Units and Dimensions

Units and Dimensions are a fundamental and essential topic in Physics. For the measurement of a physical quantity, a unit plays an important role. The 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.

Here, we will explore Units and Dimensions in detail, covering definitions, fundamental and derived units, the system of units, a list of units for physical quantities, dimensional formulas, and a comprehensive list of physical quantities along with their dimensions.

What are Units?

A unit is a standard measurement used to quantify physical quantities. It is a reference point that allows us to express and compare the magnitude of different physical properties. For example, when measuring distance, we use units like meters (m) or kilometres (km), and when measuring time, we use seconds (s).

This helps in accurately comparing and quantifying different physical phenomena. Units provide a consistent framework for communication in science, technology, and daily life.

System of Units

A system of units is a standardised collection of units used to measure different physical quantities, such as length, mass, time, temperature, and more. These systems ensure consistency and accuracy in measurements, making it easier to compare and communicate results across various fields and countries.

There are several systems of units, but the most commonly used are the International System of Units (SI) . The SI system is the modern form of the metric system and is widely used in science, engineering, and most countries around the world. It includes seven base units (such as meters for length, kilograms for mass, and seconds for time), which are used to derive other units for more complex measurements.

Any system of units includes both fundamental and derived units for all kinds of physical quantities. The commonly used systems of units are as follows:

• CGS System (Centimeter, Gram, Second): In this system, the unit of length is the centimeter, the unit of mass is the gram, and the unit of time is the second.
• FPS System (Foot, Pound, Second): In this system, the unit of length is the foot, the unit of mass is the pound, and the unit of time is the second.
• MKS System (Meter, Kilogram, Second): In this system, the unit of length is the meter, the unit of mass is the kilogram, and the unit of time is the second.
• SI System (International System of Units): The SI system consists of seven fundamental units and two supplementary units (radian and steradian) that measure plane and solid angles, respectively. In science and engineering, there are two types of units commonly used:

1. Fundamental Units
2. Derived Units

1. Fundamental Units : are independent to each other and these units are mainly used to measure the units of the fundamental physical quantities.Fundamental units are also known as base units. There are seven fundamental units available namely - meter, kilogram, second, ampere, kelvin, candela and mole. The fundamental units and their symbols in the SI system are discussed in the following table:

2. Derived Units: are those that can be expressed in terms of fundamental units. Every derived unit is originated from some physical law defining that unit. These units are essential for measuring more complex physical quantities. There are several steps involved in deriving a unit.

Step -1 : Identify the formula for the quantity.

Step -2 : Substitute the units of all involved quantities in the same system.

Step -3 : Simplify the expression to obtain the final derived unit.

✦SI Derived Units with Special Names

There are two other supplementary fundamental units, namely Radian and Steradian are two supplementary which measures plane angle and solid angle respectively:

The SI System: Coherent Measurement System

The International System of Units (SI) is the most widely used system of measurement, particularly in scientific research. It is known for being a coherent system of units, meaning that the units for derived quantities are directly related to basic, fundamental units through simple mathematical relationships.

In a coherent system, the units of derived quantities are expressed as multiples or submultiples of base units. The SI system is a coherent, rationalized extension of the MKS (Meter-Kilogram-Second) system. Historically, the ampere system (RMKSA system), developed by Prof. Giorgi, helped establish the structure of the SI system.

• Meter (m): Originally, a meter was defined as 1,650,763.73 times the wavelength of light emitted by Krypton-86. But since 1983, it’s been defined by how far light travels in a vacuum in 1/299,792,458 of a second.
• Kilogram (kg):The kilogram used to be defined by a specific platinum-iridium cylinder stored at the International Bureau of Weights and Measures in Paris.
• Second (s):A second is now defined by the number of times radiation from a cesium-133 atom oscillates—specifically, 9,192,631,770 times. This is a very precise measurement that helps keep time accurate worldwide.
• Ampere (A):The ampere is the unit of electric current and It is defined by the force that occurs when two parallel wires, placed one meter apart, carry the same current and this force is 2 × 10^-7 newtons per meter of wire length.
• Kelvin (K):The kelvin is used to measure temperature and It is defined as 1/273.16 of the temperature at the triple point of water (where water exists as a solid, liquid, and gas at the same time).
• Candela (cd):The candela measures light intensity and it is based on the brightness of a very specific source: a blackbody at the temperature where platinum solidifies under standard pressure.
• Mole (mol):A mole is used to measure the amount of substance and It is the number of atoms or molecules in 12 grams of carbon-12—about 6.022 × 10²³ entities, which is known as Avogadro’s number.

Angular Units:

• Radian (rad):The radian is the angle subtended by an arc whose length is equal to the radius of a circle. In terms of degrees, 1 radian = 57.2958 degrees (approximately).
• Steradian (sr): A steradian is the unit of solid angle. It is the angle subtended at the center of a sphere by a surface area of 1 square meter on the surface of a sphere with a radius of 1 meter.

The table below lists common macro prefixes used in the metric system to represent large-scale quantities :

Macro Prefixes

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

Micro Prefixes

✥Important Key Points

The table below presents various units used to measure length, mass, and time, along with their equivalent values.

What are Dimensions?

The dimensions of a physical quantity refer to the exponents to which the fundamental units are raised in order to express that quantity in terms of basic units.

Dimensional Formula

A dimensional formula is an expression that shows how the fundamental units (such as mass, length, time, etc.) and their respective powers are required to represent the unit of a physical quantity. The dimensional formula of a physical quantity is written by enclosing the symbols of the base quantities (e.g., mass, length, time) with appropriate exponents inside square brackets.

For example, the dimensional formula for force (F) is represented as [MLT^−2], where:

• M stands for mass,
• L stands for length,
• T stands for time.

It shows that force involves mass to the first power, length to the first power, and time to the power of -2.

Dimensional Formulas of Physical Quantities

Some of the examples of dimensional formulas are as follows:

There are some quantities which having same dimensional Formula ,

• Impulse and momentum
• Force, thrust
• Work, torque, the moment of force, energy
• Angular momentum, Planck’s constant, rotational impulse
• Stress, pressure, modulus of elasticity, energy density
• Force constant, surface tension, surface energy
• Angular velocity, frequency, velocity gradient
• Gravitational potential, latent heat
• Thermal capacity, entropy, universal gas constant, and Boltzmann’s constant
• Power, luminous flux
• Current, electric charge
• Magnetic flux, magnetic field strength

Dimensional Constants

Dimensional constants are physical constants that have dimensions (i.e., they involve some combination of fundamental units like mass, length, time, etc.). These constants, unlike dimensionless constants, are expressed with specific dimensional formulas because they are related to measurable physical quantities.

For example,

Gravitational constant (G) has dimensions [M⁻¹L³T⁻²].

Planck's constant (h) has dimensions ([M L^{^2} T^{^{-1}}].

These constants are called "dimensional" because their units can be expressed in terms of the fundamental physical quantities.

Dimensional Quantities

These are quantities that do not have any physical dimensions. They are often "pure numbers" or ratios that don't depend on any particular unit system.

These are purely numerical values and have no associated physical units.

For Examples ,

• π (Pi) – The ratio of the circumference of a circle to its diameter.
• e – The base of the natural logarithm.

Trigonometric functions like sin θ, cos θ, tan θ – These functions of angles are dimensionless because they represent ratios of sides of a right triangle or other geometric relations, and thus do not have physical units.

Dimensional Variables

Dimensional variables are physical quantities that have specific dimensions (expressed in terms of fundamental units like mass, length, time, etc.), but their values can change depending on the situation or conditions.

Examples of dimensional variables include velocity, acceleration, force, work, and power because their values can vary depending on the circumstances (speed, direction, mass, etc.).

Unit Conversions and Physical Constants

• One atmosphere = 76 cm Hg = 1.013×10⁵Pa
• Velocity of light in vacuum (c) = 3×10⁸ m/s
• Velocity of sound in air at STP = 331 m/s^-1
• Acceleration due to gravity (g) = 9.81 ms^-2
• Density of water at 4°C = 1000 kg/m^-3
• Density of air at STP = 1.293 kg/m^-3
• Avogadro's number (N) = 6.023 × 10²³/mol
• Absolute zero = -273.15°C or 0 K
• Atomic mass unit =  1.66 × 10^-27 kg
• Quantum of charge (e) =  1.602 × 10^-19 C
• Boltzmann’s constant (K) = 1.381 × 10^-23 JK^-1
• Stefan’s constant = 5.67 × 10^–8 W/m²/K⁴
• Planck’s constant (h) = 6.626 × 10^-34 Js
• Universal gas constant (R) = 8.314 J/mol–K
• Mechanical equivalent of heat (J) = 4.186 J/cal
• Permeability of free space (μ₀) =  4π × 10^-7 Hm^-1
• Permittivity of free space (ε₀) =  8.854 × 10^-12 Fm^-1
• The universal gravitational constant = 6.67 × 10^-11 Nm²kg^-2
• Speed of sound in water at 25°C = 1482 m/s
• Rydberg constant = 1.097×10⁷ m⁻¹

Unit Conversions and Physical Quantities

• 1 dyne = 10^-5 N,
• 1 kmph =  5/18 ms^-1
• 1 bar = 10⁶ dyne/cm² = 10⁵ Nm^-2 = 10⁵ pascal
• 76 cm of Hg =  1.013×10⁶ dyne/cm²  = 1.013×10⁵ pascal = 1.013 bar.
• 1 toricelli or torr =  1 mm of Hg = 1.333×10³ dyne/cm² = 1.333 millibar.
• 1 H.P = 746 watt
• 1 kilowatt hour = 36×10⁵ J
• 1 kgwt = g newton
• 1 calorie = 4.2 joule
• 1 electron volt =1.602×10^-19 joule
• 1 erg = 10^-7 joule

Conclusion

Units are the specific standard measures used to quantify physical quantities (such as meters, kilograms, and seconds), while dimensions represent the fundamental physical nature of these quantities (such as length, mass, and time). In other words, units provide a way to measure the dimensions of a quantity.

You may also read,

• System of Units
• Metric System of Measurements
• Metric Conversion Chart