Hibbeler Mechanics

Cheatsheet Content

### Fundamental Principles - **Newton's First Law:** A particle remains at rest or continues to move with constant velocity if there is no unbalanced force acting on it. - **Newton's Second Law:** The acceleration of a particle is proportional to the net force acting on it and is in the direction of this force. $\vec{F} = m\vec{a}$ - **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 $G = 6.673 \times 10^{-11} \text{ m}^3/(\text{kg} \cdot \text{s}^2)$. ### Units and Dimensions - **SI Units:** - Mass (kg) - Length (m) - Time (s) - Force (N) = kg $\cdot$ m/s$^2$ - **US Customary (FPS) Units:** - Mass (slug) - Length (ft) - Time (s) - Force (lb) - **Weight:** $W = mg$ - Earth: $g = 9.81 \text{ m/s}^2$ or $32.2 \text{ ft/s}^2$ ### Vectors - **Scalar:** Quantity with magnitude only (e.g., mass, time, temperature). - **Vector:** Quantity with both magnitude and direction (e.g., force, velocity, acceleration). - **Vector Addition (Parallelogram Law):** $$\vec{R} = \vec{A} + \vec{B}$$ - **Vector Subtraction:** $$\vec{R}' = \vec{A} - \vec{B} = \vec{A} + (-\vec{B})$$ - **Rectangular Components:** - $\vec{F} = F_x \hat{i} + F_y \hat{j} + F_z \hat{k}$ - Magnitude: $F = \sqrt{F_x^2 + F_y^2 + F_z^2}$ - Direction Cosines: $\cos\alpha = F_x/F$, $\cos\beta = F_y/F$, $\cos\gamma = F_z/F$ - Unit Vector: $\vec{u}_F = \vec{F}/F = \cos\alpha \hat{i} + \cos\beta \hat{j} + \cos\gamma \hat{k}$ - **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 the angle between two 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. - Determinant form: $$\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix}$$ - Used to calculate moment of a force. ### Force Systems - **Concurrent Forces:** All forces pass through a single point. Resultant is a single force. - **Coplanar Forces:** All forces lie in a single plane. - **Equilibrium (Static):** - Particle: $\sum \vec{F} = 0 \implies \sum F_x = 0, \sum F_y = 0, \sum F_z = 0$ - Rigid Body: $\sum \vec{F} = 0$ and $\sum \vec{M}_O = 0$ - **Moment of a Force (about point O):** $\vec{M}_O = \vec{r} \times \vec{F}$ - Magnitude: $M_O = rF\sin\theta = Fd$ (where $d$ is perpendicular distance). - Scalar form for 2D: $M_O = F_y x - F_x y$ (positive counter-clockwise). - **Varignon's Theorem:** The moment of a force about any point is equal to the sum of the moments of its components about the same point. - $\vec{M}_O = \vec{r} \times (\vec{F}_1 + \vec{F}_2) = \vec{r} \times \vec{F}_1 + \vec{r} \times \vec{F}_2$ - **Couple Moment:** Two parallel forces, equal in magnitude, opposite in direction, separated by a perpendicular distance $d$. - $\vec{M} = \vec{r} \times \vec{F}$ (independent of the point O). - Magnitude: $M = Fd$. - **Resultant of a Force System:** Can be reduced to a single resultant force $\vec{F}_R = \sum \vec{F}$ and a resultant couple moment $\vec{M}_R = \sum \vec{M} + \sum \vec{M}_c$. ### Free-Body Diagrams (FBDs) - **Steps:** 1. Isolate the body or particle. 2. Show all external forces and moments acting on the body (known and unknown). 3. Include weight, applied loads, and reaction forces/moments from supports. 4. Label forces with magnitudes and directions. - **Common Supports and Reactions:** - **Cable/Rope:** Tension $\vec{T}$ (pulling away from body). - **Smooth Surface:** Normal force $\vec{N}$ (perpendicular to surface). - **Roller/Rocker:** Normal force $\vec{N}$ (perpendicular to surface). - **Pin/Hinge (2D):** Two force components ($F_x, F_y$). - **Fixed Support (2D):** Two force components ($F_x, F_y$) and a moment ($M$). - **Two-Force Member:** A member subjected to forces at only two points. The forces must be equal, opposite, and collinear. - **Three-Force Member:** A member subjected to forces at only three points. The forces must be concurrent or parallel for equilibrium. ### Trusses - **Assumptions:** 1. Members are joined by smooth pins. 2. Loads are applied only at the joints. - **Result:** Members are either in pure tension (T) or pure compression (C). - **Method of Joints:** 1. Draw FBD of entire truss to find external reactions. 2. Draw FBD for each pin (joint). 3. Apply $\sum F_x = 0$ and $\sum F_y = 0$ to each joint. 4. Start with joints having known forces and at most two unknown member forces. - **Method of Sections:** 1. Draw FBD of entire truss to find external reactions. 2. Cut the truss through members whose forces are to be determined (max 3 unknowns). 3. Draw FBD of one section. 4. Apply $\sum F_x = 0$, $\sum F_y = 0$, and $\sum M = 0$. - **Zero-Force Members:** 1. If only two non-collinear members connect at a joint and no external load or reaction acts at that joint, both members are zero-force members. 2. If three members connect at a joint, two of which are collinear, and no external load or reaction acts at that joint, the third member is a zero-force member. ### Frames and Machines - **Frames:** Stationary structures designed to support loads. Contain at least one multi-force member. - **Machines:** Structures designed to transmit and modify forces. Contain moving parts. - **Analysis Steps:** 1. Draw FBD of the entire structure to find external reactions. 2. Disassemble the structure into its component members. 3. Draw FBD for each member, showing all internal and external forces. 4. Apply Newton's Third Law for internal forces between connected members (equal and opposite). 5. Apply equilibrium equations ($\sum F_x = 0, \sum F_y = 0, \sum M = 0$) to each member or combination of members. ### Center of Gravity and Centroids - **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: $\bar{x} = \frac{\sum \bar{x}_i A_i}{\sum A_i}$, $\bar{y} = \frac{\sum \bar{y}_i A_i}{\sum A_i}$ - **Theorems of Pappus and Guldinus:** - **Area of surface of revolution:** $A = \theta \bar{r} L$ (where $L$ is arc length, $\theta$ is angle of revolution in radians, $\bar{r}$ is distance from centroid of arc to axis of revolution). - **Volume of body of revolution:** $V = \theta \bar{r} A$ (where $A$ is area of generating shape). ### Fluid Statics - **Pressure:** $P = F/A$ - **Hydrostatic Pressure:** $P = \rho g h$ or $P = \gamma h$ (where $\rho$ is density, $\gamma$ is specific weight). - **Manometry:** Pressure difference is measured by fluid column height. - **Hydrostatic Force on Submerged Plane Surface:** - Magnitude: $F_R = P_{avg} A = (\rho g \bar{y} \sin\theta) A = P_c A$ (where $\bar{y}$ is centroid depth). - Location (Center of Pressure): $y_p = \bar{y} + \frac{I_{xx}}{\bar{y} A}$ - For horizontal surface, $y_p = \bar{y}$. - **Buoyancy (Archimedes' Principle):** The buoyant force $F_B$ on a submerged or floating body is equal to the weight of the fluid displaced by the body. - $F_B = \rho_f g V_{disp}$ - Acts through the centroid of the displaced volume (center of buoyancy). ### Moments of Inertia - **Definition:** Measure of a body's resistance to angular acceleration (mass moment of inertia) or a measure of a body's resistance to bending or buckling (area moment of inertia). - **Area Moment of Inertia (Second Moment of Area):** - $I_x = \int y^2 dA$ - $I_y = \int x^2 dA$ - Polar Moment of Inertia: $J_O = I_x + I_y = \int r^2 dA$ - **Parallel-Axis Theorem:** - $I = \bar{I} + Ad^2$ (where $\bar{I}$ 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:** - $I = \int r^2 dm$ - For composite bodies: $I = \sum (I_i + m_i d_i^2)$ ### Kinematics of a Particle - **Rectilinear Motion:** - Velocity: $v = ds/dt$ - Acceleration: $a = dv/dt = d^2s/dt^2 = v(dv/ds)$ - 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:** - $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ - $\vec{v} = \dot{x}\hat{i} + \dot{y}\hat{j} + \dot{z}\hat{k}$ - $\vec{a} = \ddot{x}\hat{i} + \ddot{y}\hat{j} + \ddot{z}\hat{k}$ - **Normal and Tangential Components:** - $\vec{v} = v \hat{u}_t$ - $\vec{a} = a_t \hat{u}_t + a_n \hat{u}_n$ - $a_t = \dot{v}$ or $v \frac{dv}{ds}$ (tangential acceleration, changes speed) - $a_n = v^2/\rho$ (normal acceleration, changes direction, $\rho$ is radius of curvature) - **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: $\sum F_x = m a_x$, $\sum F_y = m a_y$, $\sum F_z = m a_z$ - Normal-Tangential: $\sum F_t = m a_t$, $\sum F_n = m a_n = m(v^2/\rho)$ - 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:** $U_{1-2} = \int_{s_1}^{s_2} \vec{F} \cdot d\vec{r}$ - **Kinetic Energy:** $T = \frac{1}{2}mv^2$ - **Principle of Work and Energy:** $T_1 + U_{1-2} = T_2$ - **Conservative Forces:** Work done is independent of path (e.g., gravity, spring force). - Gravitational Potential Energy: $V_g = W y$ - Elastic Potential Energy: $V_e = \frac{1}{2}ks^2$ - **Conservation of Energy (for conservative systems):** $T_1 + V_1 = T_2 + V_2$ - **Impulse and Momentum:** - **Linear Momentum:** $\vec{L} = m\vec{v}$ - **Linear Impulse:** $\text{Imp}_{1-2} = \int_{t_1}^{t_2} \vec{F} dt$ - **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 (for isolated systems):** $\sum (m\vec{v})_1 = \sum (m\vec{v})_2$ - **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:** $(\vec{H}_O)_1 + \sum \int_{t_1}^{t_2} \vec{M}_O dt = (\vec{H}_O)_2$