Class 11 Physics Chapter 2: Units and Measurements — Comprehensive NCERT Study Notes

Class 11 Physics Chapter 2: Units and Measurements (मात्रक एवं मापन)

A rigorous and mathematically clear masterclass on measurement theory, fundamental units, dimensional analysis, and error propagation for CBSE, RBSE, and competitive foundation (JEE Main / NEET).

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1. Physical Quantities and Units (भौतिक राशियाँ और मात्रक)

A Physical Quantity is any property of a material or system that can be quantified by measurement. It consists of a numerical value ($n$) and a unit ($u$): $\text{Physical Quantity} = n \times u$ Since the magnitude remains constant regardless of the unit chosen: $n_1 u_1 = n_2 u_2$

The International System of Units (SI Units)

The SI system defines 7 Fundamental (Base) Quantities and 2 Supplementary Quantities:

Base Quantity	SI Unit	Symbol	Dimension
Length (लंबाई)	Metre	$\text{m}$	$[L]$
Mass (द्रव्यमान)	Kilogram	$\text{kg}$	$[M]$
Time (समय)	Second	$\text{s}$	$[T]$
Electric Current (विद्युत धारा)	Ampere	$\text{A}$	$[A]$
Thermodynamic Temperature (ताप)	Kelvin	$\text{K}$	$[K]$
Amount of Substance (पदार्थ की मात्रा)	Mole	$\text{mol}$	$[\text{mol}]$
Luminous Intensity (ज्योति तीव्रता)	Candela	$\text{cd}$	$[cd]$

Supplementary Quantities:

1. Plane Angle ($d\theta$): $\text{Radian (rad)} = \frac{\text{arc length}}{\text{radius}}$
2. Solid Angle ($d\Omega$): $\text{Steradian (sr)} = \frac{dA}{r^2}$

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2. Dimensional Analysis (विमीय विश्लेषण)

The dimensions of a physical quantity are the powers (exponents) to which the base quantities are raised to represent that quantity.

Standard Dimensional Formulas:

• Velocity ($v$): $\frac{[L]}{[T]} = [M^0 L^1 T^{-1}]$
• Acceleration ($a$): $\frac{[L T^{-1}]}{[T]} = [M^0 L^1 T^{-2}]$
• Force ($F = ma$): $[M] \times [L T^{-2}] = [M^1 L^1 T^{-2}]$
• Work / Energy ($W = F \cdot s$): $[M L T^{-2}] \times [L] = [M^1 L^2 T^{-2}]$
• Power ($P = W/t$): $\frac{[M L^2 T^{-2}]}{[T]} = [M^1 L^2 T^{-3}]$
• Pressure / Stress ($P = F/A$): $\frac{[M L T^{-2}]}{[L^2]} = [M^1 L^{-1} T^{-2}]$
• Universal Gravitational Constant ($G$): $[M^{-1} L^3 T^{-2}]$
• Planck’s Constant ($h$): $[M^1 L^2 T^{-1}]$

Principle of Homogeneity of Dimensions (विमीय समांगता का सिद्धांत)

A physical equation is dimensionally correct if and only if the dimensions of each and every term on both sides of the equation are identical. Only quantities of the same dimensions can be added or subtracted!

Application: Checking the correctness of $s = ut + \frac{1}{2}at^2$:

• LHS: $[s] = [L]$
• RHS Term 1: $[ut] = [L T^{-1}][T] = [L]$
• RHS Term 2: $[\frac{1}{2}at^2] = [L T^{-2}][T^2] = [L]$ Since every term has dimension $[L]$, the equation is dimensionally consistent.

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3. Errors in Measurement (मापन में त्रुटियाँ)

Every measurement made by any instrument contains some uncertainty called an error.

Classification of Errors:

1. Systematic Errors: Instrument imperfections, personal bias, zero errors. They can be eliminated by calibration.
2. Random Errors: Unpredictable fluctuations in environmental conditions (temperature, voltage, air currents). Minimized by taking arithmetic mean of multiple readings.

Mathematical Representation of Errors:

Let measured values be $a_1, a_2, \dots, a_n$.

• Mean Value ($a_{\text{mean}}$): $a_{\text{mean}} = \frac{1}{n} \sum_{i=1}^n a_i$
• Absolute Error ($\Delta a_i$): $\Delta a_i = |a_{\text{mean}} - a_i|$
• Mean Absolute Error ($\Delta a_{\text{mean}}$): $\Delta a_{\text{mean}} = \frac{1}{n} \sum_{i=1}^n |\Delta a_i|$
• Relative / Fractional Error: $\text{Relative Error} = \frac{\Delta a_{\text{mean}}}{a_{\text{mean}}}$
• Percentage Error: $\% \text{ Error} = \frac{\Delta a_{\text{mean}}}{a_{\text{mean}}} \times 100\%$

Combination and Propagation of Errors:

• Sum & Difference ($Z = A \pm B$): $\Delta Z = \Delta A + \Delta B$ (Absolute errors add up).
• Product & Quotient ($Z = A \cdot B$ or $Z = A / B$): $\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}$ (Relative errors add up).
• Power Rule ($Z = A^p B^q / C^r$): $\frac{\Delta Z}{Z} = p\frac{\Delta A}{A} + q\frac{\Delta B}{B} + r\frac{\Delta C}{C}$

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4. Significant Figures (सार्थक अंक)

1. All non-zero digits are significant ($189.4 \implies 4\text{ sig figs}$).
2. Zeros between non-zero digits are significant ($2008 \implies 4\text{ sig figs}$).
3. Leading zeros before the first non-zero digit are never significant ($0.0035 \implies 2\text{ sig figs}$).
4. Trailing zeros after a decimal point are significant ($4.500 \implies 4\text{ sig figs}$).
5. Exact numbers have infinite significant figures.

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