General Physics I: Classical Mechanics Encyclopedic Guide

General Physics I: Classical Mechanics Introduction

  • Definition of Physics: Physics is the most fundamental of the sciences, aimed at discovering the basic laws of the Universe.
  • Theoretical Physics: Focuses on developing theory and mathematics of physical laws.
  • Applied Physics: Focuses on applying principles to practical problems.
  • Experimental Physics: Sits at the intersection of physics and engineering; practitioners build and work with scientific equipment.
  • Major Fields:
    • Classical Mechanics: The study of motion based on Newton’s laws.
    • Electricity and Magnetism: Together considered a single related field.
    • Quantum Mechanics: Motion of bodies at atomic sizes and smaller.
    • Optics & Acoustics: The study of light and sound, respectively.
    • Thermodynamics & Statistical Mechanics: The study of the nature of heat.
    • Solid-state, Plasma, and Atomic/Nuclear Physics: Studies of specific states and scales of matter.
    • Relativity: Einstein’s theories of high-speed motion (Special) and gravity (General).

Deductive Logic and Problem-Solving

  • Methodology: Physicists start with known facts and relevant equations/definitions to deduce a conclusion.
  • Example: To find average speed if a body travels (700 \text{ m}) in (10 \text{ s}), use (v_{\text{ave}} = \text{distance} / \text{time}) to get (70 \text{ m/s}).
  • Strategies:
    • Units should be converted to SI base units immediately.
    • Derive algebraic solutions before substituting numbers to reduce round-off error.
    • Check results for reasonableness (e.g., a pendulum bob shouldn't travel (14,000 \text{ mph})).

Systems of Units and SI Standards

  • History: SI (Système International d’unités) was modernized in May 2019, redefining units based on fixed physical constants rather than physical prototypes.
  • Redefined Constants (2019):
    • Frequency: \Delta\nu(^{133}Cs)_{\text{hfs}} = 9,192,631,770 \text{ Hz}.
    • Velocity (c): Speed of light is exactly (299,792,458 \text{ m/s}).
    • Action (h): Planck constant is (6.62607015 \times 10^{-34} \text{ J s}).
    • Electric Charge (e): (1.602176634 \times 10^{-19} \text{ C}).
    • Boltzmann constant (k_B): (1.380649 \times 10^{-23} \text{ J/K}).
    • Avogadro constant (N_A): (6.02214076 \times 10^{23} \text{ mol}^{-1}).
  • Mass vs. Weight: Mass (m, measured in kg) is the amount of matter; Weight (W, measured in Newtons) is the gravitational force. W = mg, where (g = 9.80 \text{ m/s}^2).
  • Dimensional Analysis: Every term in an equation must have the same units. Arguments of functions (sin, log, exp) and exponents must be dimensionless.

Kinematics in One Dimension

  • Displacement (\Delta x): \Delta x = x2 - x1. It is the net distance, not total distance.
  • Instantaneous Velocity: v = \frac{dx}{dt}.
  • Instantaneous Acceleration: a = \frac{dv}{dt} = \frac{d^2x}{dt^2}.
  • Dot Notation: Newton's shorthand where \dot{x} = v and \ddot{x} = a.
  • Constant Acceleration Equations:
    • x(t) = \frac{1}{2}at^2 + v0t + x0
    • v(t) = at + v_0
    • v^2 = v0^2 + 2a(x - x0)
  • Geometric Interpretations:
    • Slope of (x \text{ vs. } t) is velocity.
    • Slope of (v \text{ vs. } t) is acceleration.
    • Area under (v \text{ vs. } t) is displacement.
    • Area under (a \text{ vs. } t) is change in velocity.

Vectors and Vector Algebra

  • Representations: Polar form (A\angle\theta) or Rectangular form \mathbf{A} = Ax\mathbf{i} + Ay\mathbf{j} + A_z\mathbf{k}.
  • Magnitude: |\mathbf{A}| = \sqrt{Ax^2 + Ay^2 + A_z^2}.
  • Dot Product: \mathbf{A} \cdot \mathbf{B} = AB \cos \theta = Ax Bx + Ay By + Az Bz. Result is a scalar.
  • Cross Product: \mathbf{A} \times \mathbf{B} = (AB \sin \theta) \mathbf{\hat{u}}. Result is a vector perpendicular to both \mathbf{A} and \mathbf{B}.
  • Properties: Cross product is anti-commutative: \mathbf{A} \times \mathbf{B} = -\mathbf{B} \times \mathbf{A}.

Projectile Motion

  • Independence of Motion: Horizontal and vertical components are independent. Objects fire horizontally fall at the same rate as those dropped.
  • Standard Formulae:
    • Range: R = \frac{v_0^2}{g} \sin 2\theta. Maximum at (45^\circ).
    • Time in flight: tf = \frac{2v0 \sin \theta}{g}.
    • Maximum Altitude: h = \frac{v_0^2 \sin^2 \theta}{2g}.
  • Monkey and Hunter Paradox: The bullet always hits the falling monkey because gravity accelerates both at the same rate (g), causing them to fall identical distances from their inertial paths.

Dynamics: Newton's Laws and Force

  • Newton's Laws:
    1. Law of Inertia: Bodies stay at rest/uniform motion unless a force acts.
    2. F = ma: More accurately \mathbf{F} = \frac{d\mathbf{p}}{dt}. Net force equals mass times acceleration.
    3. Action-Reaction: Forces always exist in equal and opposite pairs.
  • Friction:
    • Static Friction: fs \le \mus n.
    • Kinetic Friction: fk = \muk n.
    • Note: \mu depends on surfaces and is usually determined experimentally as (\mu = \tan \theta).
  • Hooke’s Law: \mathbf{F} = -k\mathbf{x}. (k) is the spring constant.
  • Resistive Forces in Fluids:
    • Model I (Low speed): FR = -bv. Leads to terminal velocity (v\infty = mg/b).
    • Model II (High speed): FR = \frac{1}{2} CD \rho A v^2. Terminal velocity is \sqrt{\frac{2mg}{C_D \rho A}}.

Work, Energy, and Power

  • Work: \mathbf{W} = \int \mathbf{F} \cdot d\mathbf{r}. Measured in Joules.
  • Kinetic Energy: K = \frac{1}{2}mv^2.
  • Potential Energy:
    • Gravity (Near surface): U = mgh.
    • Gravity (Planetary): U = -\frac{Gm1m2}{r}.
    • Spring: U = \frac{1}{2}kx^2.
  • Conservation of Energy: In a closed system, (K + U) remains constant.
  • Virial Theorem: For (F \propto r^n), average \langle K \rangle = \frac{n+1}{2} \langle U \rangle.
  • Power: P = \frac{dE}{dt} = \mathbf{F} \cdot \mathbf{v}. Measured in Watts.

Rotational Motion and Rigid Bodies

  • Analogy with Linear Motion: Angle (\theta), Angular Velocity (\omega), Angular Acceleration (\alpha).
  • Moment of Inertia (I): Resistance to rotation. I = \int r^2 dm.
    • Solid Sphere: I = \frac{2}{5}MR^2.
    • Thin Hoop: I = MR^2.
    • Parallel Axis Theorem: I = I_{\text{cm}} + Mh^2.
  • Torque: \mathbf{\tau} = \mathbf{r} \times \mathbf{F}. In pure rotation, \tau = I\alpha.
  • Rolling Bodies: Total kinetic energy is (K{\text{rot}} + K{\text{trans}}). Acceleration of a rolling body down an incline is a = \frac{g \sin \theta}{\beta + 1}, where (\beta = I_{\text{cm}}/MR^2).

Advanced Mechanics: Lagrangian and Hamiltonian

  • Lagrangian Mechanics: Uses (L = K - U). Solve \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{x}} \right) - \frac{\partial L}{\partial x} = 0.
  • Hamiltonian Mechanics: Uses total energy (H = K + U) as a function of position (x) and momentum (p).
    • \frac{dx}{dt} = \frac{\partial H}{\partial p} and \frac{dp}{dt} = -\frac{\partial H}{\partial x}.

Fluid Statics and Dynamics

  • Archimedes’ Principle: Buoyant force equals weight of displaced fluid. Fraction of floating body submerged is (\rhob / \rhof).
  • Bernoulli’s Equation: \frac{P}{\rho g} + \frac{v^2}{2g} + y = \text{constant}. Relates pressure, speed, and height.
  • Viscosity: Internal friction in fluids. Low Reynolds number flow follows Stokes’s Law: (F_R = 6\pi \eta r v).
  • Superfluids: Liquid Helium II exists below the lambda point (2.17 \text{ K}), exhibiting zero viscosity and the "fountain effect".

Celestial Mechanics

  • Kepler’s Laws:
    1. Orbits are ellipses with the Sun at one focus.
    2. Equal areas are swept in equal times.
    3. Period squared is proportional to semi-major axis cubed: (P^2 \propto a^3).
  • Vis Viva Equation: v = \sqrt{GM\left(\frac{2}{r} - \frac{1}{a}\right)}, giving velocity anywhere in an orbit.
  • Escape Velocity: v_e = \sqrt{\frac{2GM}{R}}. For Earth, this is (11.2 \text{ km/s}).