of generating line. - **Volume of body of revolution:** $V = \theta \bar{r} A$ - $\theta$: angle of revolution (radians), $\bar{r}$: distance from axis of revolution to centroid of generating area, $A$: area of generating area. ### Moments of Inertia #### Area Moment of Inertia - Measure of an area's resistance to bending or buckling. - **Rectangular:** - $I_x = \int y^2 dA$ - $I_y = \int x^2 dA$ - **Polar Moment of Inertia ($J_O$):** - $J_O = \int r^2 dA = \int (x^2+y^2) dA = I_x + I_y$ #### Parallel-Axis Theorem - $I = \bar{I} + Ad^2$ - $I$: moment of inertia about parallel axis. - $\bar{I}$: moment of inertia about centroidal axis. - $A$: area. - $d$: perpendicular distance between the two parallel axes. #### Radius of Gyration ($k$) - $I = k^2 A \implies k = \sqrt{I/A}$ - Represents the distance from an axis at which the entire area can be concentrated to yield the same moment of inertia. #### Product of Inertia ($I_{xy}$) - $I_{xy} = \int xy dA$ - If an axis is an axis of symmetry, then $I_{xy}$ for that axis is zero. - Parallel-Axis Theorem for Product of Inertia: $I_{xy} = \bar{I}_{xy} + \bar{x}\bar{y}A$ #### Principal Moments of Inertia - Maximum and minimum moments of inertia for an area. - Occur at principal axes, which are rotated by an angle $\theta_p$. - $\tan(2\theta_p) = \frac{-I_{xy}}{ (I_x - I_y)/2 }$ - $I_{max/min} = \frac{I_x + I_y}{2} \pm \sqrt{\left(\frac{I_x - I_y}{2}\right)^2 + I_{xy}^2}$ - At principal axes, the product of inertia $I_{x'y'}$ is zero. ### Kinematics of a Particle #### Rectilinear Motion (straight line) - **Position:** $s(t)$ - **Velocity:** $v = \frac{ds}{dt}$ - **Acceleration:** $a = \frac{dv}{dt} = \frac{d^2s}{dt^2}$ - **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 (x, y, z):** - $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ - $\vec{v} = \dot{x}\hat{i} + \dot{y}\hat{j} + \dot{z}\hat{k}$ - $\vec{a} = \ddot{x}\hat{i} + \ddot{y}\hat{j} + \ddot{z}\hat{k}$ - **Normal and Tangential Components (n, t):** - $\vec{v} = v \hat{u}_t$ - $\vec{a} = a_t \hat{u}_t + a_n \hat{u}_n$ - $a_t = \dot{v}$ (change in speed) - $a_n = \frac{v^2}{\rho}$ (change in direction, $\rho$ is radius of curvature) - **Cylindrical Components (r, $\theta$, z):** - $\vec{r} = r \hat{u}_r + z \hat{u}_z$ - $\vec{v} = \dot{r} \hat{u}_r + r\dot{\theta} \hat{u}_\theta + \dot{z} \hat{u}_z$ - $\vec{a} = (\ddot{r} - r\dot{\theta}^2) \hat{u}_r + (r\ddot{\theta} + 2\dot{r}\dot{\theta}) \hat{u}_\theta + \ddot{z} \hat{u}_z$ ### Kinetics of a Particle #### Newton's Second Law - $\sum \vec{F} = m\vec{a}$ - **Rectangular:** $\sum F_x = m a_x$, $\sum F_y = m a_y$, $\sum F_z = m a_z$ - **Normal-Tangential:** $\sum F_t = m a_t$, $\sum F_n = m a_n = m \frac{v^2}{\rho}$ - **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:** $U_{1-2} = \int_1^2 \vec{F} \cdot d\vec{r}$ - **Constant Force:** $U_{1-2} = (F\cos\theta) s$ - **Weight:** $U_g = -W \Delta y$ - **Spring:** $U_s = -\frac{1}{2} k (s_2^2 - s_1^2)$ - **Principle of Work and Energy:** $T_1 + U_{1-2} = T_2$ - $T = \frac{1}{2} m v^2$ (kinetic energy) - **Conservative Forces:** Work is path independent (gravity, spring). - **Conservation of Energy:** $T_1 + V_1 = T_2 + V_2$ - $V_g = W y$ (gravitational potential energy) - $V_e = \frac{1}{2} k s^2$ (elastic potential energy) - Only applies if only conservative forces do work. #### Impulse and Momentum - **Linear Impulse:** $\vec{I} = \int \vec{F} dt$ - **Linear Momentum:** $\vec{L} = m\vec{v}$ - **Principle of Linear Impulse and Momentum:** $m\vec{v}_1 + \sum \int_{t_1}^{t_2} \vec{F} dt = m\vec{v}_2$ - $\vec{L}_1 + \sum \vec{I}_{1-2} = \vec{L}_2$ - **Conservation of Linear Momentum:** If $\sum \vec{F} = \vec{0}$, then $m\vec{v}_1 = m\vec{v}_2$. - **Impact:** - **Coefficient of Restitution (e):** $e = \frac{(v_B)_2 - (v_A)_2}{(v_A)_1 - (v_B)_1}$ (along line of impact) - $e=1$: perfectly elastic, $e=0$: perfectly plastic. - **Angular Momentum (for a particle):** $\vec{H}_O = \vec{r} \times m\vec{v}$ - **Principle of Angular Impulse and Momentum:** $(\vec{H}_O)_1 + \sum \int_{t_1}^{t_2} \vec{M}_O dt = (\vec{H}_O)_2$ - **Conservation of Angular Momentum:** If $\sum \vec{M}_O = \vec{0}$, then $(\vec{H}_O)_1 = (\vec{H}_O)_2$. ### Kinematics of a Rigid Body #### Types of Motion - **Translation:** - **Rectilinear:** All points move in parallel straight lines. $\vec{v}_B = \vec{v}_A$, $\vec{a}_B = \vec{a}_A$. - **Curvilinear:** All points move in parallel curved paths. $\vec{v}_B = \vec{v}_A$, $\vec{a}_B = \vec{a}_A$. - **Rotation about a Fixed Axis:** - $\omega = \frac{d\theta}{dt}$ (angular velocity) - $\alpha = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}$ (angular acceleration) - $v = \omega r$ - $a_t = \alpha r$, $a_n = \omega^2 r = v^2/r$ - **Constant Angular Acceleration:** - $\omega = \omega_0 + \alpha_c t$ - $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha_c t^2$ - $\omega^2 = \omega_0^2 + 2 \alpha_c (\theta - \theta_0)$ - **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 of Zero Velocity (IC):** - Point on the body (or extension) that has zero velocity at a given instant. - Used to simplify velocity analysis for general plane motion: $v = \omega r_{IC}$. - Found at intersection of perpendiculars to velocities of two points. ### Kinetics of a Rigid Body (Plane Motion) #### Equations of Motion - **Translation:** $\sum F_x = m(\bar{a}_x)$, $\sum F_y = m(\bar{a}_y)$, $\sum M_G = 0$ (moment about center of mass G) - **Rotation about a Fixed Axis:** $\sum F_n = m(\bar{a}_n)$, $\sum F_t = m(\bar{a}_t)$, $\sum M_O = I_O \alpha$ (moment about fixed axis O) - **General Plane 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) - Alternative: $\sum M_P = \sum (\mathcal{M}_k)_P$ (moment about any point P) - $\sum (\mathcal{M}_k)_P = I_P \alpha$ if P is a fixed point or the IC. #### Mass Moment of Inertia ($\bar{I}$) - Measure of a body's resistance to angular acceleration. - $\bar{I} = \int r^2 dm$ - **Parallel-Axis Theorem:** $I_O = \bar{I} + m d^2$ - $I_O$: moment of inertia about axis O. - $\bar{I}$: moment of inertia about parallel centroidal axis. - $m$: mass. - $d$: perpendicular distance between axes. #### Work and Energy - **Kinetic Energy:** $T = \frac{1}{2} m \bar{v}^2 + \frac{1}{2} \bar{I} \omega^2$ (for general plane motion) - For fixed axis rotation: $T = \frac{1}{2} I_O \omega^2$ - **Principle of Work and Energy:** $T_1 + U_{1-2} = T_2$ - Work done by forces and moments (including spring and gravity). #### Impulse and Momentum - **Linear Impulse and Momentum:** $m\vec{\bar{v}}_1 + \sum \int \vec{F} dt = m\vec{\bar{v}}_2$ - **Angular Impulse and Momentum:** $(\vec{H}_G)_1 + \sum \int \vec{M}_G dt = (\vec{H}_G)_2$ - $(\vec{H}_G)_1 = \bar{I}\omega_1$ (for plane motion) - **Conservation of Momentum:** - Linear: If $\sum \vec{F} = \vec{0}$, then $m\vec{\bar{v}}_1 = m\vec{\bar{v}}_2$. - Angular: If $\sum \vec{M}_G = \vec{0}$, then $\bar{I}\omega_1 = \bar{I}\omega_2$.