Hibbeler Mechanics

Cheatsheet Content

### Fundamental Principles - **Newton's First Law:** A particle originally at rest, or moving in a straight line with constant velocity, tends to remain in this state provided the particle is not subjected to an unbalanced force. - **Newton's Second Law:** The acceleration of a particle is proportional to the resultant force acting on it and is in the direction of this force. $$\Sigma \vec{F} = m\vec{a}$$ - **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 $G = 66.73 \times 10^{-12} \text{ m}^3/(\text{kg} \cdot \text{s}^2)$ - **Units:** SI (kg, m, s, N) and US Customary (slug, ft, s, lb). - $1 \text{ slug} = 14.59 \text{ kg}$ - $1 \text{ lb} = 4.448 \text{ N}$ - Weight: $W = mg$, where $g = 9.81 \text{ m/s}^2$ or $32.2 \text{ ft/s}^2$. ### Force Vectors - **Scalar & Vector:** Scalar (magnitude only), Vector (magnitude & direction). - **Vector Addition (Parallelogram Law):** Resultant vector $\vec{R} = \vec{A} + \vec{B}$. - Use law of cosines ($C = \sqrt{A^2+B^2-2AB\cos c}$) and law of sines ($\frac{A}{\sin a} = \frac{B}{\sin b} = \frac{C}{\sin c}$) for magnitude and direction. - **Cartesian Vectors:** $\vec{F} = F_x\hat{i} + F_y\hat{j} + F_z\hat{k}$ - Magnitude: $F = \sqrt{F_x^2 + F_y^2 + F_z^2}$ - Direction Cosines: $\cos \alpha = \frac{F_x}{F}$, $\cos \beta = \frac{F_y}{F}$, $\cos \gamma = \frac{F_z}{F}$ - Unit Vector: $\vec{u}_F = \frac{\vec{F}}{F} = \cos \alpha \hat{i} + \cos \beta \hat{j} + \cos \gamma \hat{k}$ - **Dot Product:** $\vec{A} \cdot \vec{B} = AB\cos\theta = A_x B_x + A_y B_y + A_z B_z$ - Angle between vectors: $\theta = \cos^{-1}\left(\frac{\vec{A} \cdot \vec{B}}{AB}\right)$ - Component of $\vec{A}$ along $\vec{B}$: $A_B = \vec{A} \cdot \vec{u}_B$ ### Equilibrium of a Particle - **Free-Body Diagram (FBD):** Essential for solving equilibrium problems. Isolate the particle and show all external forces acting on it. - **Equations of Equilibrium:** - 2D: $\Sigma F_x = 0$, $\Sigma F_y = 0$ - 3D: $\Sigma F_x = 0$, $\Sigma F_y = 0$, $\Sigma F_z = 0$ ### Force System Resultants - **Moment of a Force (Scalar):** $M_O = Fd$, where $d$ is the perpendicular distance from $O$ to the line of action of $F$. - **Moment of a Force (Vector):** $\vec{M}_O = \vec{r} \times \vec{F}$ - 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: $|\vec{A} \times \vec{B}| = AB\sin\theta$ - **Moment about an Axis:** $M_{axis} = \vec{u}_{axis} \cdot (\vec{r} \times \vec{F})$ - **Couple Moment:** $\vec{M} = \vec{r} \times \vec{F}$ (independent of pivot point) - **Reduction of a Simple Distributed Loading:** - Resultant Force: $F_R = \int w(x) dx$ (area under the loading curve) - Location: $\bar{x} = \frac{\int x w(x) dx}{\int w(x) dx}$ (centroid of the area) ### Equilibrium of a Rigid Body - **Free-Body Diagram (FBD):** Show all external forces and couple moments. Identify known and unknown forces/moments. - **Equations of Equilibrium:** - 2D: $\Sigma F_x = 0$, $\Sigma F_y = 0$, $\Sigma M_O = 0$ - 3D: $\Sigma F_x = 0$, $\Sigma F_y = 0$, $\Sigma F_z = 0$, $\Sigma M_x = 0$, $\Sigma M_y = 0$, $\Sigma M_z = 0$ - **Support Reactions:** - **Roller:** One force perpendicular to surface. - **Pin/Hinge:** Two force components (x, y). - **Fixed Support:** Two force components (x, y) and one couple moment. - **Ball-and-socket:** Three force components (x, y, z). - **Journal Bearing:** Two force components (perpendicular to shaft), two couple moments (perpendicular to shaft). - **Fixed Support (3D):** Three force components (x, y, z) and three couple moments (x, y, z). ### Structural Analysis - **Trusses:** - **Assumptions:** Members are two-force members, forces applied at joints. - **Method of Joints:** Apply particle equilibrium ($\Sigma F_x=0, \Sigma F_y=0$) at each joint. - **Method of Sections:** Cut through members, apply rigid body equilibrium ($\Sigma F_x=0, \Sigma F_y=0, \Sigma M=0$) to a section. - **Frames and Machines:** - Disassemble into component parts. - Apply rigid body equilibrium to each part. - Action-reaction forces between connected parts are equal and opposite. ### Friction - **Dry Friction:** Occurs between non-lubricated surfaces. - **Static Friction:** $F_s \le \mu_s N$ (up to impending motion) - **Kinetic Friction:** $F_k = \mu_k N$ (when motion occurs) - $\mu_s > \mu_k$ - Angle of Static Friction: $\tan \phi_s = \mu_s$ - **Wedges:** Used to increase the force applied. Analyze FBDs of wedge and object. - **Belts:** $T_2 = T_1 e^{\mu_s \beta}$ for impending motion, where $\beta$ is the angle of contact in radians. ### Center of Gravity and Centroid - **Center of Gravity:** Point where the entire weight of a body can be considered to act. - $\bar{x} = \frac{\int \tilde{x} dW}{\int dW}$, $\bar{y} = \frac{\int \tilde{y} dW}{\int dW}$, $\bar{z} = \frac{\int \tilde{z} dW}{\int dW}$ - **Centroid (Area):** Geometric center of an area. - $\bar{x} = \frac{\int \tilde{x} dA}{\int dA}$, $\bar{y} = \frac{\int \tilde{y} dA}{\int dA}$ - **Composite Bodies:** - $\bar{x} = \frac{\Sigma \tilde{x}_i A_i}{\Sigma A_i}$, $\bar{y} = \frac{\Sigma \tilde{y}_i A_i}{\Sigma A_i}$, etc. - **Theorems of Pappus and Guldinus:** - **Area of Surface of Revolution (A):** $A = \theta \bar{y} L$ (for $L$ rotated about x-axis) or $A = 2\pi \bar{y} L$ (full revolution). - **Volume of Revolution (V):** $V = \theta \bar{y} A$ (for $A$ rotated about x-axis) or $V = 2\pi \bar{y} A$ (full revolution). ### Moment of Inertia - **Area Moment of Inertia:** Measures resistance to bending. - $I_x = \int y^2 dA$, $I_y = \int x^2 dA$ - Polar Moment of Inertia: $J_O = I_x + I_y = \int r^2 dA$ - **Parallel-Axis Theorem:** $I = \bar{I} + Ad^2$ - $\bar{I}$ is moment of inertia about centroidal axis, $A$ is area, $d$ is distance between parallel axes. - **Radius of Gyration:** $k = \sqrt{I/A}$ - **Mass Moment of Inertia:** Measures resistance to angular acceleration. - $I = \int r^2 dm$ - Parallel-Axis Theorem (Mass): $I = \bar{I} + md^2$ ### Kinematics of a Particle - **Rectilinear Motion:** - Velocity: $v = \frac{ds}{dt}$ - Acceleration: $a = \frac{dv}{dt} = v \frac{dv}{ds}$ - 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:** $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$, $\vec{v} = v_x\hat{i} + v_y\hat{j} + v_z\hat{k}$, $\vec{a} = a_x\hat{i} + a_y\hat{j} + a_z\hat{k}$ - **Normal and Tangential Components:** - Velocity: $\vec{v} = v\vec{u}_t$ - Acceleration: $\vec{a} = a_t\vec{u}_t + a_n\vec{u}_n$ - $a_t = \dot{v}$ (or $v \frac{dv}{ds}$), $a_n = \frac{v^2}{\rho}$ ($\rho$ is radius of curvature) - **Cylindrical Components:** $\vec{r} = r\vec{u}_r + z\vec{u}_z$ - $\vec{v} = \dot{r}\vec{u}_r + r\dot{\theta}\vec{u}_\theta + \dot{z}\vec{u}_z$ - $\vec{a} = (\ddot{r} - r\dot{\theta}^2)\vec{u}_r + (r\ddot{\theta} + 2\dot{r}\dot{\theta})\vec{u}_\theta + \ddot{z}\vec{u}_z$ ### Kinetics of a Particle - **Newton's Second Law:** $\Sigma \vec{F} = m\vec{a}$ - **Rectangular:** $\Sigma F_x = ma_x$, $\Sigma F_y = ma_y$, $\Sigma F_z = ma_z$ - **Normal & Tangential:** $\Sigma F_t = ma_t$, $\Sigma F_n = ma_n = m\frac{v^2}{\rho}$ - **Cylindrical:** $\Sigma F_r = m(\ddot{r} - r\dot{\theta}^2)$, $\Sigma F_\theta = m(r\ddot{\theta} + 2\dot{r}\dot{\theta})$ - **Work and Energy:** - **Work of a Force:** $U_{1-2} = \int_{s_1}^{s_2} \vec{F} \cdot d\vec{r}$ - Constant Force: $U_{1-2} = (F\cos\theta) \Delta s$ - Spring Force: $U_{1-2} = \frac{1}{2} k s_1^2 - \frac{1}{2} k s_2^2$ - **Kinetic Energy:** $T = \frac{1}{2} m v^2$ - **Principle of Work and Energy:** $T_1 + U_{1-2} = T_2$ - **Conservative Forces:** Gravity ($V_g = W y$), Elastic Spring ($V_e = \frac{1}{2} k s^2$) - **Conservation of Energy:** $T_1 + V_1 = T_2 + V_2$ (for conservative systems) - **Impulse and Momentum:** - **Linear Impulse:** $\vec{I} = \int \vec{F} dt$ - **Linear Momentum:** $\vec{L} = m\vec{v}$ - **Principle of Linear Impulse and Momentum:** $m\vec{v}_1 + \Sigma \int_{t_1}^{t_2} \vec{F} dt = m\vec{v}_2$ - **Conservation of Linear Momentum:** $\Sigma (m\vec{v})_1 = \Sigma (m\vec{v})_2$ (for no external impulse) - **Impact:** - Coefficient of Restitution: $e = \frac{(v_B)_2 - (v_A)_2}{(v_A)_1 - (v_B)_1}$ - $e=1$ (elastic), $e=0$ (plastic) ### Kinematics of a Rigid Body - **Types of Motion:** - **Translation:** All points have same velocity and acceleration. - Rectilinear: straight path - Curvilinear: curved path - **Rotation about a Fixed Axis:** - Angular Velocity: $\omega = \frac{d\theta}{dt}$ - Angular Acceleration: $\alpha = \frac{d\omega}{dt} = \omega \frac{d\omega}{d\theta}$ - Constant Angular Acceleration: $\omega = \omega_0 + \alpha_c t$, $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha_c t^2$, $\omega^2 = \omega_0^2 + 2 \alpha_c (\theta - \theta_0)$ - Relation to Linear: $v = \omega r$, $a_t = \alpha r$, $a_n = \omega^2 r = v^2/r$ - **General Plane Motion:** Translation + Rotation - **Relative Velocity:** $\vec{v}_B = \vec{v}_A + \vec{\omega} \times \vec{r}_{B/A}$ - **Instantaneous Center (IC):** Point with zero velocity. For pure rotation about IC. - **Relative Acceleration:** $\vec{a}_B = \vec{a}_A + \vec{\alpha} \times \vec{r}_{B/A} - \omega^2 \vec{r}_{B/A}$ ### Kinetics of a Rigid Body - **Equations of Motion (Plane Motion):** - $\Sigma F_x = m(\bar{a}_x)$ - $\Sigma F_y = m(\bar{a}_y)$ - $\Sigma M_G = \bar{I}\alpha$ (moment about center of mass G) - OR $\Sigma M_P = \bar{I}\alpha + m(\vec{r}_{G/P} \times \vec{\bar{a}})$ (moment about any point P) - **Work and Energy (Rigid Body):** - **Kinetic Energy:** $T = \frac{1}{2} m \bar{v}^2 + \frac{1}{2} \bar{I} \omega^2$ - **Principle of Work and Energy:** $T_1 + U_{1-2} = T_2$ - **Impulse and Momentum (Rigid Body):** - **Linear Impulse-Momentum:** $m(\vec{\bar{v}})_1 + \Sigma \int \vec{F} dt = m(\vec{\bar{v}})_2$ - **Angular Impulse-Momentum:** $\bar{I}\omega_1 + \Sigma \int M_G dt = \bar{I}\omega_2$ - **Conservation of Momentum:** $\Sigma (m\bar{v})_1 = \Sigma (m\bar{v})_2$ and $\Sigma (\bar{I}\omega)_1 = \Sigma (\bar{I}\omega)_2$ (no external impulse/moment)