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# Units and Dimensions - Dimensional Formula

Last updated date: 29th May 2024
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## Introduction Units and Measurement

The process of measurement is required to measure or compare physical quantities in day-to-day life. Thus, to measure each of the standard quantities, we choose certain units for them that are accepted worldwide. Other similar quantities can also be expressed in terms of these units and can be measured accordingly. The physical quantities are measured in terms of the unit and are known as a standard of that physical quantity. Thus to express any measurement done, we need the numerical value (n) and the unit (μ) of it.

Measured physical quantity = Numerical value x Unit

For example, the Length of any given rod = 6 m

where 6 is a numerical value and m (meter) is a unit of length.

### Fundamental and Derived Physical Quantities

Fundamental quantities are those elementary physical quantities that do not require any other physical quantity to express them. In other words, these cannot be resolved further in terms of any other physical quantity. Thus fundamental quantities are known as basic physical quantities. The units used to express these fundamental physical quantities are called fundamental units.

Mass, Length, and Time expressed in kilogram, meter and second respectively are fundamental units. All those physical quantities which can be expressed in terms of basic physical quantities or are derived from the combination of two or more fundamental quantities are termed as the derived physical quantities. For example, units of velocity, force are m/s and kgm/s2 respectively and they are examples of derived units.

## Unit of Fundamental and Supplementary Physical Quantities in SI System

### Definitions of Fundamental Units

There are seven fundamental units of SI. These have been defined as

• 1 meter is defined as the distance that contains 1650763.73 wavelengths of orange-red light of atom Kr-8.

• 1 kilogram is defined in terms of a cylindrical prototype of mass made of platinum and iridium alloys of height 39 mm and diameter 39 mm. Also, it can be defined as a mass of 5.0188 x 1025 atoms of carbon-12.

• 1 second is defined as the time in which a cesium atom vibrates 9192631770 times in an atomic clock.

• 1 kelvin is defined as the temperature at which (1/273.16) is part of the thermodynamics temperature of the triple point of water.

• 1 candela is (1/60) luminous intensity of an ideal source by an area of cm’ when the source is at the melting point of platinum (1760°C).

• 1 ampere is the current which is maintained in two straight parallel conductors of infinite length and almost negligible cross-section area placed one meter apart in a vacuum and producing force of a force 2 x 10-7 N per meter length between them.

• 1 mole is defined as the amount of substance of a system that contains as many as elementary entities ( atoms, molecules, ions, electrons or group of particles, as this and atoms in 0.012 kg of carbon isotope 6C12.

### System of Units

Earlier there were three different systems of units that were used in different countries. These were CGS, FPS, and MKS systems. But nowadays, the whole world is adopting the international SI system of units. In this system of units, seven quantities are taken as the base quantities.

 FundamentalQuantity System of Units CGS MKS FPS Length centimeter meter foot Mass gram kilogram pound Time second

 Physical Quantity Unit Symbol Length meter m Mass kilogram kg Time second s Quantity of substance mole mol Electric current ampere A Thermodynamic temperature kelvin K Intensity of light candela cd Plane angle Radian rad Solid angle Steradian sr

### List of SI Units

 Macro Prefixes Symbol Value Kilo K $10^3$ Mega M $10^6$ Giga G $10^9$ Tera T $10^{12}$ Peta P $10^{15}$ Exa E $10^{18}$ Zetta Z $10^{21}$ Yotta Y $10^{24}$

 Micro Prefixes Symbol Value milli m $10^{-3}$ mu 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 $10^{-24}$

## How to Write Units of Physical Quantities?

1. Unit names should not begin with a capital letter, even if they are named after scientists. For instance, newton, watt, ampere, and metre.

2. The unit should be written in full or only in agreed-upon symbols.

3. No plural form is used for units. For instance, 10 kg but not 10 kgs, and 20 w but not 20 ws.

4. No full stop or punctuation mark should be used within or after unit symbols. For instance, 10 W but not 10 W.

## What Exactly are Dimensions and Dimensional Analysis?

Dimensions: Dimensions of a physical quantity are the powers to which the fundamental units are raised to obtain one unit of that quantity.

Dimensional Analysis: Dimensional analysis is the practice of determining the dimensions of physical quantities in order to check their relationships. These dimensions are not affected by numerical multiples or constants, and any quantity in the world can be expressed as a function of the fundamental dimensions.

### Dimensional Formula

Dimensional formula of any physical quantity is an expression that represents how and which of the base quantities are included in that quantity. It is written by enclosing the symbols for base quantities with appropriate power in square brackets i.e( ).

E.g: the Dimension formula of mass is: (M)

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

When a physical quantity X depends on base dimensions M(Mass), L(Length), and T(Time), Temperature, current electricity, luminous intensity, and amount of substance with respective powers a, b and c, its dimensional formula is represented as  $M^{a} L^{b} T^{c}$.

• The dimensional formula of Velocity is $M^{0} L T^{-1}$

• Dimensional formula of Volume is $M^{0} L^{3} T^{0}$

• The dimensional formula of Force is $M L T^{-2}$

• Dimensional formula of Area is $M^{0} L^{2} T^{0}$

• The dimensional formula of Density is $M L^{-3} T^{0}$

 Physical Quantity Dimensional Equation Force (F) F = $M L T^{-2}$ Power (P) P = $M L^{2} T^{-3}$ Velocity (v) v  = $M L T^{-1}$ Density (D) D =  $M L^{3} T^{0}$ Energy (E) E =  $M L^{2} T^{-2}$ Pressure (P) P = $M L^{-1} T^{-2}$ Time Period of wave T = $M^{0} L^{0} T^{-1}$

### Some Important Conversions

 Conversion/Constant Value 1 bar $1 \times 10^6\text{ dyne/cm}^2 = 1 \times 10^5 Nm^{-2} = 1 \times 10^5 \text{ pascal}$ 76 cm of Hg $1.013 \times 10^6 \text{dyne/cm}^2 = 1.013 \times 10^5 \text{ pascal} = 1.013\text{ bar}$ 1 toricelli or torr $1 mm\text{ of Hg }= 1.333 \times 10^3\text{ dyne/cm}^2 = 1.333\text{ millibar}$ 1 kmph $5/18 ms^{-1}$ 1 dyne $1 \times 10^{-5} N$ 1 H.P 746 watt 1 kilowatt hour $36 \times 10^5 J$ 1 kgwt g newton 1 calorie 4.2 joule 1 electron volt $1.602 \times 10^{-19}$ joule 1 erg $1 \times 10^{-7}$ joule

### Some Important Physical Constants

 Conversion/Constant Value Velocity of light in vacuum (c) $3 \times 10^8 ms^{-1}$ Velocity of sound in air at STP $331 ms^{-1}$ Acceleration due to gravity (g) $9.81 ms^{-2}$ Avogadro number (N) $6.023 \times 10^{23}/mol$ Density of water at 4oC $1000 kgm^{-3}$ or $1 g/cc$ Absolute zero -273.15 deg C or 0 K Atomic mass unit $1.66 \times 10^{-27}$ kg Quantum of charge (e) $1.602 \times 10^{-19}$ C Stefan’s constant $5.67 \times 10^{-8}$ W/m$^2$/K$^4$ Boltzmann’s constant (K) $1.381 \times 10^{-23} JK^{-1}$ One atmosphere 76 cm Hg = 1.013 \times 10$^5$ Pa Mechanical equivalent of heat (J) 4.186 J/cal Planck’s constant (h) $6.626 \times 10^{-34}$ Js Universal gas constant (R) 8.314 J/mol–K Permeability of free space (μ0) $4\pi \times 10^{-7} Hm^{-1}$ Permittivity of free space (ε0) $8.854 \times 10^{-12} Fm^{-1}$ Density of air at S.T.P. $1.293 \text{kg m}^{-3}$ Universal gravitational constant $6.67 \times 10^{-11} Nm^2kg^{-2}$

### Derived SI Units with Special Names

 Physical Quantity SI Unit Symbol Frequency hertz Hz Energy joule J Force newton N Power watt W Pressure pascal Pa Electric charge orquantity of electricity coulomb C Electric potential difference and emf volt V Electric resistance ohm $ohm$ Electric conductance siemen S Electric capacitance farad F Magnetic flux weber Wb Inductance henry H Magnetic flux density tesla T Illumination lux Lx Luminous flux lumen Lm

### Dimensional Formulas for Physical Quantities

 Physical Quantity Unit Dimensional Formula Acceleration or acceleration due to gravity $ms^{-2}$ $LT^{-2}$ Angle (arc/radius) rad $M^0L^0T^0$ Angular displacement rad $M^0L^0T^0$ Angular frequency (angular displacement/time) $\text{rads}^{-1}$ $T^{-1}$ Angular impulse (torque x time) Nms $ML^2T^{-1}$ Angular momentum (Iω) $kgm^2s^{-1}$ $ML^2T^{-1}$ Angular velocity (angle/time) $\text{rads}^{-1}$ $T^{-1}$ Area (length x breadth) $m^2$ $L^2$ Boltzmann’s constant $JK^{-1}$ $ML^2T^{-2}\theta^{-1}$ Bulk modulus $Nm^{-2}, Pa$ $M^1L^{-1}T^{-2}$ Calorific value $Jkg^{-1}$ $L^2T^{-2}$ Coefficient of linear or areal or volume expansion ${}^circC^{-1}$ or $K^{-1}$ $\theta^{-1}$ Coefficient of surface tension (force/length) $Nm^{-1}$ or $Jm^{-2}$ $MT^{-2}$ Coefficient of thermal conductivity $Wm^{-1}K^{-1}$ $MLT^{-3}\theta^{-1}$ Coefficient of viscosity poise $ML^{-1}T^{-1}$ Compressibility (1/bulk modulus) $Pa^{-1}, m^2N^{-2}$ $M^{-1}LT^2$ Density (mass/volume) $kgm^{-3}$ $ML^{-3}$ Displacement, wavelength, focal length m L Electric capacitance (charge/potential) $CV^{-1}$, farad $M^{-1}L^{-2}T^4I^2$ Electric conductance (1/resistance) Ohm$^{-1}$ or mho or siemen $M^{-1}L^{-2}T^3I^2$ Electric conductivity (1/resistivity) siemen/metre or Sm$^{-1}$ $M^{-1}L^{-3}T^3I^2$ Electric charge or quantity of electric charge (current x time) coulomb IT Electric current ampere I Electric dipole moment (charge x distance) Cm LTI Electric field strength or intensity of electric field (force/charge) $NC^{-1}, Vm^{-1}$ $MLT^{-3}I^{-1}$ Electric resistance ohm $ML^2T^{-3}I^{-2}$ Emf (or) electric potential (work/charge) volt $ML^2T^{-3}I^{-1}$ Energy (capacity to do work) joule $ML^2T^{-2}$ Energy density $Jm^{-3}$ $ML^{-1}T^{-2}$ Entropy $J\theta^{-1}$ $ML^2T^{-2}\theta^{-1}$ Force (mass x acceleration) newton (N) $MLT^{-2}$ Force constant or spring constant (force/extension) $Nm^{-1}$ $MT^{-2}$ Frequency (1/period) Hz $T^{-1}$ Gravitational potential (work/mass) $Jkg^{-1}$ $L^2T^{-2}$ Heat (energy) J or calorie $ML^2T^{-2}$ Illumination (Illuminance) lux (lumen/metre2) $MT^{-3}$ Impulse (force x time) Ns or kgms$^{-1}$ $MLT^{-1}$ Inductance (L)orcoefficient of self-induction henry (H) $ML^2T^{-2}I^{-2}$ Intensity of gravitational field (F/m) $Nkg^{-1}$ $L^1T^{-2}$ Intensity of magnetization (I) $Am^{-1}$ $L^{-1}I$ Joule’s constant or mechanical equivalent of heat Jcal$^{-1}$ $M^0L^0T^0$ Latent heat (Q = mL) $Jkg^{-1}$ $M^0L^2T^{-2}$ Linear density (mass per unit length) $kgm^{-1}$ $ML^{-1}$ Luminous flux lumen or $(Js^{-1})$ 4ML^2T^{-3}$Magnetic dipole moment$Am^2L^2I$Magnetic flux (magnetic induction x area) weber (Wb)$ML^2T^{-2}I^{-1}$Magnetic induction (F = Bil)$NI^{-1}m^{-1}$or T$MT^{-2}I^{-1}$Magnetic pole strength (unit: ampere–meter) Am LI Modulus of elasticity (stress/strain)$Nm^{-2}$, Pa$ML^{-1}T^{-2}$Moment of inertia (mass x radius2)$kgm^2ML^2$Momentum (mass x velocity) kgms$^{-1}MLT^{-1}$Permeability of free space$Hm^{-1}$or$NA^{-2}MLT^{-2}I^{-2}$Permittivity of free space$Fm^{-1}$or$C^2N^{-1}m^{-2}M^{-1}L^{-3}T^4I^2$Planck’s constant (energy/frequency) Js$ML^2T^{-1}$Poisson’s ratio (lateral strain/longitudinal strain) ––$M^0L^0T^0$Power (work/time)$Js^{-1}$or watt (W)$ML^2T^{-3}$Pressure (force/area)$Nm^{-2}$or Pa$ML^{-1}T^{-2}$Pressure coefficient or volume coefficient$^circC^{-1}$or$\theta^{-1}\theta^{-1}$Pressure head m$M^0LT^0$Radioactivity disintegrations per second$M^0L^0T^{-1}$Ratio of specific heats ––$M^0L^0T^0$Refractive index ––$M^0L^0T^0$Resistivity or specific resistance$ohm - mML^3T^{-3}I^{-2}$Specific conductance or conductivity (1/specific resistance) siemen/metre or$Sm^{-1}M^{-1}L^{-3}T^3I^2$Specific entropy (1/entropy)$KJ^{-1}M^{-1}L^{-2}T^2\theta$Specific gravity (density of the substance/density of water) ––$M^0L^0T^0$Specific heat (Q = mst)$Jkg^{-1}\theta^{-1}M^0L^2T^{-2}\theta^{-1}$Specific volume (1/density)$m^3kg^{-1}M^{-1}L^3$Speed (distance/time)$ms^{-1}LT^{-1}$Stefan’s constant$Wm^{-2}\theta^{-4}ML^0T^{-3}\theta^{-4}$Strain (change in dimension/original dimension) ––$M^0L^0T^0$Stress (restoring force/area)$Nm^{-2}$or Pa$ML^{-1}T^{-2}$Surface energy density (energy/area)$Jm^{-2}MT^{-2}$Temperature$^\circ C$or θ$M^0L^0T^0\theta$Temperature gradient$^\circ Cm^{-1}$or$\theta m^{-1}M^0L^{-1}T^0\theta$Thermal capacity (mass x specific heat)$J\theta^{-1}ML^2T^{-2}\theta^{-1}$Time period second T Torque or moment of force (force x distance) Nm$ML^2T^{-2}$Universal gas constant (work/temperature)$J mol^{-1}\theta^{-1}ML^2T^{-2}θ^{-1}$Universal gravitational constant$Nm^2kg^{-2}M^{-1}L^3T^{-2}$Velocity (displacement/time)$ms^{-1}LT^{-1}$Velocity gradient (dv/dx)$s^{-1}T^{-1}$Volume (length x breadth x height)$m^3L^3$Water equivalent kg$ML^0T^0$Work (force x displacement) J$ML^2T^{-2}\$

## Quantities with the Same Dimensional Formula

When certain things in physics have similar formulas, they fall into the same category. For example:

1. Impulse and momentum.

2. Work, torque, moment of force, and energy.

3. Angular momentum, Planck’s constant, and rotational impulse.

4. Stress, pressure, modulus of elasticity, and energy density.

5. Force constant, surface tension, and surface energy.

6. Angular velocity, frequency, and velocity gradient.

7. Gravitational potential and latent heat.

8. Thermal capacity, entropy, universal gas constant, and Boltzmann’s constant.

9. Force and thrust.

10. Power and luminous flux.

## Practical Uses of Dimensional Analysis

Dimensional analysis is crucial in understanding physical quantities. Here are some ways it's applied:

1. Checking Equations: It helps ensure that a physical equation is correct.

2. Establishing Relationships: It aids in figuring out connections between different physical quantities.

3. Unit Conversion: It allows for converting the units of a physical quantity from one measurement system to another.

### Tips for the JEE Preparation

The chapter Units and dimensions are very important. It will be easier to be thorough in the chapter once you complete the syllabus. Once you complete the syllabus you will be well aware of the formulas and the chapter becomes much easier for you to understand rather than just mugging the formulas.

What the students should understand is the importance of each chapter and the topics and learn the concepts and then proceed further. When there is clarity things will fall in the right place and everything becomes very simple.

Timetable- It is very important to have a proper plan and a study strategy. Know the syllabus and make a plan in such a way that the syllabus has to be completed in a specific time so that there is enough time to practice the question papers and the mock test.

Know the Strengths and Weaknesses- While analyzing the syllabus you will come across the topics which might be easy and also difficult ones. Knowing this, devote more time to the difficult topics so that it is learned and understood clearly.

Revise the Topics- As you proceed further with the learning of the new topics it is equally important to revise the previously studied topics. So before you start any new topic, spare a few minutes to revise the topics. This will be very helpful to remember well and also easier.

Formulas- This chapter has lots of formulas and often it is a difficult task to remember those formulas thoroughly. So the best method would be to write down all the formulas in a single sheet or notes and read them out every day, this way you will be able to remember effectively. It is a known fact that when you keep reading certain things every day automatically it gets captured in your mind and it won’t get erased easily.

One more important thing to remember is to learn the formulas with understanding and just don’t memorize. Once you complete the syllabus then you can start solving the previous year's question paper. This will help you to know the exact pattern of the real exam and also the difficulty level in the exam. This will help you to prepare in those areas and be thorough with the preparation. The methods are very simple, it just helps to be focussed and the practice makes a difference.

The extensive practice of question papers will help the students to have an appropriate exam preparation with the improvement of the time management skills which is very important in exams like IIT JEE.

Solving the question papers will also help in the enhancement of speed and accuracy.

### Advantages of Studying the Real-life Examples of the Exothermic Reactions

• Science is a practical subject, especially Chemistry, that is to say, the knowledge of practical concepts of chemistry is just as important as that of its theory.

• A real-life example of Exothermic Reactions becomes rather easy to understand the theory by the real-life practical demonstration. Furthermore, it helps the students not only in understanding the concepts of Exothermic Reactions but also in remembering the same as well.

• The real-life example also makes the students understand how the everyday things that happen around are the reason for Exothermic Reactions.

## Conclusion

Understanding unit and dimension formulas is like having a universal language in the world of physics. It's a way to group similar things together, like impulse and momentum or work and energy. These formulas help us check if our physics equations make sense and even figure out new relationships between different quantities. Think of it as a toolkit that lets us speak the same "quantum language" and convert measurements from one system to another. So, whether we're exploring the mysteries of angular momentum or calculating the power of light, unit and dimension formulas are our trusty guides in the fascinating realm of physics.

## FAQs on Units and Dimensions - Dimensional Formula

1. What are the limitations of  Dimension Formulas?

This can be defined as the relationship of physical quantities. We can also define the dimensional quantities which do not have a fixed value. Now let us know some of the limitations in the dimensional formulas. The formulas cannot be derived if they contain trigonometric functions, exponential functions, and log functions. This does not contain any dimensional constants in the formula. It is not possible to know the plus and minus in two points and the relations.

2. What are the four main applications of dimensions?

The four main applications of dimensions are that it helps in the conversion of the physical quantity from one system to another. It helps in checking the accuracy of the physical relation. It helps to obtain the relations among various physical quantities. Lastly, it helps in finding the dimensions of a constant in physical relation. So focussing and learning all these applications and the concepts will help in the preparation of this chapter comprehensively. Students can refer to the Vedantu NCERT solution to have a better understanding.

3. What do you understand by Dimensional analysis?

Dimensional analysis helps in quantifying the size and shape of the objects. It helps to analyze the nature of the objects involving lengths, angles, and some of the geometrical properties like flatness and straightness. The main concepts in dimensions are that it is possible to add or subtract those quantities that have the same dimensions only. We can also conclude that the two physical quantities are said to be equal if they have the same dimensions.

4. What are the advantages of preparing the chapter from Vedantu’s NCERT solutions?

The advantages of Vedantu are as below.

1. The solutions are well organized.

2. The chapter-wise solutions are very systematic and are precise.

3. Students will know the exact format of answering.

4. This is the best guide for helping the students in every step of the preparation.

5. All the solutions are prepared by experts having excellent subject knowledge.

6. The latest curriculum of the CBSE is followed accurately.

7. Students can be confident and prepare thoroughly through Vedantu’sNCERT solutions for the chapter Units and Dimensions.

8. We can say accuracy is Vedantu.

5. What are the advantages of solving the previous year's question papers?

Solving the previous year's questions paper helps the students to have an exact idea of the real exam pattern. Students should solve at least the previous 5 years question paper so that they can improve on their time management skills and also enhance their speed and accuracy. Students can analyze the questions which are repeated. These can be more focussed and prepared accurately.  It is mandatory to be perfect in all the topics but certain topics can be given little more importance. It does not mean that you can skip any of the topics. Every topic is very essential.

6. Explain a few limitations of Dimension Formulas.

This method can be used only if dependency is of multiplication type. The formula containing exponential, trigonometric, and logarithmic functions can not be derived using this method. A formula containing more than one term which is added or subtracted likes s = ut+ ½ at2 also cannot be derived. The relation derived from this method gives no information about the dimensionless constants.

7. The ratio of the dimension Planck’s constant and that of the moment of inertia is the dimension of?

A. Velocity

B. Angular momentum

C. Time

D. Frequency

Time

8. Write some applications of Dimensional Analysis.

Following are some applications of dimensional analysis:

1. To convert a physical quantity from one system of the unit to the other.

2. To check the dimensional correctness of a given physical relation: If in a given relation, the terms of both sides have the same dimensions, then the equation is dimensionally correct. This concept is best known as the principle of homogeneity of dimensions.

3. To derive the relationship between different physical quantities: Using the principle of homogeneity of dimension the new relation among physical quantities can be derived if the dependent quantities are known.