Units and Dimensions

Units and Dimensions is a fundamental and essential topic in Physics. For the measurement of a physical quantity, Unit plays a vital role. Unit provides a complete idea about the measurement of a physical quantity. Dimension is a measure of the size or extent of a particular quantity.

In this article, we will learn Units and Dimensions related all details including definition, fundamental and derived units, system of units, List of Units of physical quantities, Dimension formula, List of physical quantities and their dimensions, etc.

What are Units?

Units are values used to measure and express various physical quantities. Unit gives us a complete and clear idea about a physical quantity. Measurement of a physical quantity consists of mainly two parts – Numeric value and unit.

Measurement of physical quantity = Numerical value x Unit

For example, A rectangular garden has 50 meter length.

Here, “50” is the numeric value or magnitude, and “meter” is the Unit of the physical quantity.

Fundamental and Derived Units

• Fundamental Units are independent to each other and these units are mainly used to measure the units of the fundamental physical quantities. There are seven fundamental units available namely – meter, kilogram, second, ampere, kelvin, candela and mole.

• Derived Units are the units which are obtained from the fundamental units. Except fundamental units, all other units are known as Derived Units. The examples of Derived units are units of area, volume, momentum, speed, velocity, density, force, energy, work etc.

Fundamental Units

Fundamental units are also known as base units. There are total of 7 base units. Fundamental units and their Symbols in SI system are discussed in the following table:

Physical Quantity	Name of Unit	Symbol
Length	meter	m
Mass	kilogram	kg
Time	second	s
Electric current	ampere	A
Thermodynamic temperature	kelvin	K
Amount of substance	mole	mol
Luminous intensity	candela	cd

System of Units

There are four system of units available namely – MKS System, CGS System, FPS System, and SI system.

MKS System: The full form of MKS System is Metre Kilogram Second. In MKS system, the units of length, mass and time are respectively metre, kilogram and second.

CGS System : The full form of CGS system is Centimetre Gram Second. CGS system is also known as Gaussian system. In CGS system, the units of length, mass and time are respectively centimetre, gram and second.

FPS System : The full form of FPS System is Foot Pound Second. FPS is also known as British system. In FPS system, the units of length, mass and time are respectively foot, pound and second.

SI System : SI system is generally known as International System of Units. It is the modification form of MKS system. SI System of unit is adopted to have same common unit world-wide.

Macro and Micro Prefixes

 Micro- is a prefix used to describe something that is small scale, while macro- is a prefix used to describe something that is large scale.

Macro Prefix	Symbol	Value
kilo	k	103
mega	M	106
giga	G	109
tera	T	1012
peta	P	1015
exa	E	1018
zetta	Z	1021
yotta	Y	1024

Micro Prefix	Symbol	Value
centi	c	10-2
milli	m	10-3
micro	μ	10-6
nano	n	10-9
pico	p	10-12
femto	f	10-15
atto	a	10-18
zepto	z	10-21
yocto	y	10-24

Units of Length, Mass and Time

The different units used to represent length, mass and time are shown below in the table:

Units of Length	Units of Mass	Units of Time
1 Angstrom = 10-10 m	1 Quintal = 102 kg	1 minute = 60 second
1 Light year = 9.46 × 1015 m	1 Metric tone = 103 kg	1 Hour = 60 minute = 3600 second
1 AU ( Astronomical Unit) = 1.5 × 1011 m	1 Atomic mass unit = 1.66 × 10-27 kg	1 Day = 24 hours = 1440 min = 86400 s
1 Mile = 1.6 km	1 Pound = 0.4537 kg	1 Lunar month = 28 days
1 Fermi = 10-15 m	1 Slug = 14.59 kg	1 Solar month = 30 or 31 days

Important Formulas of Derived Units

Following table shows some of the examples of derived units from fundamental units.

Physical Quantity	Formula	SI Units
Area of square	(side)2	m2
Area of Triangle	1/2×Base×Height	m2
Density	Mass/Volume	kg m-3
Acceleration	change in velocity/time	ms-2
Pressure	Force/Area	Nm-2 or Pascal
Work or Energy	P/E = mgh	N-m or joule
Power	Work/time	Js-1 or watt
Velocity	Displacement/Time	ms-1
Force	Mass × Acceleration	kg ms-2 or newton
Linear Momentum	Mass × Velocity	kg ms-1
Magnetic field	Force/(Electric current × Displacement)	N amp-1 m-1 or tesla or weber/m2
Impulse	Force × Time	N-s
Volume of cuboid	Length × Breadth × Height	m3

What are Dimensions?

Dimensions of a physical quantity are defined as the powers to which the fundamental quantities should be raised to represent that physical quantity. Dimension of a physical quantity is not dependent on the magnitude or the numeric value of that physical quantity.

To represent dimensions of physical quantities, we use square bracket “[ ]” around the quantity.

Dimensional Formula

Dimensional formula is nothing but an expression that shows how the fundamental units and which of the fundamental units are required to represent the unit of physical quantity.

The process of writing of a dimensional formula of a physical quantity is by enclosing the symbols of the base quantities with appropriate power in square brackets.

Dimensions of fundamental physical quantities along with their dimensional symbols are written as –

• [M] for Mass
• [L] for Length
• [T] for Time
• [I] for Electric Current
• [θ] for Temperature
• [J] for Luminous intensity
• [N] for Amount of substance

Dimensional Formulas for Physical Quantities

Some of the examples of dimensional formulas are as follows:

Physical Quantity with Formula	Dimensional Formula
Area = Length × Breadth	[L × L] = [L2] = [M0L2T0]
Volume = Length × Breadth × Height	[L × L × L] = [L3] = [M0L3T0]
Speed = Distance/Time	[L]/[T] = [LT-1] = [M0L1T-1]
Velocity = Displacement/Time	[L]/[T] = [LT-1] = [M0L1T-1]
Acceleration = Velocity/Time	[LT-1]/[T] = [LT-2] = [M0L1T-2]
Pressure = Force/Area = (Mass × Acceleration)/Area	[MLT-2]/[L2] = [ML-1T-2]
Force = Mass × Acceleration	[M] [LT-2] = [MLT-2]
Work = Force × Displacement	[MLT-2] [L] = [ML2T-2]
Kinetic Energy = 1/2 × Mass × (Speed)2	[M] [LT-1]2 = [ML2T-2]
Potential Energy = Mass × Acceleration due to gravity × Height	[M] [LT-2] [L] = [ML2T-2]
Impulse = (force x time)	[MLT-1]

Quantities Having the Same Dimensional Formula

The following physical quantities have the same Dimensional formula:

• Momentum and Impulse
• Work, torque, energy, the moment of force
• Planck’s constant, Angular momentum, rotational impulse
• Pressure, Stress, modulus of elasticity, energy density.
• Force constant, surface tension, surface energy.
• Frequency, Angular velocity, velocity gradient.
• Gravitational potential, latent heat.
• Entropy, Thermal capacity, universal gas constant and Boltzmann’s constant.
• Force, thrust.
• Power, luminous flux.

Dimensionless Quantities

Dimensionless quantities are the physical quantities which have no dimension or zero dimension. Its numeric value or magnitude is same in all system of units.

Examples : The examples of dimensionless quantities are as follows:

• Angle
• Solid angle
• Relative density
• Specific gravity

Uses of Dimension

Dimension of a physical quantity is an essential part. It has following uses –

• It is used to convert the units from one measurement system to another measurement system.

• It helps to establish relationship among various physical quantities.

• It is used to ensure an equation is homogeneous or not.

FAQs on Units and Dimensions

What are the 7 dimension units?

The seven dimension base units are as follows –

• Length – meter (m)
• Time – second (s)
• Amount of substance – mole (mole)
• Electric current – ampere (A)
• Temperature – kelvin (K)
• Luminous intensity – candela (cd)
• Mass – kilogram (kg)

What is the SI unit of speed?

The SI unit of speed is m/s or m s^-1.

What is the SI unit of temperature?

The SI unit of temperature is kelvin.

What is the SI unit for energy?

The SI unit of energy is joule.

What are Dimensional Variables?

The physical quantities which have dimensions but do not have fixed numerical value, is called the Dimensional Variables. Examples : velocity, acceleration, force, work, power, etc are the examples of Dimensional variables.

What is Dimensional Analysis?

Dimensional analysis can be defined as the process of checking relations among various physical quantities by using the dimensions of the physical quantities. These are independent of the numerical values and constants.

What are Dimensional Constants?

The physical quantities those have dimensions and have fixed values, are called dimensional constants. For Example:

• Gravitational constant (G)
• Planck’s constant (h)
• Velocity of light in a vacuum (C)
• Universal gas constant (R)

What is Law of Homogeneity of Dimensions?

Principle of Homogeneity states that dimensions of each of the terms of a dimensional equation on both sides should be the same.