Fundamental and Derived Units

Measuring physical quantities is a big part of how we make sense of the world—and it's at the core of how physics works. Whether you're figuring out how fast a car is going or how much energy your microwave uses, you're relying on a system of units to get consistent, meaningful results.At the base of this system are the fundamental units—the standard building blocks for measuring basic things like length, mass, and time. Using these, we can create derived units to measure more complex quantities like force, pressure, and energy.

1.0Definition of Physical Quantities

• In Physics, all the quantities which are used to describe the laws of physics and can be measured are called Physical Quantities.

Every physical quantity consists of two parts:

• A numerical value (magnitude)
• A unit (standard of measurement)
• For example, in "5 meters", 5 is the numerical value and meters is the unit.

 Classification On Basis Of Directional Properties

Classification on Basis of Dependency

2.0Fundamental or Base Quantities

• Fundamental or base quantities are physical quantities that are independent of other quantities and cannot be defined in terms of other physical quantities.

Seven SI Fundamental Quantities and Units:

Supplementary Units

Note: These base quantities serve as the foundational building blocks of measurement. All other physical quantities are derived by combining these fundamental units in various ways.

3.0Derived Quantities

• Derived quantities are physical quantities that are formed by combining two or more fundamental quantities. They are called "derived" because they result from applying physical laws and mathematical relationships to the basic or fundamental quantities. In other words, these quantities aren’t measured directly using a single unit but are calculated based on the fundamental units. Examples include speed (derived from distance and time), force (from mass and acceleration), and pressure (from force and area).

Note: The units used to represent these quantities are known as derived units. They are formed by combining the fundamental units according to the relationships between physical quantities.

4.0Dimensional Analysis

Dimensions:Dimensions of a physical quantity are the powers (or exponents) to which the base quantities are raised to express that quantity.

Dimensional Formula :The dimensional formula of a physical quantity is an expression that shows which base quantities are involved in that quantity and how they are combined. It is written by enclosing the symbols of the base quantities, each raised to the appropriate power, within square brackets.

Mass - $\left[M^1{L}^0{T}^0\right]$

Momentum - $\left[M^1{L}^1{T}^{-1}\right]$

Force -$\left[M^1{L}^1{T}^{-2}\right]$

Dimensional Equation: The equation obtained by equating a physical quantity with its dimensional formula is called a dimensional equation.

Dimensions of differential coefficients and integrals

In General, $\left[\frac{d^{n}y}{d{x}^n}\right]=\left[\frac{y}{x^n}\right]$and $\left[\int ydx\right]=[yx]$

• A physical quantity’s dimensions show how base quantities are multiplied together, each raised to a certain power.

Rule of Dimensions

• Only the same physical quantities can be added or subtracted.

A+B=C-D

Dimensionless Quantities

Dimensionless Quantities are:

• Ratio of physical quantities with same dimensions.
• All mathematical constants.
• All standard mathematical functions and their inputs (exponential, logarithmic, trigonometric & inverse trigonometric).

5.0Importance of Using Correct Units

• Using correct units ensures accurate communication, consistency, and reliability in measurements and calculations. It prevents errors in science, engineering, medicine, and daily life, where even small unit mistakes can lead to serious consequences. Units help give meaning to numbers and allow for proper comparison and replication of results.