Basic unit and dimension
Practical Units
For the measurement of a very large distance the following three units are used:—
1. Astronomical Unit (AU): It is the average distance of the centre of the sun from the centre of the earth.
AU = 1.5 × 1011 m
2. Light Year: It is defined as the distance travelled by light in vacuum in one year
light year = 3 ×108 (365 × 24 × 60 × 60) metre
ly = 9.4 ×1015 m
(3) Parsec: It is defined as the distance at which an arc 1AU long subtends an angle of 1.”
1 Par sec = 3.1 × 1016 m
Dimensional Analysis
Dimensions :
Dimensions of a physical quantity are defined as the power to which the fundamental units of mass, length, time etc. must be raised to represent a derived unit of the quantity.
So speed is said to posses zero dimension in mass, one dimension in length and –1 dimension in time.
Dimensional formula :
It is an expression which shows how and which of the fundamental units are required to represent the unit of a physical quantity.
is the dimensional formula of speed.
Dimensional Equation :
It is defined as an equation obtained by equating the physical quantity with its dimensional formula.
The dimensional equations of speed are
Based on dimensional analysis, physical quantities can be divided into four types—
1. Dimensional constants: Physical quantities which possess dimensions and have a constant value.
e.g., (i) Gravitational constant
(ii) Planck’s constant
2. Dimensional Variable: There are physical quantities which possess dimensions and do not have a constant value.
e.g., velocity, impulse, density.
3. Dimensionless constant: There are physical quantities which do not possess dimension and have a constant value
e.g., (a) Numerical constant 1, 2, 3
(b) Mathematical constant.
4. Dimensionless Variable: There are physical quantities which do not have a constant value. e.g. Angle, strain & specific gravity.
The principle of Homogeneity of Dimensions:
A given physical relationship is dimensionally correct if the dimensions of the various terms on either side of the relation are same. e.g., V = u + at
The dimensions of every term in the given physical relation are the same. So according to the principle of homogeneity of dimensions, the relation v = u + at is dimensionally correct.
Uses of Dimensional Equations
To convert a physical quantity from one system of units into another.
To establish a relationship between different physical quantities.
Limitations of Dimensional Analysis
Though the dimensional method is a simple and a very convenient way of finding the dependence of a physical quantity on other quantities of a given system, it has its own limitations, some of which are listed as follows:
♦In more complicated situations, it is often not easy to find out the factors on which a physical quantity will depend. In such cases, one has to make a guess which may or may not work.
♦This method gave no information about the dimensionless constant which has to be determined either by experiment or by a complete mathematical derivation.
♦This method is used only if a physical quantity varies as the product of other physical quantities. It fails if a physical quantity depends on the sum or difference of two quantities.
can not be derived by using the method of dimensions.
♦This method will not work if a quantity depends or another quantity as sine or cosine of an angle, i.e. if the dependence is a trigonometric function. The method works only if the dependence is by power functions only.
♦ This method does not give a complete information in cases where a physical quantity depends on more than three quantities because by equating the powers of M, L and T, we can obtain only three equations for the exponents. For example, if for a simple pendulum, we assume that the time period T depends on angle θ of the swing, besides the factors I, m and g; then
By equating the dimensions on both sides, we would find that a, b and c can be determined, but d remains undetermined. Besides the quantities used in Mechanics, the other fundamental quantities like current, temperature and substance are dimensionally represented as A, K or and mol respectively. Thus dimensions of electric and thermodynamic quantities can also be determined. For example:
