Dimension of Physical Quantity
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:
$$[Length]=[L]$$
$$[Mass]=[M]$$
$$[Time]=[T]$$
$$[Temperature]=[K]or[\Theta]$$
$$[Current]=[A]0r[I]$$
$$[No.of Moles]=[N]$$
 Velocity
$$ V = \frac{displacement}{time} $$
$$ V = \frac{[L]}{[T]} $$
$$=[M^0L^1T^{1}]$$
Dimensions of velocity are 0 in mass, 1 in length and 1 in time i.e. (0, 1, 1)
 Acceleration
$$ V = \frac{\text{change in velocity}}{\text{time taken}}$$
$$ V = \frac{displacement} {time \times time} $$
$$ V = \frac{[L^1]}{[T^2]} $$
$$=[M^0L^1T^{2}]$$
Dimensions of acceleration are (0, 1, 2).
 Force
$$F = mass \times acceleration$$
$$ =mass \times \frac{\text{change in velocity}}{\text{time taken}}$$
$$ = mass\times \frac{displacement} {time \times time} $$
$$ =[M^1] \frac{[L^1]}{[T^2]} $$
$$=[M^1L^1T^{2}]$$
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 [M^{0} L^{1} T^{1}]
2) Acceleration [M^{0} L^{1} T^{2}]
3) Force [M^{1} L^{1} T^{2}]
4) Energy [M^{1} L^{2} T^{2}]
5) Power [M^{1} L^{2} T^{3}]
6) Momentum [M^{1} L^{1} T^{1}]
7) Pressure [M^{1} L^{1} T^{2}]
Dimensional equation
It is the equation obtained by equating a physical quantity with its dimensional formula.
1) Velocity [V] = [M^{0} L^{1} T^{1}]
2) Acceleration[a] = [M^{0} L^{1} T^{2}]
3) Force [F] = [M^{1} L^{1} T^{2}]
4) Energy [E] = [M^{1} L^{2} T^{2}]
5) Power [P] = [M^{1} L^{2} T^{3}]
6) Momentum [P] = [M^{1} L^{1} T^{1}]
7) Pressure [P] = [M^{1} L^{1} T^{2}]
Dimensional Formulas of Some Physical Quantities
S.N 
Physical quantity 
Relation with other physical quantities 
Dimensional formula 
SIunit 
1. 
Volume 
length× breadth× height 
[L] ×[L] ×[L]= [M^{0}L^{3}T^{0}] 
m^{3} 
2. 
Velocity or speed 
\(\frac{distance}{time}\) 
= [M^{0}L^{0}T^{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 
\(\frac{force}{area}\) 
=[ML^{1}T^{2}] 
Nm^{2} or Pa (pascal) 
6. 
Work 
force × distance 
[MLT^{2}] ×[L]= [ML^{2}T^{2}] 
J (joule) 
7. 
Energy 
Work 
[ML^{2}T^{2}] 
J (joule) 
8. 
Power 
\(\frac{work}{time}\) 
=[ML^{2}T^{3}] 
W (watt) 
9. 
Gravitational constant 
\(\frac{force \times (distance)^2}{(mass)^2}\) 
[M^{1}L^{3}T^{2}] 
Nm^{2}kg^{2} 
10. 
Angle 
\(\frac{arc}{radius}\) 
Dimensionless 
rad 
11. 
Moment of inertia 
mass × (distance)^{2} 
[ML^{2}T^{0}] 
Kgm^{2} 
12. 
Angular momentum 
moment of inertia × angular velocity 
[ML^{2}T^{0}] × [T^{1}]= [ML^{2}T^{1}] 
Kgm^{2}s^{1} 
13.

Torque or couple 
force × perpendicular distance 
[MLT^{2}] ×[L]= [ML^{2}T^{2}] 
Nm 
14. 
Coefficient of viscosity 
\(\frac{force}{\text {area} \times \text {velocity gradient}}\) 
[ML^{1}T^{1}] 
Dap (Dacapoise) 
15. 
Frequency 
\(\frac{1}{second}\) 
[T^{1}] 
Hz 
Principle of homogeneity
It states that “The dimensions of fundamental quantities on a lefthand side of an equation must be equal to the dimensions of the fundamental quantities on the righthand side of that equation.”
Four Categories of Physical Quantities
Physical quantities can be categorized into four types. They are:
 Dimensional variables
Those physical quantities which have dimensions but do not have fixed value are called dimensional variables. Examples: force, work, power, velocity etc.  Dimensionless variables
Those physical quantities which have neither dimensions nor fixed value are called dimensionless variables.  Dimensional constant
Those physical quantities which possess dimensions and fixed value are called dimensional constant. Their examples are gravitational constant, velocity of light etc.  Dimensionless constant
Those physical quantities which do not possess dimensions but possess fixed value are called dimensionless constant. Examples are pi π, counting number etc.
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