Dimensional Formula

In everyday life, the technique of measuring is required to measure or compare physical quantities. As a result, we chose units that are universally accepted to measure each of the standard values.

Other related amounts can also be stated in these units and measured in the same way. The physical quantities are measured in terms of the unit and are referred to as the physical quantity’s standard. To express any measurement, we’ll need the numerical value (n) as well as the unit ().

Physical quantity to be measured = Numerical value x Unit 

Units of Measurement

There were three separate systems of units in use in various countries previously. However, the worldwide SI system of units is currently being adopted by the entire world. Seven quantities are used as the base quantities in this system of units.

Centimetre, Gram, and Second are used in the CGS System to express length, mass, and time, respectively. In the FPS (First-Person Shooter) System. The units of length, mass, and time are expressed in feet, pounds, and seconds, respectively. Length is measured in metres, mass is measured in kilograms, and time is measured in seconds in the MKS System. Length, mass, time, electric current, thermodynamic temperature, the luminous intensity and the amount of substance are measured in metres, kilograms, seconds, amperes, kelvins, moles, and candelas, respectively.

Dimensional Formula and Dimensions

The physical quantity’s dimensions are the powers to which the basic quantities are elevated to represent that amount. The dimensional formula of any physical quantity is an equation that explains how and which of the base quantities are contained in that amount. It is written by enclosing the symbols representing base amounts in square brackets with the corresponding power, i.e. ().

For example, the dimensional formula for mass is denoted by (M)

A dimensional equation is an equation derived by equating a physical quantity with its dimensional formula.

The dimensional formula for 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 is

M[a]L[b]T[c].

Advantages

The following are the advantages of the dimensional formula:

• To determine whether or not a formula is dimensionally correct
• To change the units of measurement from one system to another
• To derive physical quantity relationships based on their interdependence
• Dimensional Formulas show how to express any physical quantity in terms of fundamental units

Limitations

Dimensional formulas have a number of advantages, but they also have certain drawbacks. The following are the details:

• In the case of trigonometric, logarithmic, and exponential functions, dimensional formulas become undefined, which implies we can’t predict the nature of values using these functions
• The physical quantities that can be used in Dimensional Formula are limited to a few
• They can’t be utilised to figure out what proportionality constants to apply
• Only addition and subtraction are allowed in dimensional formulas

Some examples of calculating the dimensional formulas:

1. Dimensional formula of force- Force is defined as the mass multiplied by the acceleration, as mentioned in the formula above. The unit of mass is Kg, and the dimensions are M1 L0 T0. The unit of acceleration is m/s2, and it is expressed dimensionally as M0 L1 T2, hence the unit of force is Kgm/s2. As a result, it can be written in three dimensions.

 As, [M1] [L0] [T0] X [M0] [L1] [T0]

 ————————————

 [M0] [L0] [T2]

= Kg x m

  ———-

      s2

As a result, we get [M1] [L1] [T2] as the dimensional formula for Force.

2.     Dimensional formula of power-

Power, denoted by the symbol (P)= work x time (second) -1

Because, work (joule)= force (M x A) X displacement= [M1 L1 T-2] X [L1]

So, the dimensional formula of work= M1 L2 T-2

Since power= work done x time taken-1

Therefore, the dimensional formula of power is M1 L2 T-3

Some other dimensional formulas

• Length = [L]1
• Mass = [M]1
• Time = [T]1
• Acceleration = [LT-2]
• Angle (arc/radius) = [M0 L0 T0]
• Angular displacement = [M0 L0 T0]
• Angular frequency = [T]-1
• Angular impulse = [M1 L2 T-1]
• Angular momentum = [M1 L2 T-1]
• Angular velocity = [T-1]
• Area = [L2]
• Bulk modulus = [M1 L-1 T-2]
• Calorific value = [L2 T-2]
• Coefficient of surface tension = [M 1T-2]
• Coefficient of viscosity = [M L-1 T-1]

Conclusion

As we’ve seen, dimensional analysis can assist you in solving difficulties that arise in your daily life. While you’ll certainly learn more about dimensional analysis as you progress through your science classes, it’s especially beneficial for Biology students to do so. Some believe that dimensional analysis can assist Biology students to develop a “better feel for numbers” and make it easier for them to segue into courses like Organic Chemistry or Physics (if they haven’t already).