Hibbeler Mechanics Cheatsheet

Cheatsheet Content

### Fundamental Principles - **Newton's First Law:** A particle remains at rest or moves with constant velocity if the resultant force on it is zero. $\sum \vec{F} = 0$. - **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. $\sum \vec{F} = m\vec{a}$. - **Newton's Third Law:** For every action, there is an equal and opposite reaction. - **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: meter (m), kilogram (kg), second (s), Newton (N) - US Customary: foot (ft), slug (slug), second (s), pound (lb) ### Force Vectors #### Cartesian Vectors - **Representation:** $\vec{F} = F_x \hat{i} + F_y \hat{j} + F_z \hat{k}$ - **Magnitude:** $|\vec{F}| = \sqrt{F_x^2 + F_y^2 + F_z^2}$ - **Unit Vector:** $\hat{u}_F = \frac{\vec{F}}{|\vec{F}|} = \cos \alpha \hat{i} + \cos \beta \hat{j} + \cos \gamma \hat{k}$ - $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$ #### Dot Product - **Definition:** $\vec{A} \cdot \vec{B} = |\vec{A}||\vec{B}| \cos \theta = A_x B_x + A_y B_y + A_z B_z$ - **Angle between vectors:** $\theta = \cos^{-1} \left( \frac{\vec{A} \cdot \vec{B}}{|\vec{A}||\vec{B}|} \right)$ - **Projection of A onto B:** $\text{Proj}_B \vec{A} = (\vec{A} \cdot \hat{u}_B) \hat{u}_B$ #### Cross Product - **Definition:** $\vec{C} = \vec{A} \times \vec{B} = (A_y B_z - A_z B_y)\hat{i} - (A_x B_z - A_z B_x)\hat{j} + (A_x B_y - A_y B_x)\hat{k}$ - **Magnitude:** $|\vec{A} \times \vec{B}| = |\vec{A}||\vec{B}| \sin \theta$ - **Right-hand rule** for direction. ### Equilibrium of a Particle - **Conditions:** $\sum \vec{F} = 0$ - **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 problem-solving. Isolate the particle and show all external forces acting on it. ### Moment of a Force (Torque) - **Scalar Formulation (2D):** $M_O = Fd$ (Force magnitude $\times$ perpendicular distance from pivot to line of action of force). - **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 force about a point is equal to the sum of the moments of its components about the same point. $\vec{M}_O = \vec{r} \times (\vec{F}_1 + \vec{F}_2) = (\vec{r} \times \vec{F}_1) + (\vec{r} \times \vec{F}_2)$. ### Resultants of Force Systems - **Resultant Force:** $\vec{F}_R = \sum \vec{F}$ - **Resultant Moment:** $\vec{M}_{R_O} = \sum (\vec{r} \times \vec{F}) + \sum \vec{M}_c$ (where $\vec{M}_c$ are couple moments) - **Wrench:** A force system can be reduced to a resultant force and a resultant couple moment, where the resultant force and couple moment are collinear. ### Equilibrium of a Rigid Body - **Conditions:** $\sum \vec{F} = 0$ and $\sum \vec{M}_O = 0$ - **2D (Coplanar Force Systems):** - $\sum F_x = 0$ - $\sum F_y = 0$ - $\sum M_O = 0$ (moment about any point O) - **3D:** - $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$ - $\sum M_x = 0$, $\sum M_y = 0$, $\sum M_z = 0$ - **Supports and Reactions:** - **Roller/Smooth Surface:** Normal force (perpendicular to surface). - **Pin/Hinge:** Two force components (e.g., $R_x, R_y$). - **Fixed Support:** Two force components and a moment (e.g., $R_x, R_y, M$). - **Two-Force Member:** A body subjected to only two forces is in equilibrium if and only if the forces are collinear, equal in magnitude, and opposite in direction. - **Three-Force Member:** A body subjected to only three forces is in equilibrium if and only if the forces are concurrent (intersect at a common point) or parallel. ### Trusses, Frames, and Machines #### Trusses - **Assumptions:** Members are two-force members, forces applied at joints. - **Method of Joints:** Apply particle equilibrium ($\sum F_x=0, \sum F_y=0$) at each joint. - **Method of Sections:** Cut the truss to expose desired member forces. Apply rigid body equilibrium ($\sum F_x=0, \sum F_y=0, \sum M=0$) to the section. - **Zero-Force Members:** Identify members that carry no load under specific loading conditions (e.g., two non-collinear members meeting at an unloaded joint, third member is zero-force). #### Frames and Machines - Components are generally not two-force members. - Disassemble the structure into its component parts. - Apply rigid body equilibrium to each component, ensuring action-reaction pairs are shown correctly. ### Center of Gravity & Centroid - **Center of Gravity (CG):** Point where the entire weight of a body can be considered to act. - $\bar{x} = \frac{\sum \tilde{x} W}{\sum W}$, $\bar{y} = \frac{\sum \tilde{y} W}{\sum W}$, $\bar{z} = \frac{\sum \tilde{z} W}{\sum W}$ - **Centroid:** Geometric center of an area or volume. - **Area:** $\bar{x} = \frac{\sum \tilde{x} A}{\sum A}$, $\bar{y} = \frac{\sum \tilde{y} A}{\sum A}$ - **Volume:** $\bar{x} = \frac{\sum \tilde{x} V}{\sum V}$, $\bar{y} = \frac{\sum \tilde{y} V}{\sum V}$, $\bar{z} = \frac{\sum \tilde{z} V}{\sum V}$ - **Composite Bodies:** Divide complex shapes into simpler geometric parts. - **Pappus-Guldinus Theorems:** - **Area of surface of revolution:** $A = \theta \bar{r} L$ (where L is arc length, $\theta$ is angle of revolution in radians). - **Volume of body of revolution:** $V = \theta \bar{r} A$ (where A is area, $\theta$ is angle of revolution in radians). ### Moments of Inertia #### Area Moments of Inertia - **Definition:** $I_x = \int y^2 dA$, $I_y = \int x^2 dA$ - **Polar Moment of Inertia:** $J_O = I_x + I_y = \int r^2 dA$ - **Parallel-Axis Theorem:** $I = I_c + Ad^2$ (where $I_c$ is moment of inertia about centroidal axis, $A$ is area, $d$ is perpendicular distance between axes). - **Radius of Gyration:** $k = \sqrt{I/A}$ #### Mass Moments of Inertia - **Definition:** $I = \int r^2 dm$ - **Parallel-Axis Theorem:** $I = I_G + md^2$ (where $I_G$ is mass moment of inertia about centroidal axis, $m$ is mass, $d$ is perpendicular distance between axes). ### Kinematics of a Particle - **Rectilinear Motion:** - **Constant Acceleration:** - $v = v_0 + a_c t$ - $s = s_0 + v_0 t + \frac{1}{2} a_c t^2$ - $v^2 = v_0^2 + 2 a_c (s - s_0)$ - **General:** - $v = \frac{ds}{dt}$ - $a = \frac{dv}{dt} = \frac{d^2s}{dt^2}$ - $a ds = v dv$ - **Curvilinear Motion:** - **Rectangular Components:** $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ - $\vec{v} = \dot{x}\hat{i} + \dot{y}\hat{j} + \dot{z}\hat{k}$ - $\vec{a} = \ddot{x}\hat{i} + \ddot{y}\hat{j} + \ddot{z}\hat{k}$ - **Normal and Tangential Components:** - $v = \dot{s}$ - $a_t = \dot{v} = v \frac{dv}{ds}$ (tangential acceleration) - $a_n = \frac{v^2}{\rho}$ (normal acceleration, directed towards center of curvature, $\rho$ is radius of curvature) - $a = \sqrt{a_t^2 + a_n^2}$ - **Cylindrical Components:** $\vec{r} = r\hat{u}_r + z\hat{k}$ - $\vec{v} = \dot{r}\hat{u}_r + r\dot{\theta}\hat{u}_\theta + \dot{z}\hat{k}$ - $\vec{a} = (\ddot{r} - r\dot{\theta}^2)\hat{u}_r + (r\ddot{\theta} + 2\dot{r}\dot{\theta})\hat{u}_\theta + \ddot{z}\hat{k}$ ### Kinetics of a Particle #### Newton's Second Law - $\sum \vec{F} = m\vec{a}$ - **Rectangular:** $\sum F_x = ma_x$, $\sum F_y = ma_y$, $\sum F_z = ma_z$ - **Normal-Tangential:** $\sum F_t = ma_t$, $\sum F_n = ma_n$ - **Cylindrical:** $\sum F_r = ma_r$, $\sum F_\theta = ma_\theta$, $\sum F_z = ma_z$ #### Work and Energy - **Work of a Force:** $U_{1-2} = \int_{s_1}^{s_2} \vec{F} \cdot d\vec{r}$ - Constant Force: $U_{1-2} = F_c \cos\theta (s_2 - s_1)$ - 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:** $T_1 + \sum U_{1-2} = T_2$ - **Conservation of Energy:** $T_1 + V_1 = T_2 + V_2$ (if only conservative forces do work) - **Gravitational Potential Energy:** $V_g = Wy$ - **Elastic Potential Energy:** $V_e = \frac{1}{2} k s^2$ #### Impulse and Momentum - **Linear Momentum:** $\vec{L} = m\vec{v}$ - **Linear Impulse:** $\int_{t_1}^{t_2} \vec{F} dt$ - **Principle of Linear Impulse and Momentum:** $m\vec{v}_1 + \sum \int_{t_1}^{t_2} \vec{F} dt = m\vec{v}_2$ - **Conservation of Linear Momentum:** $\sum m\vec{v}_1 = \sum m\vec{v}_2$ (if $\sum \int \vec{F} dt = 0$) - **Impact:** - **Coefficient of Restitution:** $e = \frac{(v_B)_2 - (v_A)_2}{(v_A)_1 - (v_B)_1}$ (for central impact) - $e=1$ for perfectly elastic, $e=0$ for perfectly plastic. - **Angular Momentum:** $\vec{H}_O = \vec{r} \times m\vec{v}$ - **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:** $(\vec{H}_O)_1 = (\vec{H}_O)_2$ (if $\sum \int \vec{M}_O dt = 0$) ### Kinematics of a Rigid Body #### Types of Motion - **Translation:** - **Rectilinear:** All points move along parallel straight lines. - **Curvilinear:** All points move along parallel curved paths. - $\vec{v}_B = \vec{v}_A$, $\vec{a}_B = \vec{a}_A$ - **Rotation about a Fixed Axis:** - $\omega = \frac{d\theta}{dt}$, $\alpha = \frac{d\omega}{dt}$ - $v = \omega r$ - $a_t = \alpha r$, $a_n = \omega^2 r$ - **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):** For general plane motion, a point about which the body appears to rotate at that instant. $\vec{v} = \omega r_{IC}$. ### Kinetics of a Rigid Body - **Equations of Motion:** - $\sum \vec{F} = m\vec{a}_G$ (translational) - $\sum \vec{M}_G = I_G \vec{\alpha}$ (rotational about CG) - $\sum \vec{M}_O = I_O \vec{\alpha}$ (rotational about fixed axis O) - **Work and Energy:** - **Kinetic Energy:** - Translation: $T = \frac{1}{2} m v_G^2$ - Rotation about fixed axis O: $T = \frac{1}{2} I_O \omega^2$ - 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:** - **Linear Impulse and Momentum:** $m(\vec{v}_G)_1 + \sum \int_{t_1}^{t_2} \vec{F} dt = m(\vec{v}_G)_2$ - **Angular Impulse and Momentum:** $(H_G)_1 + \sum \int_{t_1}^{t_2} M_G dt = (H_G)_2$ - $(H_G)_1 = I_G \omega_1$ (for plane motion) - **Angular Impulse and Momentum (fixed axis O):** $(H_O)_1 + \sum \int_{t_1}^{t_2} M_O dt = (H_O)_2$ - $(H_O)_1 = I_O \omega_1$ (for fixed axis rotation)