Unit – II A
Dimensions of Physical Quantities:
- The power to which fundamental units are raised in order to obtain the unit of physical quantity is called the dimensions of that physical quantity.
- If ‘A’ is any physical quantity then the dimensions of A are represented by [A]
- Mass, length and time are represented by M, L, T respectively.
- Therefore, the dimensions of fundamental quantities are as follows
[Mass] = [M]
[Length] = [L]
[Time] = [T]
Different Types of Variables and Constants:
- The quantities like area, volume, velocity, force etc. possess dimensions and do not have a constant value. Such quantities are called dimensional variables.
Non dimensional variables:
- The quantities like strain, angle, specific gravity are ratios which are mere numbers are dimensionless quantities and are called numerics. They have neither dimension nor constant value, they are called non-dimensional variables.
- The quantities like Gravitational constant G, the velocity of light c and Planck’s constant h have dimensions and constant value. They are called dimensional constant.
- The quantities which have no dimensions but have constant value are called non-dimensional constants. 1, 2, 3, …. , θ, π are non dimensional constant.
Difference Between Dimensional Formula and Dimensional Equation:
- The dimensional formula of a physical quantity may be defined as the expression that indicates which of the fundamental units of mass, length and time enter into the derived unit of that quantity and with what powers. e.g. Dimensional formula for velocity is [M0L1T-1]
- The equation obtained, when a physical quantity is equated with its dimensional formula is known as a dimensional equation. e.g. [V] = [M0L1T-1] is a dimensional equation.
Uses of Dimensional Analysis:
To check the correctness of a physical equation:
- By the principle of homogeneity of dimensions, the dimensions of all the terms on the two sides of an equation must be the same.
To derive the form of a physical equation:
- To find the form of a physical equation, we first consider all the physical quantities on which a given physical quantity is likely to depend on. Then, by the application of the principle of homogeneity of dimensions, we eliminate those quantities on which the physical quantity does not depend.
To derive the relation between different units of different systems of a physical quantity:
- The dimensional analysis is used to find the conversion factor when the system of units is changed from one type to other.
- Let us consider a physical quantity ‘Q’ having dimensions [MxLyTz].
- Let the fundamental units of the first system be M1, L1, T1 and the fundamental units of the second system be M2, L2, T2 . Let n1 be the value of the quantity in the first system of units and n2 be the value of the quantity in the second system of units.
- Now They represent the same physical quantity, hence we can write
To find dimensions of new( whose dimensions are unknown) physical quantity.
Limitations of Dimensional Analysis:
- The constant of a physical equation can not be found using dimensional analysis. These constants are to be determined by experiments.
- The physical equation is dimensionally correct does not mean that the equation is scientifically correct.
- It is not useful when the trigonometric or exponential functions are involved.
- This method can be used only for the relations having a product or division relation. It is not useful for addition or subtraction relation. Thus it is not useful for deriving complex relations.
- This method does not give information about dimensional constant such as Universal gravitation constant G, Planck’s constant h, Rydberg’s constant R etc.
- In this method, we compare the powers of the fundamental quantities to obtain a number of independent equations for finding the unknown powers. Since the total number of such equations cannot exceed the number of fundamental quantities (3), we cannot use this method to obtain the relation if the quantity of interest depends upon more parameters than the number of fundamental quantities used.
- In many problems, it is difficult to guess the parameters on which the quantity of interest depends. This requires trained subtle and intuitive mind.
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