TMFM$ MECH

Cheatsheet Content

### Vectors and Forces - **Scalar:** Quantity with magnitude only (e.g., mass, time, length). - **Vector:** Quantity with both magnitude and direction (e.g., force, velocity, acceleration). - **Vector Addition (Parallelogram Law):** $\vec{R} = \vec{A} + \vec{B}$ - Place tails of $\vec{A}$ and $\vec{B}$ at same point. - Form a parallelogram; diagonal from common point is $\vec{R}$. - **Vector Addition (Triangle Rule):** - Place tail of $\vec{B}$ at tip of $\vec{A}$. - $\vec{R}$ connects tail of $\vec{A}$ to tip of $\vec{B}$. - **Component Resolution:** - 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: $\cos\alpha = F_x/F$, $\cos\beta = F_y/F$, $\cos\gamma = F_z/F$ - **Dot Product:** $\vec{A} \cdot \vec{B} = AB \cos\theta = A_x B_x + A_y B_y + A_z B_z$ - Used to find angle between vectors or component of a vector along a line. - **Cross Product:** $\vec{C} = \vec{A} \times \vec{B}$ - Magnitude: $C = AB \sin\theta$ - Direction: Right-hand rule. - Determinant form: $\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix}$ - Used to find moment of a force. ### Equilibrium of a Particle - **Newton's First Law:** A particle remains at rest or continues to move with constant velocity unless acted upon by an unbalanced force. - **Condition for Equilibrium:** $\sum \vec{F} = 0$ - In 2D: $\sum F_x = 0$, $\sum F_y = 0$ - In 3D: $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$ - **Free-Body Diagram (FBD):** Essential for solving equilibrium problems. - Isolate the particle. - Show all external forces acting on the particle (e.g., weights, applied forces, tensions, normal forces). - Label known and unknown forces. - Indicate coordinate system. ### Force System Resultants - **Moment of a Force (about a point O):** $\vec{M}_O = \vec{r} \times \vec{F}$ - $\vec{r}$ is position vector from O to any point on line of action of $\vec{F}$. - Magnitude: $M_O = rF\sin\theta = Fd$ (where $d$ is perpendicular distance from O to line of action of $\vec{F}$). - Direction by right-hand rule. - **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. - $\vec{M}_O = \vec{r} \times (\vec{F}_x + \vec{F}_y) = (\vec{r} \times \vec{F}_x) + (\vec{r} \times \vec{F}_y)$ - **Moment of a Force (about an axis):** $M_{axis} = \vec{u}_{axis} \cdot (\vec{r} \times \vec{F})$ - $\vec{u}_{axis}$ is unit vector along the axis. - **Couple:** Two parallel forces of equal magnitude and opposite direction, separated by a perpendicular distance $d$. - **Couple Moment:** $M = Fd$ (magnitude). - Vector form: $\vec{M} = \vec{r} \times \vec{F}$ (where $\vec{r}$ is vector from line of action of $-\vec{F}$ to line of action of $\vec{F}$). - A couple moment is a free vector (can be moved anywhere without changing its effect). - **Resultant of a Force System:** - **Resultant Force:** $\vec{F}_R = \sum \vec{F}$ - **Resultant Couple Moment:** $(\vec{M}_R)_O = \sum \vec{M}_O + \sum \vec{M}_{couples}$ - A general force system can be reduced to a single resultant force and a resultant couple moment at any arbitrary point O. ### Equilibrium of a Rigid Body - **Conditions for Equilibrium:** - $\sum \vec{F} = 0$ (sum of all external forces is zero) - $\sum \vec{M}_O = 0$ (sum of all external moments about any point O is zero) - **2D Equilibrium Equations:** - $\sum F_x = 0$ - $\sum F_y = 0$ - $\sum M_O = 0$ (sum of moments about any point O in the plane) - **3D Equilibrium Equations:** - $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$ - $\sum M_x = 0$, $\sum M_y = 0$, $\sum M_z = 0$ (sum of moments about x, y, z axes) - **Supports and Reactions:** - **Pin/Hinge:** Two force components (usually $F_x, F_y$). - **Roller:** One force component perpendicular to surface. - **Fixed Support (Cantilever):** Three force components ($F_x, F_y$) and one moment component ($M_z$) in 2D; three force and three moment components in 3D. - **Cable/Rope:** Tension force, acting along the cable. - **Smooth Surface:** Normal force, perpendicular to the surface. - **Free-Body Diagram (FBD):** Crucial for rigid body equilibrium. - Isolate the body. - Show all external forces and moments (applied loads, reactions at supports, weights). - Indicate coordinate system. ### Trusses, Frames, and Machines - **Truss:** Structure composed of slender members joined at their endpoints. - **Assumptions:** Members are two-force members (only axial force), forces applied at joints. - **Method of Joints:** 1. Draw FBD for entire truss to find external reactions. 2. Draw FBD for each joint (particle equilibrium). 3. Solve $\sum F_x = 0$, $\sum F_y = 0$ for each joint. 4. Tension (T) if force acts away from joint; Compression (C) if towards joint. - **Method of Sections:** 1. Draw FBD for entire truss to find external reactions. 2. Cut the truss through members whose forces are desired (no more than 3 members intersected by a cut, if all are non-parallel). 3. Draw FBD for one section. 4. Solve equilibrium equations ($\sum F_x = 0$, $\sum F_y = 0$, $\sum M_O = 0$) for the section. - **Zero-Force Members:** - If only two non-collinear members meet at an unloaded joint, both are zero-force members. - If three members meet at a joint, two are collinear, and the third is not loaded externally, the third is a zero-force member. - **Frames and Machines:** Structures with at least one multi-force member (member subjected to more than two forces). - **Analysis:** 1. Treat the entire structure as a rigid body to find external reactions (if determinate). 2. Disassemble the structure into its component members. 3. Draw FBD for each member. 4. Apply equilibrium equations ($\sum F_x = 0$, $\sum F_y = 0$, $\sum M_O = 0$) to each member. 5. Remember Newton's Third Law: Forces between connected members are equal and opposite. ### Friction - **Static Friction ($F_s$):** Opposes impending motion. - $0 \le F_s \le (\mu_s N)$ - $\mu_s$: Coefficient of static friction. This is a dimensionless constant that depends on the properties of the two surfaces in contact. - $N$: Normal force. - $F_s = (\mu_s N)$ at impending motion, which is the maximum static friction force the surfaces can generate. - **Kinetic Friction ($F_k$):** Opposes actual motion. - $F_k = (\mu_k N)$ - $\mu_k$: Coefficient of kinetic friction. - Always less than $\mu_s$. - **Angle of Static Friction ($\phi_s$):** $\tan\phi_s = \mu_s$ - **Angle of Kinetic Friction ($\phi_k$):** $\tan\phi_k = \mu_k$ - **Wedges:** Used to create small movements or apply large forces. Analyze as connected rigid bodies with friction. - **Friction in Bearings and Journal Bearings:** Moment due to friction. - **Rolling Resistance:** Small retarding force due to deformation at point of contact. ### Center of Gravity and Centroid - **Center of Gravity (CG):** Point where the entire weight of a body appears to act. - For discrete masses: $$\bar{x} = \frac{\sum W_i x_i}{\sum W_i}, \quad \bar{y} = \frac{\sum W_i y_i}{\sum W_i}, \quad \bar{z} = \frac{\sum W_i z_i}{\sum W_i}$$ - **Centroid:** Geometric center of an area, volume, or line. - For Area: $$\bar{x} = \frac{\int x dA}{\int dA}, \quad \bar{y} = \frac{\int y dA}{\int dA}$$ - For Composite Areas/Volumes/Lines: - For Area: $$\bar{x} = \frac{\sum \tilde{x}_i A_i}{\sum A_i}, \quad \bar{y} = \frac{\sum \tilde{y}_i A_i}{\sum A_i}$$ - Similar formulas apply for volumes ($V_i$) and lines ($L_i$), replacing $A_i$ with $V_i$ or $L_i$ respectively. - **Theorems of Pappus and Guldinus:** - **Surface Area of Revolution:** $A = \theta \bar{r} L$ (where $\bar{r}$ is the perpendicular distance from the centroid of the line to the axis of revolution, and $\theta$ is the angle of revolution in radians). - **Volume of Revolution:** $V = \theta \bar{r} A$ (where $\bar{r}$ is the perpendicular distance from the centroid of the area to the axis of revolution, and $\theta$ is the angle of revolution in radians). ### Moments of Inertia - **Moment of Inertia of an Area:** Measures resistance to angular acceleration. - $I_x = \int y^2 dA$, $I_y = \int x^2 dA$ - **Polar Moment of Inertia:** $J_O = I_x + I_y = \int r^2 dA$ - **Radius of Gyration:** $k = \sqrt{I/A}$ - $I_x = k_x^2 A$, $I_y = k_y^2 A$, $J_O = k_O^2 A$ - **Parallel-Axis Theorem:** $I = \bar{I} + Ad^2$ - $\bar{I}$: Moment of inertia about centroidal axis. - $A$: Area. - $d$: Perpendicular distance between centroidal axis and parallel axis. - **Moment of Inertia of Composite Areas:** Sum of moments of inertia of individual parts, using the parallel-axis theorem for each part to shift to the desired axis. - **Product of Inertia:** $I_{xy} = \int xy dA$ - **Principal Moments of Inertia:** Maximum and minimum moments of inertia for an area, occurring about principal axes. - $\tan 2\theta_p = \frac{-I_{xy}}{ (I_x - I_y)/2 }$ - $I_{max/min} = \frac{I_x + I_y}{2} \pm \sqrt{ \left( \frac{I_x - I_y}{2} \right)^2 + I_{xy}^2 }$ ### Virtual Work - **Principle of Virtual Work:** If a rigid body is in equilibrium, the virtual work done by all external forces acting on the body is zero for any virtual displacement consistent with the constraints. - $\delta U = \sum F \cos\theta \delta s + \sum M \delta\theta = 0$ - $\delta s$: Virtual displacement. - $\delta\theta$: Virtual rotation. - **Applications:** - Solving for equilibrium of connected bodies without needing to find internal forces (e.g., in frames and machines). - Checking stability of equilibrium. - Determining forces or moments at connections indirectly. - **Conservative Forces:** Forces for which the work done is independent of the path and depends only on the initial and final positions (e.g., gravity, spring forces). - For conservative systems, $\delta V = 0$ for equilibrium, where $V$ is the potential energy. - **Potential Energy:** - Gravity: $V_g = Wy$ - Spring: $V_e = \frac{1}{2}ks^2$ - **Stability of Equilibrium:** - Stable: $\frac{d^2V}{ds^2} > 0$ (minimum potential energy) - Unstable: $\frac{d^2V}{ds^2}