Units and Dimensions and Formulas - Notes for Quick Revision 📚

Units and Dimensions - Units and dimensions form the foundation of physics, providing a systematic way to express and measure physical quantities. They help ensure consistency in calculations and offer insights into the relationships between various physical phenomena. Understanding units and dimensions is crucial for solving problems, deriving equations, and verifying results in physics and engineering.

This article provides detailed notes on units and dimensions, covering fundamental concepts, SI units, dimensional analysis, and applications.

What are Units?

Definition: A unit is a standard quantity used to measure physical quantities.

Types of Units:

Fundamental Units:

Basic units that are independent of other units.
Example: Meter (length), kilogram (mass), second (time).

Derived Units:

Units derived from fundamental units through mathematical relationships.
Example: Newton ( $N = {kg \cdotp m/s}^{2}$ ).

System of Units

CGS System:

Fundamental units: Centimeter (cm), Gram (g), Second (s).
Example: Force is measured in dyne.

MKS System:

Fundamental units: Meter (m), Kilogram (kg), Second (s).
Example: Force is measured in Newtons.

International System of Units (SI):

A globally accepted system of units.
Fundamental SI units are used for all physical quantities.

Fundamental SI Units

Physical Quantity	Unit Name	Symbol
Length	Meter	$\text{m}$
Mass	Kilogram	$\text{kg}$
Time	Second	$\text{s}$
Electric Current	Ampere	$\text{A}$
Thermodynamic Temperature	Kelvin	$\text{K}$
Amount of Substance	Mole	$\text{mol}$
Luminous Intensity	Candela	$\text{cd}$

What are Dimensions?

Definition: Dimensions of a physical quantity express it in terms of fundamental quantities (length, mass, time, etc.).

Representation:

Length ( $L$ ), Mass ( $M$ ), Time ( $T$ ), Electric Current ( $I$ ), Temperature ( $Θ$ ), Amount of Substance ( $N$ ), Luminous Intensity ( $J$ ).

Dimensional Formula:

Expresses a physical quantity in terms of its fundamental quantities.
Example: Force $F = m a$ : $[F] = M L T^{- 2}$

Dimensional Analysis

Principle of Homogeneity

The dimensions on both sides of a valid physical equation must be the same. This principle is used for:

Checking the correctness of equations.
Deriving relationships between physical quantities.
Converting units between systems.

Dimensional Formula of Common Quantities

Physical Quantity	Formula	Dimensions
Velocity	$v = \frac{d}{t}$	$[LT^{-1}]$
Acceleration	$a = \frac{v}{t}$	$[LT^{-2}]$
Force	$F = ma$	$[MLT^{-2}]$
Work	$W = Fd$	$[ML^2T^{-2}]$
Energy	$E = W$	$[ML^2T^{-2}]$
Power	$P = \frac{W}{t}$	$[ML^2T^{-3}]$
Pressure	$P = \frac{F}{A}$	$[ML^{-1}T^{-2}]$

Applications of Dimensional Analysis

1. Verifying Equations

Example: Kinetic Energy

K E = \frac{1}{2} m v^{2}

Dimensions: $[M] [L T^{- 1}]^{2} = [M L^{2} T^{- 2}]$ .
Both sides are dimensionally consistent.

2. Deriving Relations

Example: Derive the time period of a simple pendulum:

Assume $T$ depends on length ( $L$ ) and gravity ( $g$ ): $T \propto L^{a} g^{b}$
Using dimensional analysis: $[T] = [L]^{a} [L T^{- 2}]^{b}$ Comparing dimensions: $T = 2 π \sqrt{\frac{L}{g}}$

3. Converting Units

Example: Convert 1 Newton to dyne.

$1 N = 1 0^{5} dyne$ .

Units and Dimensions - Comprehensive Notes, Formulas, and Applications

Limitations of Dimensional Analysis

Cannot determine the numerical constants in equations (e.g., $1/2$ in $KE = \frac{1}{2} mv^2$ ).
Fails for equations involving trigonometric, logarithmic, or exponential functions.
Only checks dimensional correctness, not the physical accuracy of an equation.

Significant Figures

Rules for Counting Significant Figures:

All non-zero digits are significant.
Zeros between significant digits are significant.
Leading zeros are not significant.
Trailing zeros in a decimal number are significant.

Precision vs. Accuracy

Precision: The degree of closeness among multiple measurements.
Accuracy: The degree of closeness to the true value.

Common Errors in Measurements

Systematic Errors:

Consistent and predictable errors (e.g., instrument calibration errors).

Random Errors:

Vary unpredictably in magnitude and direction.

Gross Errors:

Large errors caused by human mistakes.

FAQs About Units and Dimensions

Why are units important in physics?

Units provide a standardized way to measure and compare physical quantities, ensuring consistency in calculations and experiments.

What is the purpose of dimensional analysis?

Dimensional analysis helps verify equations, derive relationships, and convert units between different systems.

Can dimensional analysis identify all errors in equations?

No, it only verifies dimensional consistency but cannot detect numerical inaccuracies or missing constants.

How are derived units related to fundamental units?

Derived units are combinations of fundamental units, created through mathematical relationships.

What is the significance of significant figures?

Significant figures indicate the precision of a measurement, reflecting its reliability.

Understanding units and dimensions is fundamental to mastering physics. By grasping key concepts, dimensional analysis, and their applications, students can solve complex problems, verify equations, and develop a systematic approach to measurements. These notes provide a solid foundation for academic and competitive exams, ensuring clarity in this essential topic.

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