Comprehensive University Physics Study Guide

PHYSICS AND MEASUREMENTS

• Definition of Physics: Physics deals with the study of matter and energy and the relationship between them. It involves investigating laws of motion, space and time, forces, particle interactions, and electromagnetic radiation.
• Frontiers of Fundamental Science:
  • The World of the Extremely Large: The universe itself, radio telescopes, and the Big Bang.
  • The World of the Extremely Small: Subatomic particles like electrons, protons, and neutrons.
  • The World of Complex Matter: Middle-sized things from molecules to the Earth.
• Physical Quantities: These consist of a numerical magnitude and a unit. They are divided into:
  • Base Quantities: Minimal number of quantities in terms of which others are defined (Length, Mass, Time, Electric Current, Temperature, Luminous Intensity, Amount of Substance).
  • Derived Quantities: Definitions based on other physical quantities (Velocity, Force, etc.).
• System International (SI) Units:
  • Base Units: Metre ($m$), Kilogram ($kg$), Second ($s$), Ampere ($A$), Kelvin ($K$), Candela ($cd$), Mole ($mol$).
  • Supplementary Units: Purely geometrical quantities.
    • Radian ($rad$): Planar angle between two radii of a circle cutting an arc equal to the radius.
    • Steradian ($sr$): Solid angle subtended at the center of a sphere by an area equal to the square of the radius.
• Scientific Notation: Numbers expressed as $M \times 10^n$, where there is only one non-zero digit to the left of the decimal.
• Errors and Uncertainties:
  • Random Error: Occurs when repeated measurements give different values under same conditions; reduced by averaging multiple readings.
  • Systematic Error: Influences all measurements equally (e.g., zero error, poor calibration); reduced by applying correction factors or comparing with more accurate instruments.
• Significant Figures:
  • Rules: All non-zero digits are significant. Zeros between significant figures are significant. Zeros to the left are not. In decimals, zeros to the right are significant.
  • Calculations: In multiplication/division, the result holds significant figures equal to the least accurate factor. In addition/subtraction, the result holds decimal places equal to the least precise term.
• Precision vs. Accuracy:
  • Precision: Determined by the instrument's least count (absolute uncertainty).
  • Accuracy: Depends on the fractional or percentage uncertainty in the measurement.
• Dimensional Analysis: Each base quantity is assigned a symbol ($[L]$, $[M]$, $[T]$). Used to check homogeneity of equations and derive formulae.

VECTORS AND EQUILIBRIUM

• Vector Definition: Quantities requiring both magnitude and direction (e.g., Force, Velocity). Represented by bold letters ($\mathbf{A}$) or symbols with arrowheads ($\vec{A}$).
• Rectangular Coordinate System: Cartesian system with x, y, and z axes. Position vector ($\mathbf{r}$) describes a point relative to the origin: $\mathbf{r} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k}$.
• Vector Addition: Use the "head to tail rule." Resultant vector $\mathbf{R} = \mathbf{A} + \mathbf{B}$. Vector addition is commutative ($\mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A}$).
• Vector Multiplication:
  • Scalar (Dot) Product: $\mathbf{A} \cdot \mathbf{B} = AB \cos(\theta)$. Result is a scalar (e.g., Work done: $W = \mathbf{F} \cdot \mathbf{d}$).
  • Vector (Cross) Product: $\mathbf{A} \times \mathbf{B} = (AB \sin(\theta))\mathbf{n}$. Result is a vector perpendicular to the plane of $\mathbf{A}$ and $\mathbf{B}$ (e.g., Torque: $\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}$).
• Torque (Moment of Force): The turning effect of a force. Magnitude $\tau = rF \sin(\theta)$. SI Unit is Newton-metre ($N\,m$).
• Conditions of Equilibrium:
  • First Condition: The vector sum of all forces must be zero ($\sum \mathbf{F} = 0$). This ensures translational equilibrium.
  • Second Condition: The vector sum of all torques must be zero ($\sum \boldsymbol{\tau} = 0$). This ensures rotational equilibrium.

MOTION AND FORCE

• Displacement ($\mathbf{d}$): Change in position of a body ($\mathbf{d} = \mathbf{r}_2 - \mathbf{r}_1$).
• Velocity: Rate of change of displacement. $\mathbf{v}<em>{inst} = \lim</em>{\Delta t \to 0} \frac{\Delta \mathbf{d}}{\Delta t}$.
• Acceleration: Rate of change of velocity. $\mathbf{a} = \lim_{\Delta t \to 0} \frac{\Delta \mathbf{v}}{\Delta t}$.
• Velocity-Time Graphs: Slope represents acceleration. Area under the curve represents the distance covered.
• Newton's Laws:
  • 1st Law (Inertia): Body remains at rest or uniform motion unless acted on by net force.
  • 2nd Law: $\mathbf{F} = m\mathbf{a}$. Rate of change of momentum equals applied force ($\mathbf{F} = \frac{\Delta \mathbf{p}}{\Delta t}$).
  • 3rd Law: For every action, there is an equal and opposite reaction.
• Momentum ($\mathbf{p}$): $m\mathbf{v}$. SI unit is $kg\,m\,s^{-1}$ or $N\,s$.
• Impulse ($\mathbf{I}$): $\mathbf{F} \times t = \Delta \mathbf{p}$. Product of force and the short time it acts.
• Law of Conservation of Momentum: The total linear momentum of an isolated system remains constant.
• Collisions:
  • Elastic: K.E. and momentum are both conserved. Relative velocity of approach equals relative velocity of separation ($v_1 - v_2 = v'_2 - v'_1$).
  • Inelastic: Momentum conserved, K.E. is not.
• Projectile Motion: Two-dimensional motion under constant acceleration of gravity.
  • Horizontal component: $v_{ix} = v_i \cos(\theta)$ (remains constant).
  • Vertical component: $v_{fy} = v_i \sin(\theta) - g\,t$.
  • Time of flight: $t = \frac{2v_i \sin(\theta)}{g}$.
  • Maximum Height: $h = \frac{v_i^2 \sin^2(\theta)}{2g}$.
  • Horizontal Range: $R = \frac{v_i^2 \sin(2\theta)}{g}$. (Maximum at $45^\circ$).

WORK AND ENERGY

• Work ($W$): $\mathbf{F} \cdot \mathbf{d} = Fd \cos(\theta)$. SI unit is Joule ($J$).
• Power ($P$): Rate of doing work. $P = \frac{\Delta W}{\Delta t} = \mathbf{F} \cdot \mathbf{v}$. Unit is Watt ($W$). Commercial unit: Kilowatt-hour ($1\,kWh = 3.6 \times 10^6\,J$).
• Energy Types:
  • Kinetic Energy (K.E.): $\frac{1}{2}mv^2$.
  • Potential Energy (P.E.): $mgh$ (near Earth's surface).
• Work-Energy Principle: Work done on a body equals the change in its kinetic energy.
• Absolute Potential Energy: Work done to move an object from a point to infinity: $U = -\frac{G M m}{r}$.
• Escape Velocity ($v_{esc}$): Initial velocity needed to leave Earth's gravitational field: $v_{esc} = \sqrt{2gR} \approx 11\,km/s$.
• Conservation of Energy: Energy cannot be created or destroyed, only transformed.
• Non-Conventional Energy Sources: Tides, waves, solar, biomass, waste products, and geothermal energy.

CIRCULAR MOTION

• Angular Displacement ($\theta$): Angle subtended at center. $S = r\theta$.
• Angular Velocity ($\omega$): $\lim_{\Delta t \to 0} \frac{\Delta \theta}{\Delta t}$. Unit is $rad/s$.
• Relationship between Linear and Angular: $v = r\omega$; $a_t = r\alpha$.
• Centripetal Acceleration ($a_c$): Acceleration towards center: $a_c = \frac{v^2}{r} = r\omega^2$.
• Centripetal Force ($F_c$): Tension or force maintaining circular motion: $F_c = \frac{mv^2}{r} = m\omega^2r$.
• Moment of Inertia ($I$): Rotational analogue of mass: $I = \sum m_i r_i^2$. Torque $\tau = I\alpha$.
• Angular Momentum ($L$): $\mathbf{L} = \mathbf{r} \times \mathbf{p}$. For rigid bodies, $L = I\omega$.
• Conservation of Angular Momentum: If net external torque is zero, total angular momentum is constant.
• Rotational K.E.: $K.E._{rot} = \frac{1}{2}I\omega^2$.
• Satellites: Critical velocity for close orbit $v = \sqrt{gR} \approx 7.9\,km/s$. Time period $T \approx 84\,min$.
• Geostationary Satellites: Fixed relative to Earth's surface. Orbit radius $\approx 4.23 \times 10^4\,km$. Height above equator $\approx 36,000\,km$.

FLUID DYNAMICS

• Viscosity (\eta): Internal friction between fluid layers.
• Stokes' Law: Retarding drag force on a sphere: $F = 6\pi \eta r v$.
• Terminal Velocity ($v_t$): Constant velocity when drag force equals weight: $v_t = \frac{2gr^2\rho}{9\eta}$.
• Flow Types: Streamline (laminar) vs. Turbulent.
• Equation of Continuity: Based on conservation of mass: $A_1 v_1 = A_2 v_2 = \text{constant}$.
• Bernoulli's Equation: Based on conservation of energy: $P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$.
  • Torricelli's Theorem: Efflux speed $v = \sqrt{2g(h_1 - h_2)}$.
  • Venturi Effect: Pressure is low where fluid speed is high.
• Blood Flow: Systolic pressure ($120\,torr$) and Diastolic pressure ($75-80\,torr$).

OSCILLATIONS

• Simple Harmonic Motion (SHM): Acceleration proportional to displacement and towards mean position ($a = -\omega^2 x$).
• Horizontal Mass-Spring System: Period $T = 2\pi \sqrt{\frac{m}{k}}$.
• Simple Pendulum: Period $T = 2\pi \sqrt{\frac{L}{g}}$. Independent of mass.
• Energy in SHM: Total energy is constant; interchange between K.E. = $\frac{1}{2}k(x_0^2 - x^2)$ and P.E. = $\frac{1}{2}kx^2$.
• Resonance: Large amplitude oscillations when driving frequency matches natural frequency.
  • Applications: Tuning radios, microwave ovens.
  • Disadvantages: Structural failure in bridges/aircraft.
• Damping: Energy dissipation reducing amplitude over time.

WAVES

• Wave Types: Transverse (displacement perpendicular to propagation) and Longitudinal (displacement along propagation).
• Wave Equation: $v = f \lambda$.
• Speed of Sound: $v = \sqrt{\frac{E}{\rho}}$.
  • Newton's Formula: $v = \sqrt{\frac{P}{\rho}} \approx 280\,m/s$ (Inaccurate).
  • Laplace Correction: Adiabatic process assumed; $v = \sqrt{\frac{\gamma P}{\rho}} \approx 333\,m/s$.
  • Temperature Influence: $v_t = v_0 + 0.61t$.
• Superposition Principle: Resultant displacement is algebraic sum of individual wave displacements.
• Beats: Periodic variations in intensity due to interference of two waves of slightly different frequencies. Beat frequency = $f_1 - f_2$.
• Stationary Waves: Produced by two identical waves travelling in opposite directions.
  • Nodes: Zero displacement points. Antinodes: Maximum displacement points.
  • Stretched String: Frequencies $f_n = n f_1 = \frac{n}{2L} \sqrt{\frac{F}{m}}$.
• Doppler Effect: Apparent change in frequency due to relative motion.
  • Moving Observer: $f_A = f\left(\frac{v+u_o}{v}\right)$.
  • Moving Source: $f' = f\left(\frac{v}{v-u_s}\right)$.
  • Applications: Radar, Sonar, Red shift in Astronomy.

PHYSICAL OPTICS

• Huygen's Principle: Every point of a wavefront acts as a source of secondary wavelets.
• Interference (Young's Double Slit):
  • Bright Fringes (Maxima): $d \sin(\theta) = m \lambda$.
  • Dark Fringes (Minima): $d \sin(\theta) = (m + \frac{1}{2})\lambda$.
  • Fringe Spacing: $\Delta y = \frac{\lambda L}{d}$.
• Newton's Rings: Circular fringes from an air film between a lens and glass plate.
• Michelson Interferometer: Used to measure distances with high precision (L = rac{m \lambda}{2}).
• Diffraction: Bending of light around obstacles.
  • Grating formula: $d \sin(\theta) = n \lambda$.
  • X-ray Diffraction (Bragg's Law): $2d \sin(\theta) = n \lambda$.
• Polarization: Confining vibrations to one plane; proves light is a transverse wave. Use Polaroids to eliminate glare.

OPTICAL INSTRUMENTS

• Least Distance of Distinct Vision ($d$): $25\,cm$ for a normal eye.
• Magnification ($M$): Ratio of visual angle with instrument compared to naked eye.
  • Simple Microscope: $M = 1 + \frac{d}{f}$.
  • Compound Microscope: $M = \frac{q}{p}(1 + \frac{d}{f_e})$.
  • Astronomical Telescope: $M = \frac{f_o}{f_e}$. Length in normal adjustment = $f_o + f_e$.
• Resolving Power: Ability to see details. $\theta_{min} = 1.22 \frac{\lambda}{D}$.
• Speed of Light ($c$): Measured by Michelson rotating mirror: $c = 16fd$. Accepted value $3.00 \times 10^8\,m/s$.
• Fibre Optics: Light transmission via total internal reflection or continuous refraction.
  • Types: Single mode step index, Multimode step index, Multimode graded index.

THERMODYNAMICS

• Pressure of Gas: $P = \frac{2}{3} N_0 \langle \frac{1}{2} m v^2 \rangle$.
• Temperature Interpretation: Absolute temperature is proportional to average translational K.E. ($T ∝ \langle K.E. ≫$).
• 1st Law of Thermodynamics: $\Delta Q = \Delta U + \Delta W$.
• Isothermal Process: Constant temperature ($\Delta U = 0$); $PV = \text{constant}$.
• Adiabatic Process: No heat exchange ($Q = 0$); $PV^{\gamma} = \text{constant}$.
• Molar Specific Heats: $C_p - C_v = R$.
• Heat Engines: Convert thermal energy to mechanical work via a cyclic process.
  • Carnot Cycle: Four steps (Isothermal expansion, Adiabatic expansion, Isothermal compression, Adiabatic compression).
  • Efficiency: $\eta = 1 - \frac{T_2}{T_1}$.
• 2nd Law of Thermodynamics: Heat cannot flow spontaneously from cold to hot resen/oirs; single reservoir engines are impossible.
• Entropy ($S$): Measure of disorder. Change $\Delta S = \frac{\Delta Q}{T}$. Entropy of the universe always increases.