Physical World and Measurement

Physics is the study of the basic laws of nature and their manifestation in diverse natural phenomena. Its scope spans astronomical dimensions (macroscopic domain) down to subatomic particles (microscopic domain):

• Macroscopic Domain: Deals with phenomena at the astronomical, terrestrial, and laboratory scales. It includes Classical Mechanics (motion of bodies, fluids), Electrodynamics (electric and magnetic fields, electromagnetic waves), Optics (light, mirrors, lenses, interference), and Thermodynamics (heat, temperature, entropy, heat engines).
• Microscopic Domain: Deals with the constitution and structure of matter at the scale of atoms, nuclei, quarks, and photons, governed primarily by Quantum Mechanics.
• Mesoscopic Domain: Bridges the gap between macroscopic and microscopic worlds, dealing with assemblies of tens to thousands of atoms (nanotechnology, quantum dots).

All forces encountered in the universe—from muscular tension and friction to nuclear fission—originate from just four fundamental forces:

A physical quantity is any measurable property of a physical system whose value can be expressed as the product of a numerical value ($n$) and a standard unit ($u$): Physical Quantity = n × u. For a constant physical quantity, $n \propto 1/u$, meaning a larger unit yields a smaller numerical magnitude (e.g., $1 ext{ m} = 100 ext{ cm}$).

• Fundamental (Base) Quantities: Quantities that are independent of each other and cannot be defined in terms of other physical quantities (e.g., length, mass, time).
• Derived Quantities: Quantities expressed as mathematical combinations (products or powers) of base quantities (e.g., velocity, force, energy).

In November 2018, the 26th General Conference on Weights and Measures (CGPM) made a historic decision effective from 20 May 2019: all seven SI base units are now defined exclusively in terms of fixed numerical values of seven fundamental physical constants:

The SI system previously designated two supplementary units, which are now formally classified as dimensionless derived units:

• Plane Angle: Unit is radian (rad). $ heta = rac{ ext{arc length } s}{ ext{radius } r}$. Dimension: $[M^0 L^0 T^0]$. Note: $2\pi ext{ rad} = 360^\circ \implies 1 ext{ rad} pprox 57.296^\circ$.
• Solid Angle: Unit is steradian (sr). $\Omega = rac{ ext{intercepted area } A}{r^2}$. Dimension: $[M^0 L^0 T^0]$. Total solid angle around a point is $4\pi ext{ sr}$.

Practical Astronomical & Nuclear Units:

• Astronomical Unit (AU): Mean distance between Earth and Sun $pprox 1.496 imes 10^{11} ext{ m}$.
• Light Year (ly): Distance traversed by light in vacuum in 1 Julian year $= c imes 365.25 imes 86400 pprox 9.461 imes 10^{15} ext{ m}$.
• Parsec (pc): Parallactic second, distance at which an arc of length $1 ext{ AU}$ subtends an angle of 1 arcsecond $= rac{1 ext{ AU}}{1''} = rac{1.496 imes 10^{11}}{\pi / (180 imes 3600)} pprox 3.086 imes 10^{16} ext{ m} pprox 3.26 ext{ ly}$.
• Fermi / Femtometer (fm): $1 ext{ fm} = 10^{-15} ext{ m}$ (nuclear diameter scale).
• Angstrom (Å): $1 ext{ \AA} = 10^{-10} ext{ m} = 0.1 ext{ nm}$ (atomic lattice spacing scale).
• Unified Atomic Mass Unit (u or amu): $1 ext{ u} = rac{1}{12} ext{ mass of } ^{12} ext{C atom} pprox 1.66054 imes 10^{-27} ext{ kg}$.
• Chandrasekhar Limit (CSL): Largest practical unit of mass in astrophysics $= 1.4 imes ext{Solar Mass} pprox 2.8 imes 10^{30} ext{ kg}$.

The dimensions of a physical quantity are the powers (exponents) to which the base quantities must be raised to represent that quantity. The expression showing how and which base quantities enter into a unit is its dimensional formula, enclosed in square brackets $[M^a L^b T^c \dots]$.

The Principle of Homogeneity states that a physical equation is correct only if the dimensions of every term on both sides of the equation are identical. Physical quantities having different dimensions can never be added, subtracted, or equated (you cannot add 5 meters to 3 kilograms!).

For example, in the kinematic equation $s = ut + rac{1}{2}at^2$:

• $[s] = [L]$
• $[ut] = [L T^{-1}][T] = [L]$
• $[rac{1}{2}at^2] = [1][L T^{-2}][T^2] = [L]$

Since each individual term has dimensions of $[L]$, the equation is dimensionally homogeneous and consistent.

1. Checking Equation Correctness: Determine if LHS and RHS terms share matching dimensions. Note: A dimensionally correct equation is not necessarily physically correct (dimensionless constants like $1/2$ or $2\pi$ are invisible to dimensional checks), but a dimensionally incorrect equation is always physically wrong.
2. Unit System Conversion: Since physical quantity $Q = n_1 u_1 = n_2 u_2$, the numerical value in the new system is given by:
   $$n_2 = n_1 \left[rac{M_1}{M_2} ight]^a \left[rac{L_1}{L_2} ight]^b \left[rac{T_1}{T_2} ight]^c$$
3. Deriving Physical Relations: If a physical quantity depends on certain variables, we equate it to the product of powers of those variables, equate exponents of $M, L, T$ on both sides, and solve the resulting system of linear equations.

Every experimental measurement contains unavoidable uncertainty called error. It is essential to distinguish between accuracy and precision:

• Accuracy: Indicates how close the measured value is to the true, accepted value of the physical quantity. Accuracy depends on minimizing systematic errors.
• Precision: Indicates the resolution or degree of refinement of the measuring instrument, representing the closeness of repeated measurements to each other. Precision is determined by the instrument's least count.

1. Systematic Errors: Tend to be in one direction (consistently positive or consistently negative). Sources include:
   • Instrumental errors: Zero error in vernier calipers, zero mark misalignment.
   • Imperfections in experimental technique: Measuring temperature without accounting for radiation loss.
   • Personal errors: Parallax error due to careless alignment of the observer's eye.
   • Correction: Can be calculated and eliminated by calibration and improved technique.
2. Random Errors: Occur irregularly in magnitude and sign due to unpredictable fluctuations in experimental conditions (voltage fluctuations, room temperature, vibrations).
   • Statistical Treatment: Handled via the Gaussian normal distribution. By taking $n$ observations, the random error of the mean decreases by a factor of $\sqrt{n}$ ($\sigma_{ ext{mean}} = \sigma / \sqrt{n}$).
3. Gross Errors: Arise from sheer human negligence, recording improper readings, or computational blunders.

For $n$ measurements of a quantity: $a_1, a_2, \dots, a_n$:

• Mean Value (True Value approximation): $ar{a} = a_{ ext{mean}} = rac{1}{n}\sum_{i=1}^n a_i$.
• Absolute Error of each measurement: $\Delta a_i = |a_i - ar{a}|$. (Always positive magnitude).
• Mean Absolute Error: $\overline{\Delta a} = rac{1}{n}\sum_{i=1}^n \Delta a_i$. Measurement expressed as $a = ar{a} \pm \overline{\Delta a}$.
• Relative (Fractional) Error: $ ext{Relative Error} = rac{\overline{\Delta a}}{ar{a}}$.
• Percentage Error: $ ext{Percentage Error } (\delta a) = rac{\overline{\Delta a}}{ar{a}} imes 100\%$.

The significant figures of a measured quantity represent all reliably known digits plus the first uncertain digit. The standard rules are:

1. All non-zero digits are significant: $287.5 ext{ cm}$ has 4 significant figures.
2. Zeros between non-zero digits are always significant: $4008 ext{ kg}$ has 4 significant figures; $1.0003 ext{ s}$ has 5 significant figures.
3. Leading zeros (zeros to the left of the first non-zero digit) are NEVER significant: They merely position the decimal point. $0.0052 ext{ m}$ has 2 significant figures; $0.030 ext{ g}$ has 2 significant figures.
4. Trailing zeros in a number with a decimal point ARE significant: $4.800 ext{ s}$ has 4 significant figures; $35.0 ext{ mL}$ has 3 significant figures.
5. Trailing zeros in whole numbers without a decimal point are ambiguous: E.g., $4700 ext{ m}$ could have 2, 3, or 4 significant figures. Scientific notation eliminates ambiguity: $4.7 imes 10^3$ (2 sig figs), $4.70 imes 10^3$ (3 sig figs), $4.700 imes 10^3$ (4 sig figs).
6. Exact integers and conversion definitions have infinite significant figures: For instance, $2$ in $2\pi r$, or $1 ext{ inch} = 2.54 ext{ cm}$.

When dropping insignificant digits to reflect experimental precision, follow these rules:

• Case 1: If the digit to be dropped is less than 5, the preceding digit remains unchanged (e.g., $7.82$ rounded to 2 sig figs is $7.8$).
• Case 2: If the digit to be dropped is greater than 5, the preceding digit is increased by 1 (e.g., $6.87$ rounded to 2 sig figs is $6.9$).
• Case 3 (Round-to-Even Rule): If the digit to be dropped is 5 (or 5 followed by zeros):
  • If the preceding digit is even, it remains unchanged (e.g., $4.650$ rounded to 2 sig figs becomes $4.6$).
  • If the preceding digit is odd, it is increased by 1 (e.g., $4.750$ rounded to 2 sig figs becomes $4.8$).

• In Addition and Subtraction: The final calculated result must contain only as many decimal places as the measurement having the smallest number of decimal places.
  Example: $4.327 ext{ g} + 2.1 ext{ g} = 6.427 ext{ g} \implies$ rounded to $6.4 ext{ g}$ (1 decimal place).
• In Multiplication and Division: The final calculated result must contain only as many significant figures as the measurement having the smallest number of significant figures.
  Example: Mass $= 4.237 ext{ g}$ (4 sig figs), Volume $= 2.5 ext{ cm}^3$ (2 sig figs).
  $ ext{Density} = rac{4.237}{2.5} = 1.6948 ext{ g/cm}^3 \implies$ rounded to $1.7 ext{ g/cm}^3$ (2 sig figs).

Invented by Pierre Vernier in 1631, the Vernier Caliper measures external/internal lengths and depths with a precision ten times higher than a standard meter ruler.

• Principle: In a standard vernier caliper, $n$ vernier scale divisions (VSD) equal $(n-1)$ main scale divisions (MSD): $$n ext{ VSD} = (n-1) ext{ MSD} \implies 1 ext{ VSD} = rac{n-1}{n} ext{ MSD}$$
• Vernier Constant (VC) / Least Count (LC): Smallest distance directly measurable: $$ ext{VC} = 1 ext{ MSD} - 1 ext{ VSD} = 1 ext{ MSD} - rac{n-1}{n} ext{ MSD} = rac{1 ext{ MSD}}{n}$$ For a typical metric scale where $1 ext{ MSD} = 1 ext{ mm}$ and $n = 10 ext{ divisions}$: $$ ext{VC} = rac{1 ext{ mm}}{10} = 0.1 ext{ mm} = 0.01 ext{ cm}$$
• Observed Reading: $$ ext{Observed Reading} = ext{Main Scale Reading (MSR)} + ( ext{Coinciding Vernier Division } m imes ext{VC})$$
• Zero Error Correction: When jaws are fully closed:
  • Positive Zero Error: Vernier zero is to the right of main scale zero by $+p$ divisions. $ ext{Error} = +p imes ext{VC}$. Subtract from observed reading.
  • Negative Zero Error: Vernier zero is to the left of main scale zero by $-(n - q)$ divisions. $ ext{Error} = -(n - q) imes ext{VC}$. Add magnitude to observed reading.
  • Correct Reading = Observed Reading - (Zero Error with proper sign).

The Micrometer Screw Gauge operates on the principle of a precision nut and screw, translating rotational motion into infinitesimal linear displacement.

• Pitch of the Screw: The linear distance advanced by the spindle along the main pitch scale when the circular thimble is turned through one complete rotation ($360^\circ$): $$ ext{Pitch} = rac{ ext{Distance moved on linear scale}}{ ext{Number of full rotations}}$$ Common pitch values are $1.0 ext{ mm}$ or $0.5 ext{ mm}$.
• Least Count (LC): Smallest linear distance advanced when circular scale rotates by 1 division: $$ ext{LC} = rac{ ext{Pitch}}{ ext{Total number of Circular Scale Divisions } (N)}$$ For $ ext{Pitch} = 0.5 ext{ mm}$ and $N = 50$: $ ext{LC} = rac{0.5 ext{ mm}}{50} = 0.01 ext{ mm} = 0.001 ext{ cm}$.
• Observed Reading: $$ ext{Total Reading} = ext{Pitch Scale Reading (PSR)} + ( ext{Coinciding Circular Division } n imes ext{LC})$$
• Zero Error Correction: When studs touch:
  • Positive Zero Error: Circular zero line lies below the reference baseline. $ ext{Error} = +k imes ext{LC}$.
  • Negative Zero Error: Circular zero line lies above the reference baseline. $ ext{Error} = -(N - k) imes ext{LC}$.
  • True Reading = Observed Reading - (Zero Error).
• Backlash Error: Due to wear and tear of screw threads, turning the circular cap does not immediately move the screw forward. Prevention: Always rotate the ratchet head continuously in one direction while making contact.