Dimensions of physical quantities is essential to derive equations, validating the consistency of physical laws, and streamlining and resolving issues in the study of physics. Dimensions helps the students in ready for more advanced physics coursework by providing a foundation for understanding the mathematical framework that describes the physical world.
What are Physical Quantities?
Measurable features of systems or objects are known as physical quantities. These quantities can be broadly divided into two groups: derived quantities and fundamental (or base) quantities.
Fundamental Quantities: These are the quantities whose definitions don’t depend on any other quantities. There are seven basic quantities, electric current, temperature, mass, length, time, amount of substance, and intensity of light.
Derived Quantities: These are numbers that result from applying mathematical operations like multiplication and division to the basic values. For example, force is obtained from mass and acceleration, and velocity is calculated by dividing distance by time.
What are Dimensions?
Concept of “dimension” describes a physical quantity’s characteristics when stated in terms of the fundamental quantities. Dimensions of the fundamental quantities can be combined to represent any physical quantity.
Deriving relationships between various physical quantities and verifying the consistency of physical equations are two uses for this dimensional representation.Dimensions of physical quantities are expressed in terms of the following base dimensions:
Length (L)
Mass (M)
Time (T)
Electric Current (I)
Temperature (Θ)
Amount of Substance (N)
Luminous Intensity (J)
Dimensional formula, or symbolic statement showing the relationship between the dimensions of the base quantities and the derived quantity, can be used to represent any physical quantity.
Dimensional Formula and Dimensional Equation
Physical quantity’s dimensional formula is an expression that demonstrates how it depends on the fundamental quantities. It is stated as a product of fundamental quantity powers.
For example:
The dimensional formula of velocity is [M0L1T−1], meaning it depends on length (L) and time (T), but not on mass (M).
The dimensional formula of force is [M1L1T−2], as force depends on mass, length, and the square of time.
Equality of physical quantity dimensions on both sides of an equation is represented by a dimensional equation. Dimensional equation in Newton’s second rule of motion F = ma guarantees that the dimensions of both sides match. Force (F) is equal to mass (m) times acceleration (a).
Importance of Dimensions in Physics
Dimensions are useful for various reasons:
Checking the Consistency of Equations: If a certain physical equation is dimensionally consistent is one of the main applications of dimensions. Both sides of an equation such as F = ma must have the same dimensions. Dimensional homogeneity is the name given to this concept.
1.Derivation of Formulas: Relationships between physical quantities can be derived with the aid of dimensional analysis. We can frequently predict the shape of a physical equation just by knowing the dimensions of the quantities involved. It offers a trustworthy relational framework even without constants.
2. Conversion Between Units: Converting one measurement unit to another is made easier with the help of dimensions. Dimensional reasoning is used, When converting from meters to kilometers or seconds to hours. You may guarantee precise and reliable unit conversions by using the dimensions formula.
3. Prediction of Relationships Between Quantities: Dimensional analysis can be provided, possible co-relations between quantities in the absence of direct experimental data, which can aid in the development of new theories or laws.
Limitations of Dimensional Analysis
• It is unable to provide the numerical values of equations including dimensionless constants. A constant G is involved in the gravitational force equation F = Gm1m2 / r2, and its value cannot be determined from dimensions
If a physical quantity has the same measurements but depends on multiple factors, it cannot be used. Torque and energy reflect separate physical phenomena yet have the same dimensional formula [M1L2T−2].
It doesn’t provide light on a physical process’s inner mechanism.
Applications of Dimensional Analysis
Practical applications contain:
Validating Equations: Dimensional analysis is a common means used by scientists and engineers to confirm the dimensional accuracy of recently generated equations.
1.Understanding Scaling Laws: Physics provides predictable scaling of particular quantities according to their dimensions, which is useful for modeling systems with varying sizes or scales.
2. Designing Experiments: To plan experiments and determine which factors matter most in a particular physical system, dimensional analysis is in use.
Note:-
Physical quantity dimensions are fundamental to understand and use physics. It can verify the consistency of their calculations, discover new co-relations, and evaluate the accuracy of equations by translating physical data into dimensional formulas. Even with its drawbacks, dimensional analysis’s strength is vital resource for physics at all levels.
Dimension of length in velocity is = +1.
Dimension of time in velocity is = -1.
Symbolic form of dimension formula for fundamental quantities are M, L, T, K, A, cd, mol, for the fundamental quantities mass, length, time, temperature, current, luminous intensity and amount of matter respectively.
(1) numerical values are dimensionless e.g dimensional formula of two (2) is [2] = [M0L0T0]
(2) All trigonometrical function thita, sin thita, …. are dimensionless.
(3) Exponential functions are dimensionless e.g y = ex then [x] = [M0L0T0] .
(4) Logarithmic functions are dimensionless e.g
Y = loga x then [x] = [M0L0T0] , [a] = [M0L0T0] and y = loga x = [M0L0T0]
Powers or exponents of the fundamental quantities (such as length, mass, and time) that indicate how the quantity depends on these base values are referred as the dimensions of a physical quantity. They display a physical quantity’s form and nature in relation to fundamental measures.
Dimensions are vital for formula derivation, unit conversion, physical quantity prediction, and consistency checks in physical equations. They aid in guaranteeing the applicability and dimensional accuracy of the physics equations we use
Seven fundamental quantities are:
Length (L)
Mass (M)
Time (T)
Electric current (I)
Temperature (Θ)
Amount of substance (N)
Luminous intensity (J)