Derived Units & Dimensional Analysis

What Are Derived Units?

Derived units are units of measurement that are obtained by combining the base units according to the physical quantities they represent. These are used to express quantities like volume, density, velocity, and force.

• Volume: The unit is cubic meters \\(m^3\\), derived from length.
• Density: The unit is kilograms per cubic meter \\(kg/m^3\\), derived from mass and volume.
• Velocity: The unit is meters per second \\(m/s\\), derived from length and time.
• Force: The unit is newtons \\(N\\), derived from mass, length, and time, given by \\(N = kg \cdot m/s^2\\).

Dimensional Analysis

Dimensional analysis is a method used to convert one unit of measurement to another, check the consistency of equations, and derive relationships between physical quantities.

Uses of Dimensional Analysis

• Checking Equations: Ensures that both sides of an equation have the same dimensions.
• Deriving Formulas: Helps in deriving formulas by ensuring dimensional consistency.
• Unit Conversion: Facilitates conversion between different units of measurement.

Dimensional Formula

The dimensional formula of a physical quantity is an expression that shows how and which of the base quantities are included in that quantity. It is written as \\(M^aL^bT^c\\), where \\(M\\), \\(L\\), and \\(T\\) represent mass, length, and time respectively.

Examples

• Velocity: \\(\text{Dimensional formula: } [M^0L^1T^{-1}]\\)
• Force: \\(\text{Dimensional formula: } [M^1L^1T^{-2}]\\)
• Energy: \\(\text{Dimensional formula: } [M^1L^2T^{-2}]\\)