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Dimensional Analysis
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Dimensional Formula

Last Updated : 16 Apr, 2025
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Dimensional Formulas play an important role in converting units from one system to another and find numerous practical applications in real-life situations. Dimensional Formulas are a fundamental component of the field of units and measurements. In mathematics, Dimension refers to the measurement of an object's size, extent, or distance in a specific direction, such as length, width, or height, but in the context of physical quantities, the dimension signifies the exponent to which fundamental units must be raised to yield a single unit of that specific quantity.

In this article, we will discuss the introduction, definition, properties, and limitations of a Dimensional Formula and its meaning. We will also understand dimensional formulas for different physical quantities and Dimensional equations. We will also solve various examples and provide practice questions for a better understanding of the concept of this article. We have to study Dimensional Formula in Class 11.

Dimentional-Formula-(1)

Table of Content

  • What is Dimensional Formula?
  • Dimensional Formula for Various Quantities
  • Application of Dimensional Formula
  • Limitations of Dimensional Formula
  • Dimensional Formula and Dimensional Equations

What is Dimensional Formula?

The Dimensional Formula of any quantity serves as an expression that shows the powers by which fundamental units must be raised to yield a single unit of that derived quantity. These dimensional formulas play an important role in establishing relationships between variables in nearly every dimensional equation.

These formulae, also known as the Dimensional Formula of the Physical Quantity, tell us about the presence and combination of fundamental quantities within a given physical quantity. A dimensional formula is always enclosed within square brackets [ ].

Example of Dimensional Formula

Let’s suppose there is a physical quantity X that depends on the fundamental dimensions of Mass (M), Length (L), and Time (T), each with associated powers a, b, and c, then its Dimensional Formula can be expressed as follows:

Dimensional Formulae X = [MaLbTc]

Dimensional Formula for Various Quantities

The table below provides Dimensional Formulas for different physical quantities:

Physical Quantity

Unit

Dimensional Formula

Acceleration or Acceleration due to Gravity

ms-2

LT-2

Angular Displacement

rad

M0L0T0

Angular Impulse

Nms

ML2T-1

Angular Velocity (angle/time)

rads-1

T-1

Angle (Arc/Radius)

rad

M0L0T0

Angular Frequency (Angular Displacement/Time)

rads-1

T-1

Angular Momentum

kgm2s-1

ML2T-1

Boltzmann’s Constant

JK-1

ML2T-2θ-

Bulk Modulus

N/m2

ML-1T-2

Calorific Value

JKg-1

L2T-2

Coefficient of Surface Tension (Force/Length)

N/m

MT-2

Coefficient of Linear or Areal or Volume Expansion

K-1

θ-1

Coefficient of Thermal Conductivity

Wm-1K-1

MLT-3θ-1

Compressibility (1/Bulk Modulus)

m2N-2

M-1LT2

Density (Mass / Volume)

Kg/m3

ML-3

Displacement, Wavelength, Focal Length

m

L

Electric Capacitance (Charge/Potential)

farad

M-1L-2T4I2

Electric Conductivity (1/Resistivity)

Sm-1

M-1L-3T3I2

Electric Current

ampere

I

Electric Field Strength or Intensity of Electric Field (Force/Charge)

NC-1

MLT-3I-1

Emf (or) Electric Potential (Work/Charge)

volt

ML2T-3I-1

Energy Density (Energy/Volume)

Jm-3

ML-1T-2

Electric Conductance (1/Resistance)

Ohm-1

ML-1T-2T3I2

Electric Charge or Quantity of Electric Charge

coulomb

IT

Electric Dipole Moment

Cm

LTI

Electric Resistance (Potential Difference/Current)

ohm

ML2T-3I-2

Energy (Capacity to do work)

joule

ML2T-2

Entropy

Jθ–1

ML2T-2θ-1

Force

newton (N)

MLT-2

Frequency (1/period)

Hz

T-1

Force Constant or Spring Constant (Force/Extension)

Nm-1

MT-2

Gravitational Potential (Work/Mass)

J/kg

L2T-2

Heat (Energy)

J or calorie

ML2T-2

Illumination (Illuminance)

lumen/m2

MT-3

Inductance

henry (H)

ML2T-2I-2

Intensity of Magnetization (I)

Am-1

L-1I

Impulse

Ns

MLT-1

Intensity of Gravitational Field (F/m)

Nkg-1

LT-2

Joule’s Constant

Jcal-1

M0L0To

Latent Heat (Q = mL)

Jkg-1

L2T-2

Luminous Flux

Js-1

ML2T-3

Linear density (mass per unit length)

Kgm-1

ML-1

Magnetic Dipole Moment

Am2

L2I

Magnetic induction (F = Bil)

NI-1m-1

MT-2I-1

Modulus of Elasticity (Stress/Strain)

Pa


ML-1T-2

Momentum

kgms-1


MLT-1

Magnetic Flux

weber (Wb)


ML2T-2I-1

Magnetic Pole Strength

Am (ampere–meter)


LI

Moment of Inertia

Kgm2

ML2

Planck’s Constant (Energy/Frequency)

Js

ML2T-1

Power (Work/Time)

watt (W)

ML2T-3

Pressure Coefficient or Volume Coefficient

θ-1

θ-1

Permittivity of Free Space

Fm-1

M-1L-3T4I2

Poisson’s Ratio (Lateral Strain/Longitudinal Strain)

Dimensionless

M0L0T0

Pressure (Force/Area)

N/m2

ML-1T-2

Pressure Head

m

L

Radioactivity

disintegrations per second

T-1

Refractive Index

Dimensionless

M0L0T0

Specific Conductance or Conductivity (1/Specific Resistance)

Sm-1

M-1L-3T3I2

Specific Gravity (Density of the Substance/Density of Water)

Dimensionless

M0L0T0

Specific Volume (1/Density)

m3kg-1

M-1L3

Stress (Restoring Force/Area)

N/m2

ML-1T-2

Ratio of Specific Heats

Dimensionless

M0L0T0

Resistivity or Specific Resistance

Ω-m

ML3T-3I-2

Specific Entropy (1/entropy)

KJ-1

M-1L-2T2θ

Specific Heat (Q = mst)


L2T-2θ-1

Speed (Distance/Time)

m/s

LT-1

Strain (Change in Dimension/Original dimension)

Dimensionless

M0L0T0

Surface Energy Density (Energy/Area)

J/m2

MT-2

Temperature

θ

θ

Thermal Capacity

Jθ-1

ML2T-2θ-1

Torque or Moment of Force

Nm

ML2T-2

Temperature Gradient

θm-1

L-1θ

Time Period

second

T

Universal Gas Constant (Work/Temperature)

Jmol–1θ-1

ML2T-2θ-1

Universal Gravitational Constant

Nm2kg-2

M-1L3T-2

Velocity (Displacement/Time)

m/s

LT-1

Volume

m3

L3

Velocity Gradient (dv/dx)

s-1

T-1

Water Equivalent

kg

M

Work

J

ML2T-2

Decay Constant

s-1

T-1

Kinetic Energy

J

ML2T-2

Potential Energy

J

ML2T-2

Application of Dimensional Formula

Some of the common applications of dimensional formula are:

  • To verify whether a formula is dimensionally correct or not.
  • Conversion of units from one system to another for any given quantity.
  • To establish derivation between physical quantities based on mutual relationships.
  • Dimensional Formulae express every physical quantity in terms of fundamental units.

Limitations of Dimensional Formula

While Dimensional Formulas offer numerous benefits, they also come with certain limitations:

  • It's important to note that quantities like trigonometric functions, plane angles, and solid angles do not possess defined dimensional formulae since they are inherently dimensionless in nature.
  • The applicability of dimensional formulas is confined to a specific set of physical quantities.
  • They are unable to determine proportionality constants, which can be a drawback in certain situations.
  • Dimensional Formulas are primarily suitable for addition and subtraction operations, limiting their use in other mathematical operations.

Dimensional Formula and Dimensional Equations

The equations resulting from equating a physical quantity to its dimensional formula are termed Dimensional Equations. These equations are an important tool for representing physical quantities in terms of fundamental units. Dimensional formulas for specific quantities used as a foundation for establishing relationships between those quantities within any given dimensional equation.

For example, consider a physical quantity denoted as Y, which depends on the fundamental quantities M (mass), L (length), and T (time) with respective powers a, b, and c. The dimensional formula for this physical quantity [Y] can be expressed as:

[Y] = [MaLbTc]

As examples:

  • The dimensional equation for velocity 'v' is expressed as [v] = [M0L1T-1].
  • The dimensional equation for acceleration 'a' is denoted as [a] = [M0L1T-2].
  • The dimensional equation for force 'F' is given as [F] = [M1L1T-2].
  • The dimensional equation for energy 'E' is represented as [E] = [M1L2T-2].
  • These dimensional equations provide a way to understand and represent various physical quantities in terms of their fundamental units.

Read More,

  • System of Units
  • Dimensional Analysis

Solved Examples on Dimensional Formula

Example 1: Using Dimensional Formula, X= MaLbTc, find the values of a, b, and c for density.

Solution:     

To find: Values for a, b, and c

Given:

Quantity = Density

Using the Dimensional Formula,

X = MaLbTc

We know,

Density = (mass/length3)

= M/L3

= M1L-3T0

Comparing with Dimensional Formula, we get,

a = 1, b = -3, c = 0

Answer: a = 1, b = -3, c = 0

Example 2: Determine the Dimensional Formula of velocity.

Solution:     

To find: Dimensional formula of velocity

We know,

Velocity = (distance/time)

= [M0L1T-1]

Answer: Dimensional formula for velocity = [M0L1T-1]

Example 3: State and verify the formula for pressure using the Dimensional Formula analysis.

Solution: 

The formula for Pressure is given as, P = Force/Area= F/A

Using Dimensional Formula analysis,

Pressure =  Force/Area
Dimesional formula for LHS = [M1L-1T–2]
Dimesional formula for RHS = [M1L1T–2]/[L2] = [M1L-1T–2] 
Since LHS matches RHS, the given formula for Pressure is verified dimensionally.

Practice Questions on Dimensional Formula

Q1. Using Dimensional Formula, X= MaLbTc, find the values of a, b, and c for Energy.

Q2. Using Dimensional Formula, X= MaLbTc, find the values of a, b, and c for Acceleration.

Q3. Determine the Dimensional Formula of Power.

Q4. Determine the Dimensional Formula of Time period of wave.


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      It is not difficult to notice the analogous nature of rotational motion and kinematic motion. The terms of angular velocity and angular acceleration remind us of linear velocity and acceleration. So, similar to the kinematic equation of motion. Equations of rotational motion can also be defined. Suc
      6 min read
    • Dynamics of Rotational Motion
      Rigid bodies can move both in translation and rotation. As a result, in such circumstances, both the linear and angular velocities must be examined. To make these difficulties easier to understand, it is needed to separately define the translational and rotational motions of the body. The dynamics o
      9 min read
    • Angular Momentum in Case of Rotation About a Fixed Axis
      Imagine riding a bicycle. As you pedal, the wheels start spinning, and their speed depends on how fast you pedal. If you suddenly stop pedaling, the wheels keep rotating for a while before gradually slowing down. This phenomenon occurs due to rotational motion, where the spinning wheels possess angu
      7 min read

    Chapter 7 - GRAVITATION

    • Gravitational Force
      Have you ever wondered why the Earth revolves around the Sun and not the other way around? Or why does the Moon remain in orbit instead of crashing into Earth? If the Earth pulls the Moon and the Moon pulls the Earth, shouldn’t they just come together? What keeps them apart? All these questions can
      11 min read
    • Kepler's Laws of Planetary Motion
      Kepler's law of planetary motion is the basic law that is used to define the motion of planets around the stars. These laws work in parallel with Newton's Law and Gravitation Law and are helpful in studying the motion of various planetary objects. Kepeler's law provides three basic laws which are, K
      10 min read
    • Acceleration due to Gravity
      Acceleration due to gravity (or acceleration of gravity) or gravity acceleration is the acceleration caused by the gravitational force of attraction of large bodies. As we know that the term acceleration is defined as the rate of change of velocity with respect to a given time. Scientists like Sir I
      8 min read
    • What is the Acceleration due to Gravity on Earth ?
      Take something in your hand and toss it down. Its speed is zero when you free it from your grip. Its pace rises as it descends. It flies faster the longer it goes. This sounds like acceleration. Acceleration, on the other hand, implies more than just rising speed. Pick up the same object and throw i
      12 min read
    • Gravitational Potential Energy
      The energy possessed by objects due to changes in their position in a gravitational field is called Gravitational Potential Energy. It is the energy of the object due to the gravitational forces. The work done per unit mass to bring the body from infinity to a location inside the gravitational field
      13 min read
    • Escape Velocity
      Escape velocity as the name suggests, is the velocity required by an object to escape from the gravitational barrier of any celestial object. "What happens when you throw a stone upward in the air?" The stone comes back to the Earth's surface. If we throw the stone with a much higher force still it
      7 min read
    • Artificial Satellites
      When looked at the night sky many heavenly bodies like stars, moon, satellites, etc are observed in the sky. Satellites are small objects revolving or orbiting around a planet or on object larger than it. The most commonly observed and known satellite is the moon, the moon is the satellite of Earth,
      8 min read
    • Binding Energy of Satellites
      Humans learn early in life that all material items have a natural tendency to gravitate towards the earth. Anything thrown up falls to the ground, traveling uphill is much more exhausting than walking downhill, Rains from the clouds above fall to the ground, and there are several additional examples
      10 min read

    Chapter 8 - Mechanical Properties of Solids

    • Stress and Strain
      Stress and Strain are the two terms in Physics that describe the forces causing the deformation of objects. Deformation is known as the change of the shape of an object by applications of force. The object experiences it due to external forces; for example, the forces might be like squeezing, squash
      12 min read
    • Hooke's Law
      Hooke's law provides a relation between the stress applied to any material and the strain observed by the material. This law was proposed by English scientist Robert Hooke. Let's learn about Hooke's law, its application, and others, in detail in this article. What is Hooke’s Law?According to Hooke's
      10 min read
    • Stress-Strain Curve
      Stress-Strain Curve is a very crucial concept in the study of material science and engineering. It describes the relationship between stress and the strain applied on an object. We know that stress is the applied force on the material, and strain, is the resulting change (deformation or elongation)
      12 min read
    • Modulus of Elasticity
      Modulus of Elasticity or Elastic Modulus is the measurement of resistance offered by a material against the deformation force acting on it. Modulus of Elasticity is also called Young's Modulus. It is given as the ratio of Stress to Strain. The unit of elastic modulus is megapascal or gigapascal Modu
      12 min read
    • Elastic Behavior of Materials
      Solids are made up of atoms based on their atomic elasticity (or molecules). They are surrounded by other atoms of the same kind, which are maintained in equilibrium by interatomic forces. When an external force is applied, these particles are displaced, causing the solid to deform. When the deformi
      10 min read

    Chapter 9 - Mechanical Properties of Fluids

    • What is Pressure?
      Pressure is the force applied to the surface of an object per unit area over which that force is distributed. Various units are used to express pressure. Some of these derive from a unit of force divided by a unit of area; the SI unit of pressure, the pascal (Pa), for example, is one newton per squa
      9 min read
    • Streamline Flow
      The substance that can change its form under an external force is defined as fluid. Whenever an external force is applied to a fluid, it begins to flow. The study of fluids in motion is defined as fluid dynamics. Have you ever noticed a creek flowing beneath the bridge? When you see a streamline, wh
      7 min read
    • Bernoulli's Principle
      Bernoulli's Principle is a very important concept in Fluid Mechanics which is the study of fluids (like air and water) and their interaction with other fluids. Bernoulli's principle is also referred to as Bernoulli's Equation or Bernoulli Theorem. This principle was first stated by Daniel Bernoulli
      15+ min read
    • What is Viscosity?
      Viscosity is the measurement of the resistance of the flowing liquid. Let us learn more about viscosity with an example suppose we take two bowls, one bowl contains water and the other has honey in it, we drop the content of both bowls then we see that water flows much faster than honey which conclu
      12 min read
    • Surface Tension
      Surface tension is the ability of fluid surfaces to contract into the smallest possible surface area. Have you ever found that even after filling a glass full of water, you can only add a few more drops before it spills? Have you ever lost a thermometer and watched how the mercury reacts as it falls
      11 min read
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