Dimensional Analysis

Dimensions

Dimensions of a physical quantity are the powers to which fundamental quantities are to be raised to represent the quantity. The basic quantities with their symbols in square brackets are as follows:

[Length]=[L]$$$$

[Mass]=[M]$$$$

[Time]=[T]$$$$

[Temperature]=[K]or[Θ]$$\mathrm{}$$

[Current]=[A]0r[I]$$\mathrm{}$$

[No.ofMoles]=[N]$$$$

1. Velocity
   V=displacementtime$$\frac{}{}$$
   
   
   V=[L][T]$$\frac{}{}$$
   
   
   =[M0L1T−1]$${}^{\mathrm{}}{}^{\mathrm{}}{}^{\mathrm{}}$$
   
   Dimensions of velocity are 0 in mass, 1 in length and -1 in time i.e. (0, 1, -1)

2. Acceleration
   V=change in velocitytime taken$$\frac{}{}$$
   
   
   V=displacementtime×time$$\frac{}{}$$
   
   
   V=[L1][T2]$$\frac{{}^{\mathrm{}}}{{}^{\mathrm{}}}$$
   
   
   =[M0L1T−2]$${}^{\mathrm{}}{}^{\mathrm{}}{}^{\mathrm{}}$$
   
   Dimensions of acceleration are (0, 1, -2).

3. Force
   F=mass×acceleration$$$$
   
   
   =mass×change in velocitytime taken$$\frac{}{}$$
   
   
   =mass×displacementtime×time$$\frac{}{}$$
   
   
   =[M1][L1][T2]$${}^{\mathrm{}}\frac{{}^{\mathrm{}}}{{}^{\mathrm{}}}$$
   
   
   =[M1L1T−2]$${}^{\mathrm{}}{}^{\mathrm{}}{}^{\mathrm{}}$$
   
   Dimensions of force are (1, 1, -2).

Dimensional formula

It is the expression which shows how and which fundamental quantities are used in the representation of a physical quantity.

1) Velocity [M0 L1 T-1]

2) Acceleration [M0 L1 T-2]

3) Force [M1 L1 T-2]

4) Energy [M1 L2 T-2]

5) Power [M1 L2 T-3]

6) Momentum [M1 L1 T-1]

7) Pressure [M1 L-1 T-2]

Dimensional equation

It is the equation obtained by equating a physical quantity with its dimensional formula.

1) Velocity [V] = [M0 L1 T-1]

2) Acceleration[a] = [M0 L1 T-2]

3) Force [F] = [M1 L1 T-2]

4) Energy [E] = [M1 L2 T-2]

5) Power [P] = [M1 L2 T-3]

6) Momentum [P] = [M1 L1 T-1]

7) Pressure [P] = [M1 L-1 T-2]

Dimensional Formulas of Some Physical Quantities

S.N	Physical quantity	Relation with other physical quantities	Dimensional formula	SI-unit
1.	Volume	length× breadth× height	[L] ×[L] ×[L]= [M0L3T0]	m3
2.	Velocity or speed	distancetime$\frac{}{}$	= [M0L0T-1]	ms-1
3.	Momentum	mass × velocity	[M] × [LT-1]= [MLT-1]	kgms-1
4.	Force	mass × acceleration	[M] × [LT-2]= [MLT-2]	N (newton)
5.	Pressure	forcearea$\frac{}{}$	=[ML-1T--2]	Nm-2 or Pa (pascal)
6.	Work	force × distance	[MLT-2] ×[L]= [ML2T-2]	J (joule)
7.	Energy	Work	[ML2T-2]	J (joule)
8.	Power	worktime$\frac{}{}$	=[ML2T-3]	W (watt)
9.	Gravitational constant	force×(distance)2(mass)2$\frac{{}^{\mathrm{}}}{{}^{\mathrm{}}}$	[M-1L3T-2]	Nm2kg-2
10.	Angle	arcradius$\frac{}{}$	Dimensionless	rad
11.	Moment of inertia	mass × (distance)2	[ML2T0]	Kgm2
12.	Angular momentum	moment of inertia × angular velocity	[ML2T0] × [T-1]= [ML2T-1]	Kgm2s-1
13.	Torque or couple	force × perpendicular distance	[MLT-2] ×[L]= [ML2T-2]	Nm
14.	Coefficient of viscosity	forcearea×velocity gradient$\frac{}{}$	[ML-1T-1]	Dap (Dacapoise)
15.	Frequency	1second$\frac{\mathrm{}}{}$	[T-1]	Hz

Principle of homogeneity

It states that “The dimensions of fundamental quantities on a left-hand side of an equation must be equal to the dimensions of the fundamental quantities on the right-hand side of that equation.”

Four Categories of Physical Quantities

Physical quantities can be categorized into four types. They are:

1. Dimensional variablesThose physical quantities which have dimensions but do not have fixed value are called dimensional variables. Examples: force, work, power, velocity etc.
2. Dimensionless variables
   Those physical quantities which have neither dimensions nor fixed value are called dimensionless variables.
3. Dimensional constantThose physical quantities which possess dimensions and fixed value are called dimensional constant. Their examples are gravitational constant, velocity of light etc.
4. Dimensionless constantThose physical quantities which do not possess dimensions but possess fixed value are called dimensionless constant. Examples are pi π, counting number etc.