Units and dimensions

Definition

Unit: A unit is a fixed standard quantity used to express and measure a physical quantity. For example, metre is the unit of length, kilogram is the unit of mass, and second is the unit of time.

Dimension: Dimensions are the powers of fundamental physical quantities such as mass, length, time, electric current, temperature, amount of substance, and luminous intensity that describe a derived quantity. For example, the dimension of velocity is $[LT^{-1}]$, and the dimension of force is $[MLT^{-2}]$.

In simple terms, units tell us “how much,” while dimensions tell us “of what kind.”

Main Content

1. Fundamental and Derived Quantities

Fundamental quantities

• are basic physical quantities that cannot be expressed in terms of other quantities. In the SI system, the seven fundamental quantities are length, mass, time, electric current, temperature, amount of substance, and luminous intensity.
• Examples: length in metre $(m)$, mass in kilogram $(kg)$, time in second $(s)$, temperature in kelvin $(K)$.

Derived quantities

• are formed by combining fundamental quantities through mathematical relations.
• Examples: area $(m^2)$, volume $(m^3)$, velocity $(m\,s^{-1})$, acceleration $(m\,s^{-2})$, force $(kg\,m\,s^{-2})$, pressure $(N\,m^{-2})$, work $(N\,m)$, and density $(kg\,m^{-3})$.
• Fundamental quantities provide the building blocks of measurement, while derived quantities help express real-world physical phenomena.
• Every derived quantity can be represented in terms of fundamental quantities, which is why dimensional analysis is possible.

2. System of Units

• A system of units is a standardized set of units used for scientific measurement. The most widely accepted system today is the SI system (International System of Units).
• SI base units include:
  • metre $(m)$ for length
  • kilogram $(kg)$ for mass
  • second $(s)$ for time
  • ampere $(A)$ for electric current
  • kelvin $(K)$ for temperature
  • mole $(mol)$ for amount of substance
  • candela $(cd)$ for luminous intensity
• Earlier systems such as the CGS system and FPS system were also used in science and engineering.
• CGS uses centimetre, gram, and second.
• FPS uses foot, pound, and second.
• Standard units are important because they ensure uniformity, prevent confusion, and allow data to be shared globally.
• Prefixes are used with SI units to represent large or small quantities, such as kilo $(10^3)$, milli $(10^{-3})$, micro $(10^{-6})$, and mega $(10^6)$.
• Example: 1 kilometre = 1000 metre, 1 millimetre = 0.001 metre.

3. Dimensions and Dimensional Formula

• A dimensional formula expresses a physical quantity in terms of powers of fundamental dimensions such as mass $[M]$, length $[L]$, time $[T]$, current $[A]$, temperature $[\Theta]$, amount of substance $[N]$, and luminous intensity $[J]$.
• Examples of dimensional formulas:
• Length = $[L]$
• Area = $[L^2]$
• Volume = $[L^3]$
• Velocity = $[LT^{-1}]$
• Acceleration = $[LT^{-2}]$
• Force = $[MLT^{-2}]$
• Work/Energy = $[ML^2T^{-2}]$
• Power = $[ML^2T^{-3}]$
• Pressure = $[ML^{-1}T^{-2}]$
• Dimensional formula helps in understanding the nature of a physical quantity and is extremely useful in solving physics and engineering problems.
• Quantities with the same dimensions may have different units, but their dimensional form remains the same.
• Example: work and torque have the same dimensions $[ML^2T^{-2}]$, although they are physically different quantities.

Working / Process

1. Identify the physical quantity

• to be measured or analyzed, such as force, speed, pressure, or energy.

2. Express the quantity in terms of fundamental quantities

• using known physical relations. For example, force = mass × acceleration.

3. Write the unit and dimensional formula

• by substituting the base quantities:
• Force = $kg \cdot m \cdot s^{-2}$
• Dimensions of force = $[MLT^{-2}]$

Advantages / Applications

Checking correctness of equations

• Dimensional analysis helps verify whether both sides of a physical equation are dimensionally consistent. If dimensions do not match, the equation is incorrect.

Deriving relations between quantities

• Useful relations can often be obtained by comparing dimensions when the exact formula is unknown.

Unit conversion and standardization

• Units and dimensions allow conversion between different measurement systems and maintain accuracy in scientific communication.

Solving practical problems

• Engineers and scientists use dimensional concepts in mechanics, fluid flow, heat transfer, electronics, and experimental design.

Estimating physical quantities

• Dimensional reasoning can be used to approximate formulas in cases where detailed information is unavailable.

Summary

• Units are standard measures used to express physical quantities.
• Dimensions describe the physical nature of quantities in terms of fundamental quantities.
• Fundamental quantities form the basis for all derived quantities.
• Dimensional analysis is useful for checking equations, converting units, and deriving relations.