Dimensions and Dimensional Analysis in Physics

Understanding the language of physical quantities

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Executive Summary

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Physical quantities are described by seven base dimensions
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Dimensional formulae express quantities in terms of base units
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Dimensional analysis checks equation consistency and accuracy
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Applications include validating equations and deriving relationships

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Base Dimensions of Physical Quantities

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The Seven Fundamental Dimensions

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Length [L]
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Mass [M]
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Time [T]
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Electric current [A]
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Thermodynamic temperature [K]
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Luminous intensity [cd]
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Amount of substance [mol]

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Examples of Dimensional Representation

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Volume = [L] × [L] × [L] = [L³]
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Force = mass × acceleration = [M L T⁻²]
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Velocity = length/time = [L T⁻¹]

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Volume has zero mass and time dimensions
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Force has 1 mass, 1 length, -2 time dimensions
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All velocity types share [L T⁻¹]

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Dimensional Formulae and Equations

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Dimensional Formulae in Practice

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Volume: [M⁰ L³ T⁰]
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Speed: [M⁰ L T⁻¹]
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Acceleration: [M⁰ L T⁻²]
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Mass density: [M L⁻³ T⁰]

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An equation is dimensionally correct if both sides have the same dimensions.

Principle of Homogeneity of Dimensions

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Dimensional Analysis and Applications

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Key Applications of Dimensional Analysis

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Check dimensional consistency of equations
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Derive relationships between physical quantities
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Validate mathematical expressions
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Identify dimensionless quantities (e.g., angles, refractive index)

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Example: Checking Dimensional Consistency

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Equation: x = x₀ + v₀t + (1/2)at² with dimensional breakdown

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[x] = [L]
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[x₀] = [L]
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[v₀t] = [L T⁻¹][T] = [L]
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[(1/2)at²] = [L T⁻²][T²] = [L]

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Conclusions

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Dimensions describe the nature of physical quantities
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Dimensional formulae express quantities in base units
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Dimensional analysis validates equations and relationships
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Consistency is necessary but not sufficient for correctness

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References and Further Reading

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Source: Day3.pdf (Reprint 2026-27, PHYSICS) • Appendix 9: Dimensional formulae for various quantities • Recommended: Textbooks on fundamental physics and metrology

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