Fundamental And Derived Quantities

The need for a measurement system came into existence due to the need for a common measurement system throughout the world. This introduced the concepts of fundamental and derived quantities. Earlier, people used to measure through their local and traditional measurement systems.

These systems differ from place to place and unit to unit. For example, foot, centimeter, and meter are three different units for measuring length. Yard, acre, and hectare are some of the basic units to measure the area. Pound, gram, and kilogram are some common units for measuring weight. This was the basis for establishing a standard measurement unit. 

Fundamental And Derived Quantities

Those entities which can be calculated using any mathematical formula or derivation are termed a physical quantity. Also, these quantities have a unit for their measurement and conversion to another unit. According to their basic need, these physical quantities are further subdivided into two subcategories, fundamental and derived quantities. The International System of Units has categorized the units into two major categories, fundamental and derived units. According to SI units, there are seven fundamental units and the rest are derived from them. 

Fundamental Quantities

Fundamental quantities are those physical quantities that can be expressed by a single type of entity and need no other entity to derive them. For example, the quantity mass is described by the kilogram and is used to define other quantities such as force, momentum, and many more. The 7 fundamental quantities are given below

1. Mass

2. Current

3. Time

4. Temperature

5. Length

6. Substance Amount

7. Luminous Intensity

Symbol Of Fundamental Quantities

The 7 fundamental quantities have specific symbols which are used to define the dimensional formula. The symbolic representation is different for different quantities and is mainly used to define the derived quantities. Let’s get to know more about it.

1. Mass: The mass is denoted by M. According to the SI unit system, the standard unit of mass is a kilogram and symbolized as kg.

2. Current: The current is denoted by I and has the SI unit Coulomb.

3. Temperature: The SI unit of temperature is Kelvin and is denoted by the symbol K

4. Substance amount: The amount of any substance is defined in the terms of moles as SI unit and has the symbol mol.

5. Luminous intensity: The luminous intensity is measured in the terms of Candela as the SI unit. And have a symbol Cd.

6. Length: The length is depicted by the symbol L and has the SI unit meter.

7. Time: The SI unit of time is second and is denoted by the symbol T

Derived Quantities

Those physical quantities which can be expressed in the terms of fundamental quantities are commonly known as the derived quantities. For example, velocity is a derived quantity that depends on length and time.

SI Units

The international system of units is popularly known as the SI Unit. Due to differences in languages throughout the world, a common standard of measurement was not introduced. This led to a quiet disturbance in communication and measurement. The SI unit was established and is governed by the General Conference of Weight and Measures. Furthermore, this system is accepted by many countries throughout the world and is a widely adopted system.

Dimensional Formula

Let us now take a look at how to find the dimensional formula of the derived quantities using the 7 fundamental quantities. Let’s take some examples while keeping in mind the symbolic representation of the fundamental quantities.

Example 1: Find the dimensional formula of the Gravitation constant G?

As we know that 

F = G m1 m2 / r²

where F is the gravitational force

G is the gravitational constant

m1 and m2 are the mass of two bodies

r is the distance between the two objects

Let us assume the dimensional formula of G as MxLyTz.

Also, we know that the dimensional formula of F is MLT-2, m1 and m2 are M and r² is L².

Putting the above in the formula, we get

MLT-2 = MxLyTz . [M] . [M] / [L²]

Also, MLT-2 = MxLyTz . [M²] / [L²]

[MLT-2]. [M-2 ]. [L²] = MxLyTz

MxLyTz  = M-1L3T-2

From the above, we get, x = -1, y = 3, z= -2, and hence the dimensional formula for gravitational constant G is M-1L3T-2.

Example 2: Find the dimensional formula of Pressure?

We know that 

Pressure = Thrust / Area – equation 1

Where thrust is a type of force and force is the product of acceleration and mass of the body.

Thrust or Force = Mass × Acceleration

Acceleration = Rate of change of velocity / time

Now put all the values in the pressure formula equation 1. Let the formula for pressure be MxLyTz

Pressure = Mass × Acceleration / Area

MxLyTz = [M].[LT-2] / [L2]

MxLyTz = ML-1T-2

From the above equation, we have x = 1, y = -1, and z = -2

And the dimensional formula for pressure is ML-1T-2

Conclusion

The fundamental and derived quantities are the base for defining the standard units of each physical quantity. Earlier, People did not use a standard measurement system, and their units vary from region to region. This led to the setup of the International System of Units which resulted in the standardization of units. The derived quantities are extracted from the 7 fundamental quantities.