Hibbeler Mechanics (2 Pages)

Cheatsheet Content

### Vectors - **Scalar:** A quantity with magnitude only (e.g., mass, length, time). - **Vector:** A quantity with both magnitude and direction (e.g., force, velocity, acceleration). - **Vector Operations:** - **Addition (Triangle Rule):** $\vec{R} = \vec{A} + \vec{B}$ - **Addition (Parallelogram Law):** Same as triangle rule, but visualizes as a parallelogram. - **Subtraction:** $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$ - **Components of a Vector:** - **Cartesian Vector Form:** $\vec{F} = F_x \hat{i} + F_y \hat{j} + F_z \hat{k}$ - **Magnitude:** $|\vec{F}| = \sqrt{F_x^2 + F_y^2 + F_z^2}$ - **Direction Cosines:** $\cos\alpha = \frac{F_x}{|\vec{F}|}$, $\cos\beta = \frac{F_y}{|\vec{F}|}$, $\cos\gamma = \frac{F_z}{|\vec{F}|}$ (where $\alpha, \beta, \gamma$ are angles with x, y, z axes). - **Unit Vector:** $\hat{u} = \frac{\vec{F}}{|\vec{F}|} = \cos\alpha \hat{i} + \cos\beta \hat{j} + \cos\gamma \hat{k}$ - **Dot Product (Scalar Product):** - $\vec{A} \cdot \vec{B} = |\vec{A}||\vec{B}|\cos\theta$ - $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z$ - **Applications:** - Angle between two vectors: $\theta = \arccos\left(\frac{\vec{A} \cdot \vec{B}}{|\vec{A}||\vec{B}|}\right)$ - Component of $\vec{A}$ along $\vec{B}$: $A_B = \vec{A} \cdot \hat{u}_B$ - **Cross Product (Vector Product):** - $\vec{C} = \vec{A} \times \vec{B}$ - **Magnitude:** $|\vec{C}| = |\vec{A}||\vec{B}|\sin\theta$ (direction by Right-Hand Rule) - **Cartesian Form:** $$\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix}$$ - **Applications:** Moment of a force. ### Force Systems - **Concurrent Forces:** All forces pass through a single point. - **Equilibrium:** $\sum \vec{F} = 0 \implies \sum F_x = 0, \sum F_y = 0, \sum F_z = 0$ - **Moment of a Force (Torque):** Tendency of a force to rotate a body about an axis or point. - **Scalar Form:** $M_O = Fd$ (where $d$ is perpendicular distance from $O$ to line of action of $F$). - **Vector Form:** $\vec{M}_O = \vec{r} \times \vec{F}$ (where $\vec{r}$ is position vector from $O$ to any point on line of action of $F$). - **Moment about an Axis:** $M_{axis} = \hat{u}_{axis} \cdot (\vec{r} \times \vec{F})$ - **Couple:** Two parallel forces, equal in magnitude, opposite in direction, and separated by a perpendicular distance $d$. - **Moment of a Couple:** $M = Fd$ (vector is $\vec{M} = \vec{r} \times \vec{F}$, where $\vec{r}$ is from negative to positive force). - A couple's moment is a free vector (independent of reference point). - **Resultant of a Force System:** - **Equivalent Force-Couple System:** A force system can be reduced to a single resultant force $\vec{R} = \sum \vec{F}$ acting at a point $P$, and a resultant couple moment $\vec{M}_R = \sum \vec{M}_{P} + \sum \vec{M}_{couples}$. - If $\vec{R}$ and $\vec{M}_R$ are perpendicular, they can be reduced to a single resultant force (wrench). ### Equilibrium of a Rigid Body (2D) - **Free-Body Diagram (FBD):** Essential for solving equilibrium problems. Isolate the body and show all external forces and moments acting on it. - **Equations of Equilibrium:** 1. $\sum F_x = 0$ 2. $\sum F_y = 0$ 3. $\sum M_O = 0$ (Sum of moments about any point $O$ in the plane is zero). - **Common Supports and Reactions (2D):** - **Roller:** One unknown force, perpendicular to surface. - **Pin (Hinge):** Two unknown forces, $F_x$ and $F_y$. - **Fixed Support (Cantilever):** Three unknowns, $F_x$, $F_y$, and a moment $M$. - **Cable/Rope:** One unknown force, acting along the cable. - **Smooth Surface:** One unknown force, normal to surface. ### Equilibrium of a Rigid Body (3D) - **Equations of Equilibrium:** 1. $\sum F_x = 0$ 2. $\sum F_y = 0$ 3. $\sum F_z = 0$ 4. $\sum M_x = 0$ 5. $\sum M_y = 0$ 6. $\sum M_z = 0$ (Sum of forces and moments about x, y, z axes are zero). - **Common Supports and Reactions (3D):** - **Ball-and-Socket Joint:** Three unknown forces ($F_x, F_y, F_z$). - **Fixed Support:** Three unknown forces ($F_x, F_y, F_z$) and three unknown moments ($M_x, M_y, M_z$). - **Journal Bearing (no axial thrust):** Two unknown forces ($F_x, F_y$) and two unknown moments ($M_x, M_y$) if shaft is free to rotate. - **Thrust Bearing:** Three unknown forces ($F_x, F_y, F_z$) and two unknown moments ($M_x, M_y$). - **Smooth Pin/Hinge:** Two unknown forces ($F_x, F_y$) and one unknown moment ($M_z$) if axis aligned with Z. ### Trusses - **Truss:** A structure composed of slender members joined at their ends. Members are assumed to be two-force members (only axial force, tension or compression). - **Assumptions:** 1. Members are joined by smooth pins. 2. Loads are applied only at the joints. - **Method of Joints:** - Isolate each joint and draw its FBD. - Apply $\sum F_x = 0$ and $\sum F_y = 0$ to each joint. - Start with joints having at most two unknown member forces. - **Method of Sections:** - Cut the truss into two sections by passing a section through no more than three members where forces are unknown. - Draw FBD of one section. - Apply $\sum F_x = 0$, $\sum F_y = 0$, and $\sum M_O = 0$. - **Zero-Force Members:** - If only two non-collinear members connect at a joint and no external load or support reaction is applied to the joint, then both members are zero-force members. - If three members connect at a joint, two of which are collinear, and no external load or support reaction is applied, then the third member is a zero-force member. ### Frames and Machines - **Frames:** Structures designed to support loads and are usually stationary, composed of multi-force members. - **Machines:** Structures containing moving parts, designed to transmit or modify forces. - **Analysis Procedure:** 1. Draw FBD of the entire frame/machine to find external reactions. 2. Disassemble the frame/machine into its individual members. 3. Draw FBD for each member, showing all forces (internal and external) acting on them. 4. Apply Newton's Third Law (action-reaction pairs) at all connections. 5. Apply the equations of equilibrium ($\sum F_x=0, \sum F_y=0, \sum M_O=0$) to each member or combination of members. ### Friction - **Static Friction ($F_s$):** Opposes impending motion. $F_s \le \mu_s N$, where $\mu_s$ is coefficient of static friction. - **Kinetic Friction ($F_k$):** Opposes motion when sliding occurs. $F_k = \mu_k N$, where $\mu_k$ is coefficient of kinetic friction. - **Angle of Static Friction ($\phi_s$):** $\tan \phi_s = \mu_s$. The resultant of the normal and static friction forces acts at this angle from the normal. - **Wedges:** Used to increase the force applied to a body or to slightly move or adjust heavy loads. Analyze by drawing FBDs of the wedge and the body it acts upon, considering friction at all contact surfaces. - **Square-Threaded Screws:** Used for power transmission or to apply large forces. - **Moment to raise load:** $M = W r \tan(\theta_L + \phi_s)$ - **Moment to lower load:** $M = W r \tan(\phi_s - \theta_L)$ - Where $W$ is load, $r$ is mean radius of thread, $\theta_L$ is lead angle, $\phi_s$ is static friction angle. - **Lead Angle:** $\tan \theta_L = \frac{L}{2\pi r}$ ($L$ = lead, distance screw advances per revolution). ### Center of Gravity and Centroid - **Center of Gravity (CG):** Point where the entire weight of a body can be considered to act. - $x_{CG} = \frac{\sum W_i x_i}{\sum W_i}$, $y_{CG} = \frac{\sum W_i y_i}{\sum W_i}$, $z_{CG} = \frac{\sum W_i z_i}{\sum W_i}$ - **Centroid:** Geometric center of an area or volume. - **Area:** $x_C = \frac{\sum A_i x_i}{\sum A_i}$, $y_C = \frac{\sum A_i y_i}{\sum A_i}$ - **Volume:** $x_C = \frac{\sum V_i x_i}{\sum V_i}$, $y_C = \frac{\sum V_i y_i}{\sum V_i}$, $z_C = \frac{\sum V_i z_i}{\sum V_i}$ - **Theorems of Pappus and Guldinus:** - **Surface Area ($A$):** $A = 2\pi \bar{y} L$ (where $\bar{y}$ is distance from centroid of curve to axis of revolution, $L$ is length of curve). - **Volume ($V$):** $V = 2\pi \bar{y} A$ (where $\bar{y}$ is distance from centroid of area to axis of revolution, $A$ is area). ### Moments of Inertia - **Moment of Inertia of an Area ($I$):** Measure of an area's resistance to angular acceleration. - $I_x = \int y^2 dA$, $I_y = \int x^2 dA$ - **Polar Moment of Inertia ($J_O$):** $J_O = \int r^2 dA = I_x + I_y$ - **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 parallel axes). - **Product of Inertia ($I_{xy}$):** $I_{xy} = \int xy dA$ - **Principal Moments of Inertia:** Maximum and minimum moments of inertia for an area, occurring on principal axes. - $\tan 2\theta_p = \frac{-I_{xy}}{\frac{I_x - I_y}{2}}$ - $I_{max/min} = \frac{I_x + I_y}{2} \pm \sqrt{\left(\frac{I_x - I_y}{2}\right)^2 + I_{xy}^2}$ - **Radius of Gyration ($k$):** $k = \sqrt{\frac{I}{A}}$ ### Virtual Work - **Principle of Virtual Work:** If a body is in equilibrium, the total virtual work done by all external forces acting on the body is zero for any virtual displacement consistent with the constraints. - $\delta U = \sum F_i \delta s_i + \sum M_j \delta \theta_j = 0$ - Used to determine equilibrium configuration or unknown forces without dismembering the structure. - **Virtual Displacement:** An imaginary, infinitesimal displacement that is consistent with the system's constraints. - **Conservative Forces:** Forces for which the work done is independent of the path. - **Potential Energy ($V$):** For a conservative force, $F = -\frac{dV}{ds}$. - **Gravitational Potential Energy:** $V_g = Wy$ - **Elastic Potential Energy (Spring):** $V_e = \frac{1}{2}ks^2$ - **Principle of Potential Energy:** A body is in equilibrium if the first variation of its total potential energy is zero ($\delta V = 0$). - **Stable Equilibrium:** $\delta^2 V > 0$ (minimum potential energy). - **Unstable Equilibrium:** $\delta^2 V