Hibbeler Mechanics Cheatsheet

Cheatsheet Content

### Vectors and Forces - **Scalar:** Quantity with magnitude only (mass, time, length). - **Vector:** Quantity with magnitude and direction (force, velocity, acceleration). - **Vector Addition (Parallelogram Law):** $\vec{R} = \vec{A} + \vec{B}$ - Head-to-tail rule. - Resultant (R), Law of Cosines: $R = \sqrt{A^2 + B^2 - 2AB \cos C}$ - Law of Sines: $\frac{A}{\sin a} = \frac{B}{\sin b} = \frac{C}{\sin c}$ - **Cartesian Vectors:** - $\vec{F} = F_x \hat{i} + F_y \hat{j} + F_z \hat{k}$ - Magnitude: $|\vec{F}| = \sqrt{F_x^2 + F_y^2 + F_z^2}$ - Unit Vector: $\hat{u}_F = \frac{\vec{F}}{|\vec{F}|} = \cos\alpha \hat{i} + \cos\beta \hat{j} + \cos\gamma \hat{k}$ - Direction Cosines: $\cos\alpha = \frac{F_x}{|\vec{F}|}$, etc. - **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 angle between vectors or projection of one vector onto another. - Projection of $\vec{A}$ onto $\vec{B}$: $A_B = \vec{A} \cdot \hat{u}_B$ ### Equilibrium of a Particle - **Newton's First Law:** $\sum \vec{F} = 0$ - **2D Equilibrium:** - $\sum F_x = 0$ - $\sum F_y = 0$ - **3D Equilibrium:** - $\sum F_x = 0$ - $\sum F_y = 0$ - $\sum F_z = 0$ - **Free-Body Diagram (FBD):** Essential for solving equilibrium problems. Show all external forces acting on the particle. ### Force System Resultants - **Moment of a Force (Torque):** Tendency of a force to rotate a body about a point or axis. - Scalar: $M_O = Fd$ (d = perpendicular distance from O to force line of action). - Vector: $\vec{M}_O = \vec{r} \times \vec{F}$ - $\vec{r}$ is position vector from O to any point on line of action of $\vec{F}$. - $\vec{r} \times \vec{F} = (r_y F_z - r_z F_y)\hat{i} + (r_z F_x - r_x F_z)\hat{j} + (r_x F_y - r_y F_x)\hat{k}$ - **Moment about an Axis:** $M_{axis} = \hat{u}_{axis} \cdot (\vec{r} \times \vec{F})$ - **Couple Moment:** Two parallel forces, equal in magnitude, opposite in direction, separated by distance 'd'. - $M = Fd$ (magnitude) - Vector: $\vec{M} = \vec{r} \times \vec{F}$ (where $\vec{r}$ connects points on lines of action of the two forces). - **Resultant of a Force System:** - $\vec{F}_R = \sum \vec{F}$ - $\vec{M}_{R_O} = \sum (\vec{r} \times \vec{F}) + \sum \vec{M}_{couples}$ - **Wrench:** A force-couple system that is equivalent to any general force system. ### Equilibrium of a Rigid Body - **Conditions for Equilibrium:** - $\sum \vec{F} = 0$ (Translational equilibrium) - $\sum \vec{M}_O = 0$ (Rotational equilibrium about any point O) - **2D Equations:** - $\sum F_x = 0$ - $\sum F_y = 0$ - $\sum M_O = 0$ - **3D Equations:** - $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$ - $\sum M_x = 0$, $\sum M_y = 0$, $\sum M_z = 0$ - **Supports and Reactions:** - **Roller:** One unknown force (perpendicular to surface). - **Pin/Hinge:** Two unknown forces ($F_x, F_y$). - **Fixed Support:** Three unknown forces ($F_x, F_y$) and one unknown moment ($M_z$) in 2D; Three forces and three moments in 3D. - **Two-Force Member:** A member subjected to forces at only two points. Forces must be equal, opposite, and collinear. - **Three-Force Member:** A member subjected to forces at three points. Forces must be concurrent or parallel. ### Trusses, Frames, and Machines - **Trusses:** Members connected at their ends by pins to form a stable structure. All members are two-force members (axial compression or tension). - **Method of Joints:** Apply $\sum F_x = 0, \sum F_y = 0$ at each joint. - **Method of Sections:** Cut through members, then apply rigid body equilibrium equations ($\sum F_x = 0, \sum F_y = 0, \sum M_O = 0$) to a section. - **Frames and Machines:** Structures containing multi-force members. - Disassemble the structure into its component parts. - Draw FBD for each component. - Apply rigid body equilibrium for each component. - Action-reaction forces between connected members are equal and opposite. ### Center of Gravity and Centroid - **Center of Gravity (CG):** Point where the entire weight of a body appears to act. - $\bar{x} = \frac{\sum W_i x_i}{\sum W_i}$ (similarly for $\bar{y}, \bar{z}$) - **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: $\bar{x} = \frac{\sum \bar{x}_i A_i}{\sum A_i}$ (similarly for $\bar{y}$) - **Theorems of Pappus and Guldinus:** - **Area of surface of revolution:** $A = \theta \bar{r} L$ (where L is arc length, $\bar{r}$ is perpendicular distance from centroid of arc to axis of revolution, $\theta$ is angle in radians). - **Volume of body of revolution:** $V = \theta \bar{r} A$ (where A is area, $\bar{r}$ is perpendicular distance from centroid of area to axis of revolution). ### Moments 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:** $J_O = \int r^2 dA = I_x + I_y$ - **Parallel-Axis Theorem:** $I = I_c + Ad^2$ - $I_c$ is moment of inertia about centroidal axis. - $A$ is area. - $d$ is perpendicular distance between the parallel axes. - **Radius of Gyration:** $k = \sqrt{\frac{I}{A}}$ - **Mass Moment of Inertia:** Measures resistance to angular acceleration. - $I = \int r^2 dm$ - For composite bodies: $I = \sum I_i$ (if all about the same axis) - Parallel-Axis Theorem for Mass: $I = I_c + md^2$ ### Kinematics of a Particle (Rectilinear Motion) - **Position:** $s(t)$ - **Velocity:** $v = \frac{ds}{dt}$ - **Acceleration:** $a = \frac{dv}{dt} = \frac{d^2s}{dt^2}$ - **Relationship:** $a ds = v dv$ - **Constant Acceleration ($a_c$):** - $v = v_0 + a_c t$ - $s = s_0 + v_0 t + \frac{1}{2} a_c t^2$ - $v^2 = v_0^2 + 2 a_c (s - s_0)$ ### Kinetics of a Particle - **Newton's Second Law:** $\sum \vec{F} = m\vec{a}$ - **Rectangular Coordinates:** $\sum F_x = ma_x$, $\sum F_y = ma_y$, $\sum F_z = ma_z$ - **Normal and Tangential Coordinates:** - $\sum F_t = ma_t = m\frac{dv}{dt}$ (tangential acceleration, changes speed) - $\sum F_n = ma_n = m\frac{v^2}{\rho}$ (normal acceleration, changes direction, $\rho$ is radius of curvature) - **Cylindrical Coordinates:** - $\sum F_r = m(\ddot{r} - r\dot{\theta}^2)$ - $\sum F_\theta = m(r\ddot{\theta} + 2\dot{r}\dot{\theta})$ - $\sum F_z = m\ddot{z}$ - **Work and Energy:** - **Work of a Force:** $U_{1-2} = \int_{s_1}^{s_2} \vec{F} \cdot d\vec{r} = \int_{s_1}^{s_2} F \cos\theta ds$ - **Work of Weight:** $U_{1-2} = -W \Delta y$ - **Work of a Spring:** $U_{1-2} = \frac{1}{2} k (s_1^2 - s_2^2)$ - **Principle of Work and Energy:** $T_1 + U_{1-2} = T_2$ - Kinetic Energy: $T = \frac{1}{2} mv^2$ - **Conservation of Energy (Conservative Forces):** $T_1 + V_1 = T_2 + V_2$ - Potential Energy: $V = V_g + V_e$ - Gravitational Potential Energy: $V_g = Wy$ - Elastic Potential Energy: $V_e = \frac{1}{2} ks^2$ - **Impulse and Momentum:** - **Linear Impulse:** $\vec{I} = \int_{t_1}^{t_2} \vec{F} dt$ - **Linear Momentum:** $\vec{p} = m\vec{v}$ - **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:** $\sum (m\vec{v})_1 = \sum (m\vec{v})_2$ (when sum of external impulses is zero). - **Impact:** - **Coefficient of Restitution:** $e = \frac{(v_B)_2 - (v_A)_2}{(v_A)_1 - (v_B)_1}$ (for central impact) - $e=1$ (perfectly elastic), $e=0$ (perfectly plastic) - **Angular Momentum:** - **For a particle:** $\vec{H}_O = \vec{r} \times m\vec{v}$ - **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:** $(\vec{H}_O)_1 = (\vec{H}_O)_2$ (when sum of external moments about O is zero). ### Kinematics of a Rigid Body - **Translational Motion:** All points have the same velocity and acceleration. - **Rotational Motion about a Fixed Axis:** - Angular Position: $\theta$ - Angular Velocity: $\omega = \frac{d\theta}{dt}$ - Angular Acceleration: $\alpha = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}$ - Relations: $\alpha d\theta = \omega d\omega$ - **Constant Angular Acceleration ($\alpha_c$):** - $\omega = \omega_0 + \alpha_c t$ - $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha_c t^2$ - $\omega^2 = \omega_0^2 + 2 \alpha_c (\theta - \theta_0)$ - Velocity of a point P: $\vec{v}_P = \vec{\omega} \times \vec{r}_P$ - Acceleration of a point P: $\vec{a}_P = \vec{\alpha} \times \vec{r}_P + \vec{\omega} \times (\vec{\omega} \times \vec{r}_P) = \vec{\alpha} \times \vec{r}_P - \omega^2 \vec{r}_P$ - Tangential: $(a_P)_t = \alpha r_P$ - Normal: $(a_P)_n = \omega^2 r_P = v_P^2/r_P$ - **General Plane Motion:** Combination of translation and rotation. - **Relative Velocity:** $\vec{v}_B = \vec{v}_A + \vec{v}_{B/A}$ where $\vec{v}_{B/A} = \vec{\omega} \times \vec{r}_{B/A}$ - **Instantaneous Center (IC) of Zero Velocity:** Point on the body (or extension) that has zero velocity at that instant. - $\vec{v}_P = \omega r_P$ (magnitude), direction perpendicular to $r_P$. - **Relative Acceleration:** $\vec{a}_B = \vec{a}_A + \vec{a}_{B/A}$ where $\vec{a}_{B/A} = \vec{\alpha} \times \vec{r}_{B/A} - \omega^2 \vec{r}_{B/A}$ ### Kinetics of a Rigid Body - **Equations of Motion:** - **Translation:** $\sum \vec{F} = m\vec{a}_G$ - **Rotation about Fixed Axis ($O$):** $\sum M_O = I_O \alpha$ - **General Plane Motion:** - $\sum F_x = m(a_G)_x$ - $\sum F_y = m(a_G)_y$ - $\sum M_G = I_G \alpha$ (moment about center of mass G) - OR $\sum M_P = I_P \alpha + m a_G d$ (moment about any point P, if P is not G) - **Work and Energy (Rigid Body):** - **Kinetic Energy:** $T = \frac{1}{2} mv_G^2 + \frac{1}{2} I_G \omega^2$ - **Principle of Work and Energy:** $T_1 + U_{1-2} = T_2$ - **Work of a couple:** $U_{1-2} = \int_{\theta_1}^{\theta_2} M d\theta$ - **Impulse and Momentum (Rigid Body):** - **Linear Impulse and Momentum:** $m(\vec{v}_G)_1 + \sum \int \vec{F} dt = m(\vec{v}_G)_2$ - **Angular Impulse and Momentum:** $(H_G)_1 + \sum \int M_G dt = (H_G)_2$ - $(H_G) = I_G \omega$ (for rotation about G) - **Angular Impulse and Momentum about a Fixed Axis O:** $(H_O)_1 + \sum \int M_O dt = (H_O)_2$ - $(H_O) = I_O \omega$ - **Conservation of Momentum:** If external impulses/moments are zero, linear/angular momentum is conserved.