# Unit and Dimensions-Dimension Formula

## 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, Length of any given rod = 6 m

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

### Fundamental and Derived Physical Quantities

Fundamental quantities are those elementary physical quantities which 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, metre 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.

### Definitions of Fundamental Units

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

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

2. 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.

3. 1 second is defined as the time in which a caesium atom vibrates 9192631770 times in an atomic clock.

4. 1 kelvin is defined as the temperature at which (1/273.16) part of the thermodynamics temperature of the triple point of water.

5. 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).

6. 1 ampere is the current which is maintained in two straight parallel conductors of infinite length and almost negligible cross-section area placed one metre apart in a vacuum and producing force of a force 2 x 10-7 N per metre length between them.

7. 1 mole is defined as the amount of substance of a system which 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 which 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.

In CGS System Centimetre, Gram and Second are used for expressing length, mass and time respectively. In the FPS System. Foot, pound and second are used to express the quantities length, mass and time respectively. While in MKS System, Length is expressed in metre, mass is expressed in kilogram and time in second.

In Standard units system Length, mass, time, electric current, thermodynamic temperature, Amount of substance and luminous intensity are expressed in units: metre, kilogram, second, ampere, kelvin, mole and candela respectively.

### Dimensions and Dimensional Formula

Dimensions of the physical quantity are the power to which the base quantities are raised to represent that quantity. Dimensional formula of any physical quantity is an expression which 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: 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 [MaLbTc].

• Dimensional formula of Velocity is [M0LT-1]

• Dimensional formula of Volume is [M0L3T0]

• Dimensional formula of Force is [MLT-2]

• Dimensional formula of Area is [M0L2T0]

• Dimensional formula of Density is [ML-3T0]

 Physical Quantity Dimensional Equation Force (F) [F] = [M L T-2] Power (P) [P] = [M L2 T-3] Velocity (v) [v]  = [M L T-1] Density (D) [D] = [M L3 T0] Energy (E) [E] = [M L2 T-2] Pressure (P) [P] = [M L-1 T-2] Time Period of wave [T] = [M0L0 T-1]

Q1. Explain a few limitations of Dimension Formulas.

Ans: 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.

Q2. 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

Ans: Time

Q3. Write some applications of Dimensional Analysis.

Ans: 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 dimensional correctness of a given physical relation: If in a given relation, the terms of both the 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.