By the end of this chapter, the student should be able to
:
- define the term dimensions of a
physical quantity.
- check for dimensional consistency
of equations.
- use dimensional analysis to
eliminate wrong equations from a set of given equations.
- use graphical methods to identify the correct equation
out of the dimensionally consistent ones.
- use dimensional analysis to
establish a relation between given quantities.
- solve problems involving
dimensions
Physical quantities are divided into two groups:
(i) Fundamental quantities.
(ii) Derived quantities.
Fundamental quantities are those which can not be expressed
in terms of any other quantities e.g mass (M), time (T),
length (L) and temperature (q).
Derived quantities are those which can be expressed in
terms of the fundamental quantities of mass, length, and
time.
Examples:
Area = (length)2 , Volume =(length)3
velocity = length ,
Density = mass ,Force
= mass x length
time
(length)3
(time)2
Dimensions of a physical quantity show
the way the physical quantity is related to the fundamental
quantities of mass, length and time.
The symbol [ ], is read as dimensions.
[Force] = MLT-2 , [Density] = ML-3
[Pressure] = [force/Area] = MLT-2/L2
= M'L-1T-2
The term dimensions of a physical quantity, actually refer
to the powers to which fundamental quantities are raised.
Example: The dimensions of pressure are 1 in mass, -1 in
length and -2 in time.
N.B: Quantities like refractive index, strain, relative
density and efficiency of a machine which possess no units
are also dimensionless quantities.
Trigonometrical ratios, indices, logarithms and all pure
numbers like p are also dimensionless.
Applications
of method of dimensions
Dimensional analysis can be used:
(i) to check the validity of equations. Obviously wrong
equations can be eliminated from a set of possible equations.
(ii) to deduce admissable relationships between the variables
of a physical system (or derive equations).
Checking the validity of equations
In a correct equation, the units on the left hand side
(L.H.S) must balance with the units on the right hand side
(R.H.S). Likewise the [L.H.S] = [R.H.S] in a correct equation.
All correct equations must be dimensionally consistent.
N.B: All correct equations must be dimensionally consistent
but not all dimensionally consistent equations are correct.
Dimensional consistency therefore can be used to eliminate
the wrong equations but cannot be used to prove the correctness
of an equation.
