Units and Dimensions - Dimensional Formula

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.

List of SI Units

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 \[M L^{-3} T^{0}\]

Some Important Conversions

Some Important Physical Constants

Derived SI Units with Special Names

Dimensional Formulas for Physical Quantities

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.

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