. How can the method of dimensions be used to deduce a relation among the physical
quantities ? Explain it with the help of a suitable example
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In engineering and science,dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric charge) and units of measure (such as miles vs. kilometres, or pounds vs. kilograms) and tracking these dimensions as calculations or comparisons are performed. The conversion of units from one dimensional unit to another is often easier within the metric or SI system than in others, due to the regular 10-base in all units. Dimensional analysis, or more specifically the factor-label method, also known as the unit-factor method, is a widely used technique for such conversions using the rules of algebra.[1][2][3]
The concept of physical dimension was introduced by Joseph Fourier in 1822.[4] Physical quantities that are of the same kind (also called commensurable) (e.g., length or time or mass) have the same dimension and can be directly compared to other physical quantities of the same kind, even if they are originally expressed in differing units of measure (such as yards and metres). If physical quantities have different dimensions (such as length vs. mass), they cannot be expressed in terms of similar units and cannot be compared in quantity (also called incommensurable). For example, asking whether a kilogram is larger than an hour is meaningless.
Any physically meaningful equation (and any inequality) will have the same dimensions on its left and right sides, a property known as dimensional homogeneity. Checking for dimensional homogeneity is a common application of dimensional analysis, serving as a plausibility check on derived equations and computations. It also serves as a guide and constraint in deriving equations that may describe a physical system in the absence of a more rigorous derivation.
Example: Derive the formula for centripetal force F acting on a particle moving in a uniform circle.
As we know, the centripetal force acting on a particle moving in a uniform circle depends on its mass m, velocity v and the radius r of the circle. Hence, we can write
Hence,
F = ma vb rc
Writing the dimensions of these quantities,
Dimensional Formula Example 01
As per the principle of homogeneity, we can write,
a = 1, b + c = 1 and b = 2
Solving the above three equations we get, a = 1, b = 2 and c = -1.
Hence, the centripetal force F can be represented as,
F=Km^1v^2r^-1
F=k×mv^2÷r
Explanation:
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