Explain the difference between fundamental units and derived units

A fundamental unit is a basic, independent unit that is not derived from other units. A derived unit is formed by combining two or more fundamental units using mathematical operations. Example: A meter is a fundamental unit of length. Speed (m/s) is a derived unit because it is calculated as distance (meter) divided by time (second).

Can a derived quantity ever become a fundamental quantity? Why or why not?

Generally, no — derived quantities are dependent on fundamental quantities. However, definitions and scientific understanding can evolve. For example, historically, temperature was considered a derived quantity, but in SI, kelvin is treated as a fundamental unit. So, while rare, scientific consensus can change categorizations.

A physical quantity has units of m². What is it likely to represent? Is it fundamental or derived?

It likely represents area, which is a derived quantity (length × width). The unit m² is derived from the fundamental unit of length (meter).

Why is dimensional analysis useful when working with derived units?

Dimensional analysis is a useful tool that helps check the correctness of equations, convert between unit systems, derive formulas, and ensure consistent units throughout calculations. It helps prevent errors and supports clear, logical problem-solving in science and engineering.

A unit has dimensions of [M⁰L⁰T⁰]. What kind of quantity does it represent?

A unit with zero dimension in all quantities is dimensionless. Examples: Angle (radian) Refractive index Strain These are pure numbers or ratios with no physical dimensions.

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JEE Physics

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

Scalar	Vector
The physical quantities which have only magnitude but no direction are called scalar quantities.	The physical quantities which have both magnitude and direction are called vector quantities.
Example: mass, distance, density, volume, time, etc.	Example: Displacement, Velocity, Force etc.

Classification on Basis of Dependency

If the physical quantities are independent of each other, they are called the fundamental quantities
If the physical quantities are dependent of each other, they are called the derived quantities

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:

Quantity	SI unit	Symbol
Mass	Kilogram	Kg
Length	Metre	m
Time	Second	s
Electric Current	Ampere	A
Temperature	Kelvin	K
Amount of Substance	Mole	mol
Luminous Intensity	Candela	cd

Supplementary Units

Quantity	Name of Unit
Plane Angle	Radian
Solid Angle	Steradian

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.

Category	Definition	Example	Conversion / Note
Coherent Derived Units	Derived from base units without any numerical factor.	1 newton (N) = 1 kg·m/s²	Directly derived from SI base units.
Non-Coherent Units	Involve a numerical factor relative to SI base units. Used for convenience.	1 liter (L) = 0.001 m³	Not directly derived from base units.

Derived Quantity	Formula	SI Unit Symbol
Area	Length × Length	$m^{2}$
Volume	Length × Width × Height	$m^{3}$
Speed	$\frac{D i s t an ce}{T im e}$	m/s
Acceleration	$\frac{Sp ee d}{T im e}$	$\frac{m}{s ^{2}}$
Force	Mass × Acceleration	N = kg·m/s²
Pressure	$\frac{F orce}{A re a}$	Pa = N/m²
Energy or Work	Force × Distance	J = N·m
Power	$\frac{W or k}{T im e}$	$W = \frac{J}{s}$
Electric Charge	Current × Time	C = A·s
Electric Potential	$\frac{E n er g y}{C ha r g e}$	$V = \frac{J}{C}$
Frequency	$\frac{1}{T im e}$	$Hz = \frac{1}{s}$

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- $[M^{1} L^{0} T^{0}]$
Momentum: $[M^{1} L^{1} T^{- 1}]$
Force: $[M^{1} L^{1} T^{- 2}]$

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

Mass	$[M^{1} L^{0} T^{0}]$
Length	$[M^{0} L^{1} T^{0}]$
Time	$[M^{0} L^{0} T^{1}]$
Temperature	$[M^{0} L^{0} T^{0} K^{1}]$
Current	$[M^{0} L^{0} T^{0} A^{1}]$
Amount of Substance	$[M^{0} L^{0} T^{0} m o l^{1}]$
Luminous Intensity	$[M^{0} L^{0} T^{0} C d^{1}]$

Dimensions of differential coefficients and integrals

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

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.