Physical World and Measurement

Physical quantities are measured in units. The seven SI base (fundamental) quantities are: length (metre, m), mass (kilogram, kg), time (second, s), electric current (ampere, A), temperature (kelvin, K), amount of substance (mole, mol), and luminous intensity (candela, cd). Memory aid: 'Mary Kept Saving All The Money Carefully' (Metre, Kg, Second, Ampere, Kelvin/Temp, Mole, Candela). Derived units are combinations of base units, e.g., force = kg m s^-2 (newton), energy = kg m^2 s^-2 (joule). Supplementary units: radian (plane angle) and steradian (solid angle), now treated as dimensionless derived units. A complete set of base + derived units forms a 'system of units' (CGS, MKS, SI).

SI prefixes scale units: tera (10^12), giga (10^9), mega (10^6), kilo (10^3), milli (10^-3), micro (10^-6), nano (10^-9), pico (10^-12), femto (10^-15). Useful astronomical/atomic units: 1 light year = 9.46 x 10^15 m; 1 parsec = 3.08 x 10^16 m = 3.26 light years; 1 astronomical unit (AU) = 1.496 x 10^11 m; 1 angstrom = 10^-10 m; 1 fermi = 10^-15 m. Mass: 1 atomic mass unit (u) = 1.66 x 10^-27 kg; 1 quintal = 100 kg; 1 metric tonne = 1000 kg. Tip: parsec > light year > AU. Remember 1 parsec is the distance at which 1 AU subtends 1 arcsecond.

Parallax measures large distances. If a distant object is viewed from two points separated by basis b, and the parallax angle is theta (in radians), distance D = b / theta. Example: The Moon is observed from two points on Earth 6400 km apart, with parallax angle 1.5 degrees. Convert: theta = 1.5 x (pi/180) = 0.0262 rad. D = b/theta = 6.4 x 10^6 / 0.0262 = 2.44 x 10^8 m. Remember theta MUST be in radians (arc = radius x angle). For angular diameter alpha of a planet of diameter d at distance D: d = alpha x D, with alpha in radians.

Dimensions express a physical quantity in terms of base quantities M (mass), L (length), T (time), A (current), K (temperature). Key formulae: velocity [LT^-1], acceleration [LT^-2], force [MLT^-2], work/energy [ML^2T^-2], power [ML^2T^-3], momentum/impulse [MLT^-1], pressure/stress [ML^-1T^-2], density [ML^-3], frequency [T^-1], surface tension [MT^-2], coefficient of viscosity [ML^-1T^-1], Planck's constant [ML^2T^-1], gravitational constant G [M^-1L^3T^-2]. Memory tip: energy and torque share [ML^2T^-2] but torque is a vector and energy a scalar. Angular momentum and Planck's constant both have [ML^2T^-1].

Uses: (1) checking dimensional correctness of equations (principle of homogeneity - all terms must have same dimensions); (2) deriving relations among quantities; (3) converting units from one system to another. Limitations: (1) cannot determine dimensionless constants (like 1/2, pi, 2); (2) cannot derive relations involving sum/difference of terms; (3) fails for trigonometric, exponential, logarithmic functions; (4) cannot work if a quantity depends on more than 3 factors with M, L, T. Key rule: arguments of sin, cos, log, e^x are always dimensionless. Quantities with same dimensions but different nature: work and torque; stress and pressure and Young's modulus.

To convert a quantity from one system to another: n1[M1^a L1^b T1^c] = n2[M2^a L2^b T2^c], so n2 = n1 (M1/M2)^a (L1/L2)^b (T1/T2)^c. Example: Convert 1 joule to erg. Joule = [ML^2T^-2], so a=1, b=2, c=-2. n2 = 1 x (kg/g)^1 (m/cm)^2 (s/s)^-2 = 1 x (1000)(100^2)(1) = 1000 x 10000 = 10^7. So 1 J = 10^7 erg. Always raise the ratio of OLD to NEW unit to the power of the dimension.

Errors are systematic (instrumental, imperfect technique, personal - one-directional, correctable) or random (irregular, due to fluctuating conditions). Absolute error = |true value - measured value|. Mean absolute error = average of absolute errors. Relative error = mean absolute error / mean value. Percentage error = relative error x 100. Rules for combining: (1) Sum/difference (Z = A +/- B): absolute errors add, deltaZ = deltaA + deltaB. (2) Product/quotient (Z = AB or A/B): relative errors add, deltaZ/Z = deltaA/A + deltaB/B. (3) Power (Z = A^n): deltaZ/Z = n(deltaA/A). Memory: for products/powers, ADD the fractional errors, multiplying by the power.

Significant figures convey precision. Rules: (1) all non-zero digits are significant; (2) zeros between non-zero digits are significant (1002 has 4); (3) leading zeros are NOT significant (0.005 has 1); (4) trailing zeros after a decimal ARE significant (2.300 has 4); (5) trailing zeros in a number without a decimal are ambiguous (use scientific notation). In addition/subtraction, the result keeps the least number of DECIMAL PLACES. In multiplication/division, the result keeps the least number of SIGNIFICANT FIGURES. Rounding: if the digit to drop is 5 with nothing after, round to make the preceding digit even.

A physical quantity P = a^3 b^2 / (sqrt(c) d). The percentage errors in a, b, c, d are 1%, 3%, 4%, 2% respectively. Find max % error in P. Using power rule, deltaP/P = 3(da/a) + 2(db/b) + (1/2)(dc/c) + (dd/d). Substitute: = 3(1) + 2(3) + (1/2)(4) + 1(2) = 3 + 6 + 2 + 2 = 13%. Always multiply each fractional error by the magnitude of its power (sqrt means power 1/2) and ADD them all (errors always add for max error, never subtract).

Physics studies matter and energy from the microscopic (10^-14 m, nuclei) to macroscopic (10^26 m, universe) and across times from 10^-22 s to 10^17 s. There are four fundamental forces: (1) Gravitational force - weakest, infinite range, always attractive, relative strength 10^-39; (2) Electromagnetic force - between charges, infinite range, ~10^-2; (3) Weak nuclear force - in beta decay, very short range (10^-16 m), 10^-13; (4) Strong nuclear force - strongest, binds nucleons, short range (10^-15 m), strength = 1. Order of strength: Strong > Electromagnetic > Weak > Gravitational. Unification of forces is a major goal of physics.

Time was historically based on Earth's rotation, but now the SI second is defined using the cesium-133 atomic clock: 1 second = 9,192,631,770 periods of radiation from the transition between two hyperfine levels of cesium-133. Atomic clocks are extremely accurate (uncertainty ~1 part in 10^13). Range of time intervals: lifespan of most unstable particle ~10^-24 s, age of universe ~10^17 s (about 4 x 10^17 s). Memory aid: cesium clock frequency is about 9.19 x 10^9 Hz. Quartz clocks use piezoelectric oscillation; atomic clocks set the global time standard (UTC).

Accuracy = closeness of a measurement to the true value. Precision = closeness of repeated measurements to each other (resolution/repeatability). They are independent: a measurement can be precise but inaccurate (consistent but wrong, e.g., systematic error) or accurate but imprecise. Example: true length 3.678 cm. Instrument A reads 3.5 cm (low precision); instrument B reads 3.38 cm consistently (high precision, low accuracy due to systematic error). Higher precision means smaller least count. Remember: precision relates to the instrument's least count; accuracy relates to systematic errors. Good measurement needs both.