Fundamental Concepts in Mechanics: Motion, Forces, Energy, and Rotational Dynamics

Representing Motion

Displacement, Velocity, and Acceleration

Understanding motion begins with the concepts of displacement, velocity, and acceleration. These quantities describe how an object's position changes over time.

• Displacement (Δx, Δy): The change in position of an object. $\Delta x = x_f - x_i$ $\Delta y = y_f - y_i$

• Time Interval (Δt): The difference between the final and initial time. $\Delta t = t_f - t_i$

• Constant Velocity: If velocity is constant, displacement is given by: $\Delta x = v_x \Delta t$ $\Delta y = v_y \Delta t$

Motion in One and Two Dimensions

Kinematic Equations for Constant Acceleration

When acceleration is constant, the following kinematic equations describe the motion:

• $v_{xf} = v_{xi} + a_x \Delta t$

• $x_f = x_i + v_{xi} \Delta t + \frac{1}{2} a_x \Delta t^2$

• $v_{xf}^2 = v_{xi}^2 + 2a_x \Delta x$

• Similarly for the y-direction:

• $v_{yf} = v_{yi} + a_y \Delta t$

• $y_f = y_i + v_{yi} \Delta t + \frac{1}{2} a_y \Delta t^2$

• $v_{yf}^2 = v_{yi}^2 + 2a_y \Delta y$

Acceleration due to gravity: $g = 9.8\ \mathrm{m/s^2}$

Vectors and Trigonometry in Physics

Right Triangle Relationships

Trigonometric functions relate the sides of a right triangle to its angles, which is essential for resolving vectors into components.

• Pythagoras Theorem: $c^2 = a^2 + b^2$

• Sine: $\sin(\theta) = \frac{b}{c}$

• Cosine: $\cos(\theta) = \frac{a}{c}$

• Tangent: $\tan(\theta) = \frac{b}{a}$

Vector Components

Any vector can be broken into perpendicular components, typically along the x and y axes.

• $A_x = A \cos \theta$

• $A_y = A \sin \theta$

• $\theta = \tan^{-1} \left( \frac{A_y}{A_x} \right)$

Forces and Newton's Laws of Motion

Net Force and Free Body Diagrams

Forces are vector quantities that can be combined to produce a net force. Newton's Second Law relates net force to acceleration:

• Net Force: $\vec{F}_{net} = \vec{F}_1 + \vec{F}_2 + \cdots + \vec{F}_N$

• Newton's Second Law: $\vec{F}_{net} = m \vec{a}$

• Free Body Diagrams: Diagrams that show all forces acting on an object, represented as a point.

• Action-Reaction Pairs: Forces always come in pairs, equal in magnitude and opposite in direction, but act on different bodies.

Friction

• Static friction: $f_s \leq \mu_s N$

• Kinetic friction: $f_k = \mu_k N$

• Rolling friction: $f_R = \mu_R N$

Circular Motion, Orbits, and Gravity

Circular Motion

• Centripetal Acceleration: $a_c = \frac{v^2}{R}$

• Centripetal Force: $F_c = m a_c$

• Period: $T = \frac{2\pi R}{v}$

• Frequency: $f = \frac{1}{T}$

• Speed in terms of frequency: $v = 2\pi R f$

Universal Gravitation

• $F_{12} = F_{21} = \frac{G m_1 m_2}{r^2}$

• Gravitational acceleration at a planet's surface: $g = \frac{GM}{R^2}$

• For satellites in circular orbits: $v = \sqrt{\frac{GM}{r}}$

• Orbital period: $T^2 = \frac{4\pi^2 r^3}{GM}$

Rotational Motion

Angular Quantities and Kinematics

Rotational motion is described using angular displacement, velocity, and acceleration, analogous to linear motion.

• Arc length: $s = R \theta$ (θ in radians)

• Angular velocity: $\omega = \frac{\Delta \theta}{\Delta t}$

• Angular acceleration: $\alpha = \frac{\Delta \omega}{\Delta t}$

• Linear velocity: $v = r \omega$

• Linear acceleration: $a = r \alpha$

• Rotational kinematic equations: Same as linear, with $v \rightarrow \omega$ and $a \rightarrow \alpha$

Torque and Moment of Inertia

• Torque: $\tau = r F_\perp = r F \sin \phi$

• Newton's Second Law (Rotational): $\tau = I \alpha$

• Moment of Inertia: $I = \sum m_i R_i^2$

Equilibrium and Elasticity

Conditions for Equilibrium

• $\sum F_x = 0$ (no net force in x-direction)

• $\sum F_y = 0$ (no net force in y-direction)

• $\sum \tau = 0$ (no net torque)

Hooke's Law and Young's Modulus

• Hooke's Law: $F_{sp} = -k \Delta x = -k (L_0 - L)$

• Stress: $\frac{F}{A}$

• Strain: $\frac{\Delta L}{L}$

• Young's Modulus: $\frac{F}{A} = Y \left( \frac{\Delta L}{L} \right)$

Momentum

Impulse and Conservation of Momentum

• Impulse: $J = F_{avg} \Delta t = \Delta p$ (also area under F vs. t curve)

• Linear momentum: $\vec{p} = m \vec{v}$

• Angular momentum: $L = I \omega$

• Conservation of momentum: $\vec{p}_i = \vec{p}_f$ or $L_i = L_f$

• In 2D: $p_{1xi} + p_{2xi} + \cdots = p_{1xf} + p_{2xf} + \cdots$ and $p_{1yi} + p_{2yi} + \cdots = p_{1yf} + p_{2yf} + \cdots$

Energy and Work

Work, Energy, and Power

• Work: $W = F_{\parallel} d = F d \cos \theta$

• Gravitational potential energy: $U_g = mgh$

• Spring potential energy: $U_s = \frac{1}{2} k x^2$

• Kinetic energy (translation): $K = \frac{1}{2} m v^2$

• Kinetic energy (rotation): $K = \frac{1}{2} I \omega^2$

• Energy conservation: $\Delta E = \Delta K + \Delta U_g + \Delta U_s + \Delta E_{th} + \Delta E_{chem} + \cdots = W$

• For an isolated system: $W = 0$

• Thermal energy (friction): $\Delta E_{th} = f_k \Delta x$

• Thermal energy (drag): $\Delta E_{th} = D \Delta x$

• Work-energy theorem: $\Delta K + \Delta U + \Delta E_{th} = W$

• Elastic collisions (object 2 initially at rest):

  • $(v_{1x})_f = \frac{m_1 - m_2}{m_1 + m_2} (v_{1x})_i$

  • $(v_{2x})_f = \frac{2m_1}{m_1 + m_2} (v_{1x})_i$

• Power: $P = \frac{\Delta E}{\Delta t}$ (energy transformed), $P = \frac{W}{\Delta t}$ (work done)