All physical quantities can be expressed in terms of seven fundamental units called seven dimensions of physical world. Thus, a physical quantity can be expressed in terms of these fundamental units or dimensions. A physical quantity expressed in terms of fundamental dimension is termed as dimensional formula of that physical quantity.

The dimensional formulae of force, energy, compressibility and capacity of a capacitor, are [MLT^-2],

[ML²T²], [M^-1 LT²) and [M^-1L^-2T⁴A²] respectively,

Applications of dimensional analysis
   
To find the unit of a physical quantity

For example, G = [M^-1L³T²]. Its SI unit is m³kg^-1s^-2 or Nm² kg^-2

To convert a physical quantity from on system of units to another system of units

It is based on the fact that

n₁u₁ = n₂u₂

We illustrate by an example. Let us convert g from SI system to FPS system

We know [g] = [Lt²]

n₂ = n₁ [L₁/L₂] [T₁/T₂]^-2 1 ft = 30.48 cm

= 9.8 [100/30.48] [1/1]^-2 1 m = 100 cm

= 32.2 fts^-2

To check the correctness of a given physical relation

It is based on principle of homogeneity that is, the dimensions on two sides be same for a given relation.

For example

F = mv²/r where, [F] = [MLT^-2]; [v] [LT^-1]

Therefore, LHS = [MLT^-2]

RHS = [M] {LT^-1]² / [L] = [MLT^-2]

To derive a relation let us consider an example. Derive plank’s length in terms of G, c and h

L = f (G, e, h) L = KGx cy hz

[L] = [M^-1 L³ T^-2] x [LT^-2] y [ML² T^-1] z

-x + z =), 3x + y + 2z = 1 and – 2x –y –z = 0

X = ½, y = -3/2 and z = 1/2

Thus, L = K √GH/c³.

If K = 1 then L ≃ 10^-35 m.

The importance of plank’s length is yet to be established.

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