Units & Measurements
Master systems of units, fundamental and derived SI units, astronomical distance units, scientific measuring instruments, dimensional formulas, and scalar vs vector quantities.
1. Systems of Units & Base SI Units
Measurement requires a standard unit. Historically, several systems existed, which are compared below:
| System |
Length Unit |
Mass Unit |
Time Unit |
Notes |
| CGS |
Centimeter (cm) |
Gram (g) |
Second (s) |
French/Gaussian metric system. |
| FPS |
Foot (ft) |
Pound (lb) |
Second (s) |
British imperial system. |
| MKS |
Meter (m) |
Kilogram (kg) |
Second (s) |
Standard metric precursor to SI. |
| SI |
Meter (m) |
Kilogram (kg) |
Second (s) |
Modern International System (standardized in 1960). |
The SI system consists of **7 Base Units** (Meter, Kilogram, Second, Ampere, Kelvin, Mole, Candela) and **2 Supplementary Units** (Radian for plane angle, Steradian for solid angle).
2. Astronomical & Special Units
For celestial and microscopic measurements, standard units are impractical. The following special units are frequently tested in exams:
- Astronomical Unit (AU): The average distance between the Earth and the Sun.
1 AU ≈ 1.496 × 10¹¹ m
- Light Year (LY): The distance traveled by light in a vacuum in one year.
1 LY = 9.46 × 10¹⁵ m ≈ 63,241 AU
- Parsec (pc): Parallactic second, the largest unit of distance in astronomy.
1 pc ≈ 3.08 × 10¹⁶ m ≈ 3.26 Light Years
- Angstrom (Å): Used to measure atomic sizes and wavelengths.
1 Å = 10⁻¹⁰ m
- Fermi / Femtometer (fm): Used to measure nuclear radii.
1 fm = 10⁻¹⁵ m
- Chandrasekhar Limit (CSL): Largest unit of mass.
1 CSL = 1.4 × Mass of Sun
- Dobson Unit (DU): Measures the thickness of the ozone layer.
3. Derived Units & Dimensional Formulas
Derived units are combinations of base SI units. Below are the core physical quantities and their dimensional formulas tested in AHC exams:
| Physical Quantity |
Formula / SI Unit |
Dimensional Formula |
| Velocity |
Displacement / Time (m/s) |
[M⁰ L¹ T⁻¹] |
| Acceleration |
Velocity / Time (m/s²) |
[M⁰ L¹ T⁻²] |
| Force |
Mass × Acceleration (Newton, N) |
[M¹ L¹ T⁻²] |
| Work / Energy |
Force × Displacement (Joule, J) |
[M¹ L² T⁻²] |
| Power |
Work / Time (Watt, W) |
[M¹ L² T⁻³] |
| Pressure / Stress |
Force / Area (Pascal, Pa) |
[M¹ L⁻¹ T⁻²] |
| Frequency |
1 / Time Period (Hertz, Hz) |
[M⁰ L⁰ T⁻¹] |
4. Scientific Measuring Instruments
UPPSC and AHC exams frequently ask about scientific instruments and their applications:
| Instrument |
Usage |
| Actinometer |
Measures the heating power of radiation. |
| Altimeter |
Measures altitude (used in aircrafts). |
| Anemometer |
Measures wind velocity and direction. |
| Bolometer |
Measures electromagnetic/heat radiation. |
| Crescograph |
Measures plant growth (invented by J.C. Bose). |
| Fathometer |
Measures the depth of the ocean. |
| Hygrometer |
Measures humidity/moisture in the air. |
| Hydrometer |
Measures specific gravity/relative density of liquids. |
| Pyrometer |
Measures very high temperatures. |
| Sphygmomanometer |
Measures human blood pressure. |
| Sonar |
Detects underwater objects using ultrasonic waves. |
| Polygraph |
Records physiological changes to detect lies. |
5. Scalar vs Vector Quantities
Physical quantities are categorized based on direction dependency:
- Scalar Quantities: Have magnitude only, no direction. They are added by simple algebraic rules. Examples: Mass, distance, speed, time, energy, power, volume, temperature, electric current, work.
- Vector Quantities: Have both magnitude and direction, and obey vector addition laws. Examples: Displacement, velocity, acceleration, force, momentum, impulse, weight, torque, electric field.
Note on Electric Current: It has both magnitude and direction, but it is scalar because it does not follow vector laws (e.g., current at junctions simply adds up algebraically).
Historical Progression of Metrology
- 1795 — Birth of the Metric System: France officially adopts the decimal metric system, defining the meter (length) and gram (mass), laying the foundation of modern metrology.
- 1875 — Treaty of the Meter: Signed by 17 nations, establishing the International Bureau of Weights and Measures (BIPM) to ensure global standards.
- 1960 — Establishment of the SI System: The 11th General Conference on Weights and Measures (CGPM) formally adopts the International System of Units (SI) with 6 base units (Mole added later in 1971).
- 2019 — Modern Constants-Based Redefinition: BIPM redefines base units (including the kilogram, ampere, kelvin, and mole) in terms of fundamental physical constants (e.g., Planck constant, elementary charge).
Key Questions & Answers
- What is the SI unit of plane angle and solid angle?
- Plane angle is measured in **Radian (rad)**. Solid angle is measured in **Steradian (sr)**. Both are supplementary dimensionless units.
- Which instrument is used to measure plant growth, and who invented it?
- The **Crescograph**, invented by the famous Indian scientist **Jagadish Chandra Bose** (J.C. Bose).
- What is the Chandrasekhar Limit and its physical value?
- It is the largest practical unit of **mass** (limit for white dwarf stars). It equals **1.4 times the mass of the Sun** (approx. 2.8 × 10³⁰ kg).
- Is electric current a scalar or a vector quantity?
- Electric current is a **scalar quantity**. Although it has both magnitude and direction, it does not obey the laws of vector addition.
Memory Aids
- Mnemonic 1: The 7 SI Base Units: Remember the fundamental base units: • **Let**: **L**ength (Meter) • **Mass**: **M**ass (Kilogram) • **Time**: **T**ime (Second) • **Current**: **E**lectric Current (Ampere) • **Temper**: **T**emperature (Kelvin) • **Light**: Luminous **I**ntensity (Candela) • **Amount**: **A**mount of Substance (Mole)
- Mnemonic 2: Difference Between Hydrometer and Hygrometer: Avoid the common exam trap: • **d**: Hy**d**rometer measures relative **d**ensity (specific gravity) of liquids. • **g**: Hy**g**rometer measures humidity (water vapor in **g**as/air).
- Mnemonic 3: Astronomical Distance Scale: Order of magnitude from largest to smallest: • **P**: **P**arsec (3.26 LY / 3.08 × 10¹⁶ m) • **L**: **L**ight Year (9.46 × 10¹⁵ m) • **A**: **A**stronomical Unit (1.496 × 10¹¹ m)
Common Exam Traps
- Trap 1: Believing that a 'Light Year' is a unit of time due to the word 'year'. Remember, a light year is a unit of **astronomical distance** (the distance light travels in one vacuum year).
- Trap 2: Confusing supplementary units with dimensional units. Radian and Steradian are **dimensionless** (their dimensional formula is [M⁰ L⁰ T⁰]), yet they have units.
- Trap 3: Confusing the usage of Pyrometer and Thermometer. A pyrometer is used for measuring **extremely high temperatures** (often without direct contact), while a standard thermometer is for normal ranges.