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अध्याय 1: मात्रक और मापन (Units and Measurements)

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CBSE • कक्षा XI • Physics • अध्याय 1

अनुमानित समय: 45 Mins

प्रगति: अध्ययनरत

मात्रक और मापन (Units and Measurements)

In Class 11 Physics, "Units and Measurements" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.

अध्याय CBT ऑनलाइन टेस्ट दें सभी अध्याय देखें

प्रस्तावना अध्याय अवधारणाएँ अध्याय सारांश स्व-मूल्यांकन अभ्यास टेस्ट दें

📏 Have You Ever Wondered?

How do astrophysicists measure the distance to a galaxy billions of light-years away without leaving Earth, or how do particle physicists measure the ...

How do astrophysicists measure the distance to a galaxy billions of light-years away without leaving Earth, or how do particle physicists measure the mass of a Higgs boson down to $10^{-25}\text{ kg}$? Dimensional analysis and precision measurement form the bedrock of experimental physics.

यह अध्याय क्यों महत्वपूर्ण है

In Class 11 Physics, "Units and Measurements" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.

अध्ययन से पूर्व (आवश्यक ज्ञान)

Physical quantities and SI units from Class 9.
Scientific notation.
Significant figures.

इस अध्याय के लक्ष्य

Define Fundamental (7 SI base units) and Derived physical quantities.
Apply Dimensional Analysis to check the dimensional consistency of physical equations.
Derive relations among physical quantities using the Principle of Homogeneity of Dimensions.
Distinguish between Systematic errors, Random errors, and Least count errors.
Calculate Absolute error, Relative error, Percentage error, and apply significant figure arithmetic.

अध्याय रूपरेखा एवं प्रगति

1 1. SI Base Units & Dimensional Form...

2 2. Principle of Homogeneity & Appli...

3 3. Error Analysis & Significant Fig...

सम्पूर्ण सैद्धांतिक एवं वैचारिक अध्ययन

1. SI Base Units & Dimensional Formulas

The International System of Units (SI) defines 7 Fundamental Units: Length ($m$), Mass ($kg$), Time ($s$), Electric Current ($A$), Thermodynamic Temperature ($K$), Amount of Substance ($mol$), and Luminous Intensity ($cd$). The Dimensions $[M^a L^b T^c]$ represent the powers to which base quantities are raised (e.g. Force $= [M L T^{-2}]$, Energy $= [M L^2 T^{-2}]$).

2. Principle of Homogeneity & Applications

Principle of Homogeneity: An equation is physically valid only if the dimensions of every term on both sides are strictly identical:
• Checking equation correctness (e.g. $s = ut + \frac{1}{2}at^2$).
• Converting units from one system to another ($n_1 u_1 = n_2 u_2$).
• Deriving relationships between physical variables.

3. Error Analysis & Significant Figures

Absolute Error: $\Delta a_i = |a_{\text{mean}} - a_i|$.
Relative Error: $\frac{\Delta a_{\text{mean}}}{a_{\text{mean}}}$. Percentage Error $= \frac{\Delta a_{\text{mean}}}{a_{\text{mean}}} \times 100\%$.
Combination of Errors: In a product/quotient $Z = A^p B^q / C^r$, maximum fractional error is: $$\mathbf{\frac{\Delta Z}{Z} = p\frac{\Delta A}{A} + q\frac{\Delta B}{B} + r\frac{\Delta C}{C}}$$

चित्रात्मक व्याख्या एवं मॉडल

Units and Measurements Master Matrix

Conceptual framework, core mechanisms, and analytical relationships

Academic Architecture

1. SI Base Units & Dimensional Formulas • 2. Principle of Homogeneity & Applications

अध्याय का सार संक्षेप एवं 10 मुख्य निष्कर्ष

मुख्य बिंदु 1

Seven SI Base Quantities: Mass, Length, Time, Current, Temperature, Mole, Candela.

मुख्य बिंदु 2

Homogeneity of Dimensions: Fundamental physical rule requiring dimensional equality across all equation terms.

मुख्य बिंदु 3

Dimensional Limitations: Cannot determine dimensionless constants or handle trigonometric/exponential functions.

मुख्य बिंदु 4

Significant Figures: Meaningful digits conveying the precision of an experimental measurement.

मुख्य बिंदु 5

Percentage Error: Standard normalized error metric in laboratory experimental physics.

स्व-मूल्यांकन अभ्यास (Check Your Understanding)

मूल वैचारिक स्पष्टता की जांच के लिए नैदानिक प्रश्न। पहले स्वयं हल करें, फिर उत्तर देखें।

Check the dimensional correctness of the equation: $\frac{1}{2}mv^2 = mgh$.

उत्तर एवं व्याख्या देखें

उत्तर: LHS: $[M][LT^{-1}]^2 = [M L^2 T^{-2}]$. RHS: $[M][LT^{-2}][L] = [M L^2 T^{-2}]$. Since LHS and RHS have identical dimensions $[M L^2 T^{-2}]$, the equation is dimensionally correct.
Dimensionally correct ([M L^2 T^-2]).

Find the dimensions of universal gravitational constant $G$ in Newton's law $F = G \frac{m_1 m_2}{r^2}$.

उत्तर एवं व्याख्या देखें

उत्तर: $G = \frac{F r^2}{m_1 m_2} = \frac{[M L T^{-2}][L^2]}{[M^2]} = [M^{-1} L^3 T^{-2}]$.
[M^-1 L^3 T^-2].

The period of oscillation of a simple pendulum is $T = 2\pi \sqrt{l/g}$. If $l$ is measured with 1% error and $g$ with 2% error, find the percentage error in $T$.

उत्तर एवं व्याख्या देखें

उत्तर: $T \propto l^{1/2} g^{-1/2} \implies \frac{\Delta T}{T} = \frac{1}{2}\frac{\Delta l}{l} + \frac{1}{2}\frac{\Delta g}{g} = \frac{1}{2}(1\%) + \frac{1}{2}(2\%) = 0.5\% + 1.0\% = 1.5\%$.
1.5%.

State two limitations of dimensional analysis.

उत्तर एवं व्याख्या देखें

उत्तर: (1) It cannot determine the value of dimensionless proportionality constants (like $2\pi$), (2) It fails when a physical quantity depends on trigonometric, logarithmic, or exponential functions.
Cannot find dimensionless constants or handle trig functions.

How many significant figures are in the numbers: (i) 0.0025, (ii) 2.050, (iii) 5000?

उत्तर एवं व्याख्या देखें

उत्तर: (i) 0.0025 has 2 significant figures, (ii) 2.050 has 4 significant figures, (iii) 5000 has 1 significant figure (without a decimal point).
(i) 2, (ii) 4, (iii) 1.

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