Derived Quantities & Dimensional Analysis

Understanding Derived Quantities

Derived quantities are physical quantities that are not fundamental but are derived from fundamental quantities. Examples include:

• Volume: Derived from length. Its unit is cubic meters \\(m^3\\).
• Density: Derived from mass and volume. Its unit is kilograms per cubic meter \\(\text{kg/m}^3\\).
• Speed: Derived from distance and time. Its unit is meters per second \\(\text{m/s}\\).

Dimensional Analysis

Dimensional analysis is a technique used to check the validity of equations and derive units for derived quantities. It involves comparing the dimensions on both sides of an equation.

Checking Equation Validity

To check if an equation is dimensionally consistent, ensure that the dimensions on both sides match. For example, in the equation for speed:

$$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$$

The dimensions are:

• Speed: \\[ \text{LT}^{-1} \\
• Distance: \\[ \text{L} \\
• Time: \\[ \text{T} \\

Thus, the equation is dimensionally consistent as both sides have the dimension \\(\text{LT}^{-1}\\).

Deriving Units

To derive units for a derived quantity, express it in terms of fundamental quantities. For example, to derive the unit of density:

\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]

The units are:

• Mass: \\[ \text{kg} \\
• Volume: \\[ \text{m}^3 \\

Thus, the unit of density is \\(\text{kg/m}^3\\).