What is a physical quantity?

Physics describes observable phenomena through quantities that can be measured and compared. A measurement consists of a numerical value and a unit. Writing merely “5” for a length is incomplete; “5 m” communicates the quantity. Physical quantities may be fundamental (independent base quantities) or derived (formed using relationships among base quantities).

For UPSC General Science, questions often ask whether a unit is fundamental or derived, whether an equation is dimensionally consistent, and whether accuracy should be confused with precision.

• Base quantities in SI are length, mass, time, electric current, thermodynamic temperature, amount of substance and luminous intensity.
• Derived quantities include speed, acceleration, force, pressure, work, power and electric charge.
• A dimension describes the powers of fundamental quantities; a unit supplies the agreed scale of measurement.

Seven SI base units and important derived units

The International System of Units (SI) provides a coherent standard for scientific measurements. The base units are not interchangeable with familiar derived units such as newton or joule. The SI unit of mass is the kilogram, not the gram; kelvin is written K without a degree sign.

Remember to distinguish a unit symbol from a physical quantity: the symbol A can denote ampere (unit of current) in a table of units, while I often denotes electric current in equations.

• Force: newton (N) = kg·m·s⁻²; energy or work: joule (J) = N·m.
• Power: watt (W) = J·s⁻¹; pressure: pascal (Pa) = N·m⁻².
• Electric charge: coulomb (C) = A·s; potential difference: volt (V) = J·C⁻¹.

SI prefixes, scientific notation and conversions

Prefixes express factors of ten. Common large prefixes are kilo (10³), mega (10⁶) and giga (10⁹); small prefixes include milli (10⁻³), micro (10⁻⁶) and nano (10⁻⁹). Scientific notation writes a nonzero value as a coefficient between 1 and 10 multiplied by a power of ten.

Conversion becomes easy when the dimension is raised to the appropriate power: 1 km² = 10⁶ m², not 10³ m²; 1 cm³ = 10⁻⁶ m³. Read the exponent on the whole unit before converting.

• 1 km = 1,000 m; 1 mm = 10⁻³ m; 1 μm = 10⁻⁶ m.
• 1 hour = 3,600 s; 72 km/h = 20 m/s.
• In general, km/h → m/s: multiply by 5/18; m/s → km/h: multiply by 18/5.

Accuracy, precision, uncertainty and errors

Accuracy means closeness of a measured value to the true or accepted value; precision concerns how closely repeated measurements agree. Results can be highly precise but inaccurate because of a systematic bias. Random error typically causes scatter; systematic error consistently shifts readings in one direction.

Absolute error is the magnitude of deviation from a reference value. If observations are repeated and the accepted value is unavailable, an introductory treatment often uses deviations from their arithmetic mean to estimate the mean absolute error. Relative error is absolute error divided by a suitable reference value; percentage error multiplies this ratio by 100.

• For observations 9.8, 10.0, 10.2 cm, mean = 10.0 cm and mean absolute deviation = (0.2 + 0 + 0.2)/3 ≈ 0.133 cm.
• Relative mean error ≈ 0.133/10.0 = 0.0133; percentage ≈ 1.33%.
• For small independent uncertainties, a maximum-error estimate in multiplication or division adds the fractional/percentage contributions; powers multiply relative uncertainty by the absolute exponent.
• An instrument’s limited resolution can create uncertainty even when readings are repeatable.

Significant figures and rounding

Significant figures indicate the meaningful digits supported by a measurement. All nonzero digits are significant; zeros between nonzero digits count; leading zeros do not count. Trailing zeros after a decimal point generally count. In whole numbers without explicit decimal or scientific notation, trailing zeros may be ambiguous.

For multiplication and division, retain as many significant figures as the least precise input. For addition and subtraction, retain the least precise decimal place. Round at the end of a calculation to reduce cumulative rounding error.

• 0.00450 has three significant figures: 4, 5 and the final 0.
• 2.300 has four significant figures; 70.0 has three.
• 2.4 × 3.21 = 7.704, reported as 7.7 (two significant figures).
• 10.25 + 0.3 = 10.55, conventionally reported to one decimal place as 10.6.

Dimensions and dimensional formulas

In basic mechanics, quantities are represented using mass [M], length [L] and time [T]. Thus velocity is [LT⁻¹], acceleration is [LT⁻²], and force, from F = ma, is [MLT⁻²]. Other areas also require dimensions of electric current and temperature.

A dimensionless quantity has all fundamental exponents zero. Examples include strain and the coefficient of friction. Angular measure in radians is treated as dimensionless in SI, though the named unit rad is used to avoid confusion.

Dimensional homogeneity and its limitations

The principle of dimensional homogeneity says that terms added or equated in a valid physical equation must have the same dimensions. For s = ut + ½at², both ut and at² have dimension [L], matching displacement. This is a useful quick rejection test in multiple-choice questions.

Dimensional analysis may reveal the form of a relationship. For a simple pendulum, assuming the period depends on length l and acceleration due to gravity g, dimensions imply T ∝ √(l/g). However, dimensions cannot determine the numerical constant 2π, nor distinguish every physically different formula with the same dimensions.

• A dimensionally incorrect equation must be physically incorrect.
• A dimensionally correct equation may still be physically incorrect.
• Trigonometric and exponential function arguments must be dimensionless.
• Two different quantities can share dimensions: work and torque both involve N·m but represent different physical concepts.

Least count: vernier calipers and screw gauge

Least count is the smallest interval that an instrument is designed to resolve. In a standard vernier example with one main scale division (MSD) = 1 mm and ten vernier divisions spanning nine MSD, one vernier division = 0.9 mm, so least count = 1 − 0.9 = 0.1 mm. Instruments of other designs have different least counts.

For a micrometer screw gauge, least count = pitch divided by number of circular-scale divisions. If pitch = 0.5 mm per revolution and the circular scale has 50 equal divisions, least count = 0.01 mm. Apply the sign of zero correction appropriately: corrected reading = observed reading − zero error (with sign).

• Never assume every vernier caliper has 0.1 mm least count: inspect its scale design.
• Always write a final measurement using the correct unit and resolution.
• A positive zero error requires subtraction; a negative zero error requires addition in the signed-error formula.

Essential formula sheet

Use these equations only when their physical assumptions are satisfied. Symbols and units are explained in the chapter.

Solved numericals and applications

Work each question once without looking at the solution; then compare your method with the steps.

UPSC-style practice MCQs with explanations

These are original practice questions written for competitive-examination revision. They are not presented as verbatim UPSC previous-year questions.

Last-minute revision and common traps

• Mass is measured in kilograms in SI; weight is a force measured in newtons.
• Trailing zeros can be significant in decimals; leading zeros are placeholders.
• The least count depends on an instrument’s actual scale, not just its name.
• Dimensionally equivalent quantities need not have the same physical meaning.
• Always apply squared/cubed unit conversions to the conversion factor.
• Significant-figure rules differ for addition and multiplication.