Hibbeler Engineering Mechanics

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1. General Principles Newton's Laws: 1st: A particle remains at rest or continues to move with constant velocity if there is no unbalanced force acting on it. 2nd: $\vec{F} = m\vec{a}$ (for constant mass) 3rd: Action-reaction forces between two particles are equal in magnitude and opposite in direction. Units: SI: mass (kg), length (m), time (s), force (N) US Customary: force (lb), length (ft), time (s), mass (slug) Conversion: $1 \text{ slug} = 1 \text{ lb} \cdot \text{s}^2 / \text{ft}$ Gravitational Acceleration ($g$): SI: $9.81 \text{ m/s}^2$ US Customary: $32.2 \text{ ft/s}^2$ Weight ($W$): $W = mg$ 2. Force Vectors 2.1 2D Force Systems Rectangular Components: $F_x = F \cos \theta$, $F_y = F \sin \theta$ Magnitude: $F = \sqrt{F_x^2 + F_y^2}$ Direction: $\theta = \arctan(F_y / F_x)$ Unit Vector: $\vec{u} = \vec{F} / F = (\cos \theta)\vec{i} + (\sin \theta)\vec{j}$ 2.2 3D Force Systems Rectangular Components: $\vec{F} = F_x\vec{i} + F_y\vec{j} + F_z\vec{k}$ Magnitude: $F = \sqrt{F_x^2 + F_y^2 + F_z^2}$ Coordinate Direction Angles: $\cos \alpha = F_x/F$, $\cos \beta = F_y/F$, $\cos \gamma = F_z/F$ Identity: $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$ Unit Vector: $\vec{u} = \frac{\vec{F}}{F} = \cos \alpha \vec{i} + \cos \beta \vec{j} + \cos \gamma \vec{k}$ Position Vector: $\vec{r} = (x_B - x_A)\vec{i} + (y_B - y_A)\vec{j} + (z_B - z_A)\vec{k}$ Force from Position Vector: $\vec{F} = F (\vec{r} / r)$ 2.3 Dot Product $\vec{A} \cdot \vec{B} = AB \cos \theta$ $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z$ Angle between Vectors: $\theta = \arccos \left( \frac{\vec{A} \cdot \vec{B}}{AB} \right)$ Projection of $\vec{A}$ onto $\vec{B}$: $A_B = \vec{A} \cdot \vec{u}_B$ 3. Equilibrium of a Particle Condition: $\sum \vec{F} = \vec{0}$ 2D Equations: $\sum F_x = 0$, $\sum F_y = 0$ 3D Equations: $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$ Free-Body Diagram (FBD): Essential for problem-solving. 4. Force System Resultants 4.1 Moment of a Force Scalar (2D): $M_O = Fd$ (force $\times$ perpendicular distance) Vector (3D): $\vec{M}_O = \vec{r} \times \vec{F}$ $$ \vec{M}_O = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ r_x & r_y & r_z \\ F_x & F_y & F_z \end{vmatrix} $$ Varignon's Theorem: The moment of a force about a point is equal to the sum of the moments of its components about the point. 4.2 Moment of a Couple $\vec{M} = \vec{r} \times \vec{F}$ (where $\vec{r}$ is from negative to positive force) Magnitude: $M = Fd$ 4.3 Reduction of a Simple Distributed Loading Resultant Force ($F_R$): Area under the loading curve. Location ($\bar{x}$): Centroid of the area under the loading curve. 5. Equilibrium of a Rigid Body 5.1 2D Equilibrium Equations: $\sum F_x = 0$ $\sum F_y = 0$ $\sum M_O = 0$ (moment about any point $O$) Supports & Reactions: Roller/Rocker: 1 unknown (normal force $\perp$ surface) Pin/Hinge: 2 unknowns ($F_x, F_y$) Fixed Support: 3 unknowns ($F_x, F_y, M$) 5.2 3D Equilibrium Equations: $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$ $\sum M_x = 0$, $\sum M_y = 0$, $\sum M_z = 0$ Supports & Reactions: Ball & Socket: 3 forces ($F_x, F_y, F_z$) Journal Bearing: 2 forces ($\perp$ shaft) and 2 moments (if not self-aligning) Fixed Support: 3 forces, 3 moments 6. Trusses, Frames, and Machines 6.1 Trusses Assumptions: Members are two-force members. Loads applied at joints. Joints are pinned. Methods: Method of Joints: Apply particle equilibrium ($\sum F_x=0, \sum F_y=0$) at each joint. Method of Sections: Cut through members, apply rigid body equilibrium ($\sum F_x=0, \sum F_y=0, \sum M=0$) to a section. Zero-Force Members: Often at unloaded joints connecting two non-collinear members, or at joints with three members where two are collinear and no external load. Determinacy: $b+r = 2j$ (statically determinate), $b+r > 2j$ (statically indeterminate) $b$: number of members, $r$: number of reactions, $j$: number of joints 6.2 Frames and Machines Components are multi-force members. Disassemble the structure and draw FBD for each component. Action-reaction forces between connected members. 7. Internal Forces Normal Force ($N$): Perpendicular to cross-section. Shear Force ($V$): Parallel to cross-section. Bending Moment ($M$): Caused by forces tending to bend the member. Sign Convention: Normal: Tension (+), Compression (-) Shear: Up on left, down on right (+) Moment: Concave up (smiley face) (+) Relations: $w = dV/dx$ (distributed load and shear) $V = dM/dx$ (shear and moment) $\Delta V = \int w(x) dx$ $\Delta M = \int V(x) dx$ 8. Friction Static Friction ($F_s$): $F_s \le \mu_s N$ (prevents motion) Kinetic Friction ($F_k$): $F_k = \mu_k N$ (occurs during motion) $\mu_s > \mu_k$ Angle of Static Friction: $\phi_s = \arctan(\mu_s)$ Wedges: Often involve friction on multiple surfaces. Cables & Belts (Flexible Belts): $T_2 = T_1 e^{\mu \beta}$ ($\beta$ in radians) 9. Center of Gravity & Centroid Center of Gravity ($\bar{x}, \bar{y}, \bar{z}$): $\bar{x} = \frac{\sum \tilde{x} W}{\sum W}$, $\bar{y} = \frac{\sum \tilde{y} W}{\sum W}$, $\bar{z} = \frac{\sum \tilde{z} W}{\sum W}$ (for composite bodies) Centroid of Area ($\bar{x}, \bar{y}$): $\bar{x} = \frac{\int \tilde{x} dA}{\int dA}$, $\bar{y} = \frac{\int \tilde{y} dA}{\int dA}$ For composite areas: $\bar{x} = \frac{\sum \tilde{x} A}{\sum A}$, $\bar{y} = \frac{\sum \tilde{y} A}{\sum A}$ Centroid of Line ($\bar{x}, \bar{y}$): $\bar{x} = \frac{\sum \tilde{x} L}{\sum L}$, $\bar{y} = \frac{\sum \tilde{y} L}{\sum L}$ Theorems of Pappus-Guldinus: Area of surface of revolution ($A = \theta \bar{r} L$) Volume of body of revolution ($V = \theta \bar{r} A$) 10. Moments of Inertia 10.1 Area Moments of Inertia $I_x = \int y^2 dA$, $I_y = \int x^2 dA$ Polar Moment of Inertia: $J_O = \int r^2 dA = I_x + I_y$ Parallel-Axis Theorem: $I = \bar{I} + Ad^2$ $\bar{I}$: moment of inertia about centroidal axis $A$: area, $d$: distance between parallel axes Radius of Gyration: $k = \sqrt{I/A}$ 10.2 Mass Moments of Inertia $I_x = \int (y^2 + z^2) dm$, $I_y = \int (x^2 + z^2) dm$, $I_z = \int (x^2 + y^2) dm$ For 2D bodies (thin plates) about $z$-axis: $I_z = \int r^2 dm$ Parallel-Axis Theorem: $I = \bar{I} + md^2$ 11. Virtual Work Principle of Virtual Work: $\delta U = 0$ for equilibrium. $\delta U = \sum F_i \delta r_i + \sum M_j \delta \theta_j = 0$ For a system in equilibrium, the total virtual work done by all active forces during any virtual displacement is zero. Used for systems with multiple interconnected bodies. Stability: Stable: $\Delta V_g > 0$ (potential energy minimum) Unstable: $\Delta V_g Neutral: $\Delta V_g = 0$ (potential energy constant) 12. Kinematics of a Particle 12.1 Rectilinear Motion $v = ds/dt$ $a = dv/dt = d^2s/dt^2$ $a ds = v dv$ Constant Acceleration: $v = v_0 + a_c t$ $s = s_0 + v_0 t + \frac{1}{2} a_c t^2$ $v^2 = v_0^2 + 2 a_c (s - s_0)$ 12.2 Curvilinear Motion Rectangular Components: $\vec{r} = x\vec{i} + y\vec{j} + z\vec{k}$ $\vec{v} = \dot{x}\vec{i} + \dot{y}\vec{j} + \dot{z}\vec{k}$ $\vec{a} = \ddot{x}\vec{i} + \ddot{y}\vec{j} + \ddot{z}\vec{k}$ Normal & Tangential Components: $v = \dot{s}$ $a_t = \dot{v} = v dv/ds$ (rate of change of speed) $a_n = v^2/\rho$ (rate of change of direction, $\rho$ = radius of curvature) $\vec{a} = a_t \vec{u}_t + a_n \vec{u}_n$ Cylindrical Components: $\vec{r} = r\vec{u}_r + z\vec{k}$ $\vec{v} = \dot{r}\vec{u}_r + r\dot{\theta}\vec{u}_\theta + \dot{z}\vec{k}$ $\vec{a} = (\ddot{r} - r\dot{\theta}^2)\vec{u}_r + (r\ddot{\theta} + 2\dot{r}\dot{\theta})\vec{u}_\theta + \ddot{z}\vec{k}$ 12.3 Relative Motion $\vec{r}_B = \vec{r}_A + \vec{r}_{B/A}$ $\vec{v}_B = \vec{v}_A + \vec{v}_{B/A}$ $\vec{a}_B = \vec{a}_A + \vec{a}_{B/A}$ 13. Kinetics of a Particle 13.1 Equation of Motion $\sum \vec{F} = m\vec{a}$ 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: $\sum F_r = m(\ddot{r} - r\dot{\theta}^2)$, $\sum F_\theta = m(r\ddot{\theta} + 2\dot{r}\dot{\theta})$, $\sum F_z = m\ddot{z}$ 13.2 Work & Energy Work of a Force: $U_{1-2} = \int \vec{F} \cdot d\vec{r}$ Work of Gravity: $U_g = -W \Delta y$ Work of Spring: $U_s = \frac{1}{2} k (s_1^2 - s_2^2)$ Kinetic Energy: $T = \frac{1}{2} m v^2$ Principle of Work & Energy: $T_1 + \sum U_{1-2} = T_2$ Power: $P = \vec{F} \cdot \vec{v} = Fv \cos \theta$ Conservative Forces: Gravity, Elastic Spring. Non-conservative: Friction, Drag. Conservation of Energy: $T_1 + V_1 = T_2 + V_2$ (only for conservative forces) $V = V_g + V_e = W y + \frac{1}{2} k s^2$ 13.3 Impulse & Momentum Linear Momentum: $\vec{L} = m\vec{v}$ Linear Impulse: $\text{Imp} = \int \vec{F} dt$ Principle of Linear Impulse & Momentum: $m\vec{v}_1 + \sum \int \vec{F} dt = m\vec{v}_2$ Conservation of Linear Momentum: $m_A\vec{v}_{A1} + m_B\vec{v}_{B1} = m_A\vec{v}_{A2} + m_B\vec{v}_{B2}$ (when $\sum \vec{F}_{ext} = 0$) Impact: Coefficient of Restitution ($e$): $e = \frac{(v_B)_2 - (v_A)_2}{(v_A)_1 - (v_B)_1}$ $e=1$: elastic, $e=0$: plastic Angular Momentum: $(\vec{H}_O)_1 + \sum \int \vec{M}_O dt = (\vec{H}_O)_2$ $\vec{H}_O = \vec{r} \times m\vec{v}$ Conservation of Angular Momentum: $(\vec{H}_O)_1 = (\vec{H}_O)_2$ (when $\sum \vec{M}_O = \vec{0}$) 14. Planar Kinematics of a Rigid Body Translation: All points have same velocity and acceleration. Rotation About a Fixed Axis: $\omega = d\theta/dt$, $\alpha = d\omega/dt = \omega d\omega/d\theta$ $v = r\omega$, $a_t = r\alpha$, $a_n = r\omega^2$ Absolute General Plane Motion Analysis: Use position vectors, differentiate. Relative-Motion Analysis: $\vec{v}_B = \vec{v}_A + \vec{v}_{B/A} = \vec{v}_A + \vec{\omega} \times \vec{r}_{B/A}$ $\vec{a}_B = \vec{a}_A + \vec{a}_{B/A} = \vec{a}_A + \vec{\alpha} \times \vec{r}_{B/A} - \omega^2 \vec{r}_{B/A}$ Instantaneous Center of Zero Velocity (IC): Point on body with zero velocity. $\vec{v}_P = \omega r_{P/IC}$. 15. Planar Kinetics of a Rigid Body Equations of Motion: $\sum F_x = m(\bar{a}_x)$ $\sum F_y = m(\bar{a}_y)$ $\sum M_G = \bar{I}\alpha$ (moment about center of mass $G$) Alternatively: $\sum M_P = \sum (\mathcal{M}_k)_P$ (moment about any point $P$) Work & Energy: $T = \frac{1}{2} m \bar{v}^2 + \frac{1}{2} \bar{I} \omega^2$ Principle of Work & Energy: $T_1 + \sum U_{1-2} = T_2$ Conservation of Energy: $T_1 + V_1 = T_2 + V_2$ Impulse & Momentum: $\sum \int \vec{F} dt = m(\vec{v}_G)_2 - m(\vec{v}_G)_1$ $\sum \int M_G dt = \bar{I}\omega_2 - \bar{I}\omega_1$ Alternatively for any point $P$: $\sum \int M_P dt = (H_P)_2 - (H_P)_1$ $(H_P)_1 = (\bar{I}\omega)_1 + m(\vec{r}_{G/P} \times \vec{v}_G)_1$ 16. 3D Kinematics of a Rigid Body Angular Velocity: $\vec{\omega}$ Angular Acceleration: $\vec{\alpha} = d\vec{\omega}/dt$ Velocity of a Point: $\vec{v}_B = \vec{v}_A + \vec{\omega} \times \vec{r}_{B/A}$ Acceleration of a Point: $\vec{a}_B = \vec{a}_A + \vec{\alpha} \times \vec{r}_{B/A} + \vec{\omega} \times (\vec{\omega} \times \vec{r}_{B/A})$ Moving Reference Frame: $\vec{v}_B = \vec{v}_A + \vec{\Omega} \times \vec{r}_{B/A} + (\vec{v}_{B/A})_{xyz}$ $\vec{a}_B = \vec{a}_A + \vec{\dot{\Omega}} \times \vec{r}_{B/A} + \vec{\Omega} \times (\vec{\Omega} \times \vec{r}_{B/A}) + 2\vec{\Omega} \times (\vec{v}_{B/A})_{xyz} + (\vec{a}_{B/A})_{xyz}$ $\vec{\Omega}$: angular velocity of moving frame $(\vec{v}_{B/A})_{xyz}$, $(\vec{a}_{B/A})_{xyz}$: relative velocity/acceleration in moving frame $2\vec{\Omega} \times (\vec{v}_{B/A})_{xyz}$: Coriolis acceleration 17. 3D Kinetics of a Rigid Body Equations of Motion: $\sum \vec{F} = m\vec{a}_G$ $\sum \vec{M}_G = \vec{\dot{H}}_G$ For body with fixed point $O$: $\sum \vec{M}_O = \vec{\dot{H}}_O$ Angular Momentum about G: $\vec{H}_G = I_x \omega_x \vec{i} + I_y \omega_y \vec{j} + I_z \omega_z \vec{k}$ (principal axes) Euler's Equations of Motion: $\sum M_x = I_x \dot{\omega}_x - (I_y - I_z)\omega_y \omega_z$ $\sum M_y = I_y \dot{\omega}_y - (I_z - I_x)\omega_z \omega_x$ $\sum M_z = I_z \dot{\omega}_z - (I_x - I_y)\omega_x \omega_y$ Gyroscopic Motion: Precession, Nutation, Spin.