Comprehensive Physics Study Notes: Units, Measurement, Mechanics, and Gravitation

The Physical World and the Scientific Method

• Definition of Science: Science is a systematic attempt to understand natural phenomena in as much detail and depth as possible, and use the knowledge so gained to predict, modify and control phenomena.
• Etymology: The word 'Science' originates from the Latin verb 'Scientia' meaning 'to know'. Sanskrit 'Vijnan' and Arabic 'Ilm' convey the same meaning of 'knowledge'.
• The Scientific Method: Involves several interconnected steps:
  • Systematic observations.
  • Controlled experiments.
  • Qualitative and quantitative reasoning.
  • Mathematical modelling.
  • Prediction and verification or falsification of theories.
• Dynamics of Science: Science is ever dynamic with no 'final' theory or unquestioned authority. Theories must account for new, more precise results.
  • Example (Copernicus to Kepler): Nicholas Copernicus proposed circular orbits in the heliocentric theory; Johannes Kepler replaced them with elliptical orbits to better fit data collected by Tycho Brahe.
  • Example (Quantum Reform): Newtonian mechanics could not explain atomic phenomena, and wave theory failed the photoelectric effect, leading to Quantum Mechanics.
• Principles of Physics:
  • Unification: Attempting to explain diverse physical phenomena in terms of a few concepts and laws (e.g., Newton's law of gravitation governing falling apples and planetary motion).
  • Reductionism: Deriving properties of a complex system from its constituent parts (e.g., thermodynamics interpreted via kinetic theory/molecular constituents).

Units and Measurement

• Measurement: Comparison with an arbitrarily chosen, internationally accepted reference standard called a unit.
• Systems of Units:
  • CGS: Centimetre, gram, second.
  • FPS (British): Foot, pound, second.
  • MKS: Metre, kilogram, second.
  • SI (Système Internationale d’ Unites): Recommended in 1971 by the General Conference on Weights and Measures.
• SI Base Quantities and Units:
  • Length: metre ($m$). Length of light path in vacuum during $1/299,792,458$ of a second.
  • Mass: kilogram ($kg$). Mass of the international prototype (platinum-iridium alloy) at Sevres, France.
  • Time: second ($s$). Duration of $9,192,631,770$ periods of radiation from cesium-133 transition.
  • Electric Current: ampere ($A$).
  • Thermodynamic Temperature: kelvin ($K$). Fraction $1/273.16$ of the triple point of water.
  • Amount of Substance: mole ($mol$). Elementary entities equal to atoms in $0.012$ kg of carbon-12.
  • Luminous Intensity: candela ($cd$).
• Supplementary Units:
  • Plane Angle: radian ($rad$). Ratio of arc length to radius ($d\theta = ds/r$).
  • Solid Angle: steradian ($sr$). Ratio of intercepted area to square of radius ($d\Omega = dA/r^2$).
• Length Measurement (Indirect):
  • Parallax Method: Used for Large Distances ($D$).
  • Equation: $D = b/\theta$ where $b$ is the basis (distance between observation points) and $\theta$ is the parallax angle in radians.
  • Molecular size estimation: Uses oleic acid on water. Thickness $t = V/A$ where $V$ is volume and $A$ is area.
• Common Length Units:
  • $1\,\text{fermi} = 10^{-15}\,m$
  • $1\,\text{angstrom} (\text{Å}) = 10^{-10}\,m$
  • $1\,\text{astronomical unit} (AU) = 1.496 \times 10^{11}\,m$
  • $1\,\text{light year} (ly) = 9.46 \times 10^{15}\,m$
  • $1\,\text{parsec} = 3.08 \times 10^{16}\,m$
• Errors in Measurement:
  • Accuracy: How close the measurement is to the true value.
  • Precision: The resolution or limit to which the quantity is measured.
  • Systematic Errors: Tend to be in one direction (Instrumental, imperfection in technique, personal error).
  • Random Errors: Occur irregularly in sign and size.
  • Least Count Error: Error associated with the resolution of the instrument.
• Mathematical Error Relations:
  • Absolute Error: $|\Delta a_i| = |a_i - a_{mean}|$
  • Relative Error: $\frac{\Delta a_{mean}}{a_{mean}}$
  • Percentage Error: $\delta a = (\frac{\Delta a_{mean}}{a_{mean}}) \times 100\%$
  • Combination of Errors:
    • Sum/Difference ($Z = A \pm B$): $\Delta Z = \Delta A + \Delta B$
    • Product/Quotient ($Z = AB$ or $A/B$): $\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}$
    • Power ($Z = A^k$): $\frac{\Delta Z}{Z} = k \frac{\Delta A}{A}$
• Significant Figures: Reliable digits plus the first uncertain digit.
  • Trailing zeros with a decimal point are significant (e.g., $4.700$ has 4).
  • Trailing zeros without a decimal point are not significant (e.g., $12300$ has 3).
• Dimensions: Powers to which base quantities ($[M], [L], [T], [A], [K], [mol], [cd]$) are raised.
  • Principle of Homogeneity: Physical quantities can be added/subtracted only if they have the same dimensions.

Kinematics: Motion in a Straight Line

• Frame of Reference: A coordinate system ($x, y, z$) along with a clock.
• Path Length vs. Displacement:
  • Path Length: Total distance covered (scalar).
  • Displacement: Change in position ($\Delta x = x_2 - x_1$) (vector).
• Velocity:
  • Average Velocity: $v_{avg} = \Delta x / \Delta t$ (Slope of line joining two points on an $x-t$ graph).
  • Average Speed: Total path length / Total time interval.
  • Instantaneous Velocity: $v = dx/dt$ (Limit of average velocity as time becomes infinitesimal).
• Acceleration:
  • $a = dv/dt = d^2x/dt^2$.
  • Area under a $v-t$ graph represents displacement during the time interval.
• Kinematic Equations for Uniform Acceleration ($a =$ constant):
  • $v = v_0 + at$
  • $x = v_0t + \frac{1}{2}at^2$
  • $x = x_0 + v_0t + \frac{1}{2}at^2$
  • $v^2 = v_0^2 + 2ax$
  • $v^2 = v_0^2 + 2a(x - x_0)$
• Free Fall: Motion under gravity with $a = -g = -9.8\,m/s^2$.
• Relative Velocity in 1D: Velocity of B relative to A: $v_{BA} = v_B - v_A$.

Kinematics: Motion in a Plane

• Vectors: Quantities with magnitude and direction obeying triangle/parallelogram law.
  • Unit Vector: $\mathbf{\hat{n}} = \mathbf{A}/|\mathbf{A}|$ (magnitude 1). $\mathbf{\hat{i}}, \mathbf{\hat{j}}, \mathbf{\hat{k}}$ for $x, y, z$.
  • Vector Components: $\mathbf{A} = A_x \mathbf{\hat{i}} + A_y \mathbf{\hat{j}}$; where $A_x = A \cos(\theta)$ and $A_y = A \sin(\theta)$.
  • Resultant Magnitude: $R = \sqrt{A^2 + B^2 + 2AB \cos(\theta)}$.
  • Law of Sines: $\frac{R}{\sin(\theta)} = \frac{A}{\sin(\beta)} = \frac{B}{\sin(\alpha)}$.
• Projectile Motion:
  • Horizontal velocity: $v_x = v_{0x} = v_0 \cos(\theta_0)$. (Constant).
  • Vertical velocity: $v_y = v_{0y} - gt = v_0 \sin(\theta_0) - gt$.
  • Equation of Trajectory (Parabola): $y = \tan(\theta_0)x - \frac{g x^2}{2 (v_0 \cos(\theta_0))^2}$.
  • Time of Flight: $T_f = \frac{2 v_0 \sin(\theta_0)}{g}$.
  • Horizontal Range: $R = \frac{v_0^2 \sin(2\theta_0)}{g}$. (Max range at $\theta = 45^{\circ}$).
  • Maximum Height: $h_m = \frac{(v_0 \sin(\theta_0))^2}{2g}$.
• Uniform Circular Motion:
  • Centripetal Acceleration ($a_c$): Directed toward center. $a_c = v^2 / R = \omega^2 R$.
  • Angular speed ($\omega$): $\omega = d\theta / dt$.
  • Relation: $v = \omega R$.
  • Frequency ($\nu$): $1/T$. $\omega = 2\pi\nu$.

Laws of Motion

• Newton's First Law (Law of Inertia): Every body continues in its state of rest or uniform motion unless compelled by an external force ($F_{net} = 0 \rightarrow a = 0$).
• Newton's Second Law: The rate of change of momentum is proportional to the applied force.
  • Momentum $\mathbf{p} = m \mathbf{v}$.
  • $\mathbf{F} = d\mathbf{p}/dt = m\mathbf{a}$.
  • Force is a vector; it acts locally and defines acceleration "here and now".
• Impulse: Product of force and time duration. $\mathbf{I} = \mathbf{F} \Delta t = \Delta \mathbf{p}$.
• Newton's Third Law: To every action, there is always an equal and opposite reaction ($\mathbf{F}_{AB} = -\mathbf{F}_{BA}$).
• Conservation of Momentum: In an isolated system (no external force), total linear momentum is conserved.
• Common Forces:
  • Static Friction ($f_s$): Resists impending motion. $f_s \leq \mu_s N$.
  • Kinetic Friction ($f_k$): Resists actual relative motion. $f_k = \mu_k N$.
  • Rolling Friction: Resistance to rolling; much smaller than $\mu_k$.
• Circular Motion Dynamics:
  • Centripetal Force: $F_c = mv^2 / R$.
  • Level road max speed: $v_{max} = \sqrt{\mu_s Rg}$.
  • Banked road optimum speed (no friction): $v_0 = \sqrt{Rg \tan(\theta)}$.
  • Banked road max speed: $v_{max} = [Rg \frac{\mu_s + \tan(\theta)}{1 - \mu_s \tan(\theta)}]^{1/2}$.

Work, Energy and Power

• Scalar Product (Dot Product): $\mathbf{A} \cdot \mathbf{B} = AB \cos(\theta)$.
  • $\mathbf{\hat{i}} \cdot \mathbf{\hat{i}} = 1$; $\mathbf{\hat{i}} \cdot \mathbf{\hat{j}} = 0$.
• Work ($W$): $W = \mathbf{F} \cdot \mathbf{d} = F d \cos(\theta)$.
  • Variable Force: $W = \int_{x_i}^{x_f} F(x)\,dx$.
• Kinetic Energy ($K$): $K = \frac{1}{2} m v^2$.
• Work-Energy Theorem: $W_{net} = \Delta K = K_f - K_i$.
• Potential Energy ($V$): Energy by virtue of position. Defined for conservative forces as $F(x) = -dV/dx$.
  • Gravitational PE: $V(h) = mgh$.
  • Spring PE (Hooke’s Law): $V(x) = \frac{1}{2} k x^2$.
• Conservation of Mechanical Energy: If only conservative forces do work, $E = K + V =$ constant.
• Non-conservative forces: Friction converts mechanical energy into heat ($E_f - E_i = W_{nc}$).
• Einstein's Relation: $E = mc^2$ ($1\,kg$ equivalent to $9 \times 10^{16}\,J$).
• Power ($P$): Time rate of doing work. $P = dW/dt = \mathbf{F} \cdot \mathbf{v}$.
  • Unit: watt ($W$). $1\,hp = 746\,W$. $1\,kWh = 3.6 \times 10^6\,J$.
• Collisions:
  • Elastic: Both momentum and kinetic energy are conserved ($v_{1f} = [\frac{m_1 - m_2}{m_1 + m_2}]v_{1i}$ when $m_2$ is at rest).
  • Inelastic: Momentum conserved, KE not conserved.
  • Completely Inelastic: Bodies move together after collision ($v_f = \frac{m_1 v_{1i}}{m_1 + m_2}$).

System of Particles and Rotational Motion

• Rigid Body: Unchanging shape; distances between all pairs of particles remain constant.
• Centre of Mass (CM): Point where entire mass moves as if concentrated.
  • $\mathbf{R} = \frac{\sum m_i \mathbf{r}_i}{M}$.
  • For continuous mass: $\mathbf{R} = \frac{1}{M} \int \mathbf{r}\,dm$.
  • Motion: $M \mathbf{A} = \mathbf{F}_{ext}$.
• Vector Product (Cross Product): $\mathbf{a} \times \mathbf{B} = \mathbf{c}$, where $|\mathbf{c}| = ab \sin(\theta)$.
  • Direction given by right-handed screw rule.
  • $\mathbf{\hat{i}} \times \mathbf{\hat{j}} = \mathbf{\hat{k}}$ (cyclical order).
• Rotational variables:
  • Angular Velocity: $\mathbf{v} = \mathbf{\omega} \times \mathbf{r}$.
  • Torque (Moment of force): $\mathbf{\tau} = \mathbf{r} \times \mathbf{F}$.
  • Angular Momentum (Moment of momentum): $\mathbf{l} = \mathbf{r} \times \mathbf{p}$.
  • Relation: $\mathbf{\tau} = d\mathbf{L}/dt$.
• Equilibrium of Rigid Body:
  • Translational: $\sum \mathbf{F}_i = 0$.
  • Rotational: $\sum \mathbf{\tau}_i = 0$.
• Moment of Inertia ($I$): Rotational analogue of mass. $I = \sum m_i r_i^2$.
  • Fixed axis KE: $K = \frac{1}{2} I \omega^2$.
  • Radius of Gyration ($k$): $I = Mk^2$.
• Theorems of MI:
  • Perpendicular Axis (Planar bodies): $I_z = I_x + I_y$.
  • Parallel Axis: $I_{any} = I_{CM} + Ma^2$ (where $a$ is distance from CM axis).
• Rolling Motion (without slipping): $\mathbf{v}_{cm} = R \omega$.
  • Total KE: $K = \frac{1}{2} M v_{cm}^2 + \frac{1}{2} I \omega^2 = \frac{1}{2} M v^2 (1 + \frac{k^2}{R^2})$.

Gravitation

• Kepler’s Laws:
  1. Orbits: Planets move in elliptical orbits with the Sun at one focus.
  2. Areas: A line joining planet and Sun sweeps equal areas in equal times ($dA/dt = L/2m =$ constant).
  3. Periods: $T^2 \propto a^3$ ($a =$ semi-major axis).
• Universal Law of Gravitation: $F = G \frac{m_1 m_2}{r^2}$.
  • $G = 6.673 \times 10^{-11}\,N\cdot m^2\,kg^{-2}$.
• Acceleration due to gravity ($g$): $g = G M_E / R_E^2 \approx 9.8\,m/s^2$.
  • With height ($h$): $g(h) \approx g(1 - \frac{2h}{R_E})$.
  • With depth ($d$): $g(d) = g(1 - \frac{d}{R_E})$.
• Gravitational Potential Energy ($V$): $V(r) = -G \frac{M m}{r}$. (Zero at infinity).
• Escape Speed ($v_e$): Minimum speed to leave gravitational field. $v_e = \sqrt{2 G M_E / R_E} = \sqrt{2 g R_E}$. ($11.2\,km/s$ for Earth).
• Satellites:
  • Orbital Speed: $V = \sqrt{G M_E / (R_E + h)}$.
  • Time Period: $T = 2\pi (R_E+h)^{3/2} / (G M_E)^{1/2}$. (~85 min for close orbits).
  • Energy: Total $E = K + V = -G M m / [2(R_E+h)]$. (KE = |Total E|; PE = 2 × Total E).
  • Geostationary: Orbit in equatorial plane with $T = 24$ hours ($h \approx 35,800\,km$).
  • Polar: Low altitude (~500-800 km) passing over the poles.
• Weightlessness: State of free fall where normal reaction from support is zero.