Units and Measurements
मात्रक एवं मापन
CBSE LevelJEE / NEET LevelDifficulty: Moderate
Overview
Physics is a science of measurement. This chapter shows how a physical quantity is expressed as a number and a unit, how units are standardised in the SI system, and how we check, derive and report results using dimensions, significant figures and error analysis.
You will learn
SI units and base quantities · dimensional formulae · dimensional analysis and its limits · significant figures · absolute, relative and percentage error · error propagation.
Why it matters
CBSE: about 3–5 marks every year (dimension checks, significant figures, error numericals). Competitive: dimensions and error questions appear in JEE Main and NEET almost every session, and every later chapter depends on correct units.
Part 1 of 4. Practice Zone, marks-wise questions, flashcards and the chapter test follow in the next parts.
Chapter Theory (मूल अवधारणाएँ)
1. Physical Quantities and Units (भौतिक राशि एवं मात्रक)
Definition: A physical quantity is any quantity that can be measured. Its value = numerical magnitude × unit. Fundamental (base) quantities are independent; derived quantities are built from them.
Simple idea: Saying "5" means nothing; "5 m" does. The unit tells what was counted.
Significance: a bigger unit gives a smaller number, so n ∝ 1/u. SI unit: depends on the quantity. Dimensions: not applicable (pure idea).
Common mistake: writing a result without a unit or mixing CGS and SI in one formula.
CBSE: define fundamental and derived units. Competitive: unit conversion by n₁u₁ = n₂u₂.
Application: the Mars Climate Orbiter was lost in 1999 due to a pound-force and newton unit mix-up.
Hint: n ∝ 1/u.
Answer: It falls (inch is the larger unit); 254 cm = 100 inch.
2. SI System (अंतर्राष्ट्रीय मात्रक पद्धति)
Definition: The SI (Système International) has seven base units and two supplementary quantities (plane angle, solid angle).
| Quantity | Unit | Symbol | Dimension |
|---|---|---|---|
| Length (लंबाई) | metre | m | L |
| Mass (द्रव्यमान) | kilogram | kg | M |
| Time (समय) | second | s | T |
| Electric current | ampere | A | A |
| Temperature | kelvin | K | K |
| Amount of substance | mole | mol | mol |
| Luminous intensity | candela | cd | cd |
Common mistake: treating gram as the SI base unit (it is kilogram) or calling radian a base unit.
CBSE: list the base units with symbols. Competitive: since 2019 all SI units are defined using fixed constants (c, h, e, k, NA).
3. Dimensions and Dimensional Analysis (विमा)
Definition: The dimensions of a quantity are the powers to which the base quantities must be raised to represent it. Written in square brackets.
Uses: (1) check a formula, (2) convert units, (3) derive a relation. Principle of homogeneity: every term of a physical equation has the same dimensions; only like quantities add or subtract.
Limitations: cannot give dimensionless constants (like 2π or ½); cannot handle sin, log or exp of dimensional arguments; fails if a quantity depends on more than three variables of the same dimension; cannot tell if the relation is a sum.
Common mistake: a dimensionally correct equation is not always physically correct (e.g. s = ut + ½at² vs s = ut + at²; both are homogeneous).
CBSE: dimensional formula of force, work, pressure. Competitive: deriving formulas, finding dimensions of constants (G, h).
Application: engineers use dimensionless numbers (Reynolds number) for scaling models.
4. Significant Figures (सार्थक अंक)
Definition: All reliably known digits plus the first uncertain digit in a measurement.
Rules: (1) all non-zero digits count; (2) zeros between non-zero digits count; (3) leading zeros do not (0.00520 has 3); (4) trailing zeros after the decimal point count (2.500 has 4); (5) powers of 10 do not count (4.50×10³ has 3); (6) exact numbers have unlimited figures.
Arithmetic: in addition/subtraction keep the least number of decimal places; in multiplication/division keep the least number of significant figures.
Rounding: digit > 5 round up; < 5 drop; exactly 5 → make the preceding digit even.
Common mistake: counting leading zeros, or rounding at each intermediate step.
CBSE: count figures and round results. Competitive: the ambiguity of 1200 m; use 1.2×10³ or 1.200×10³.
5. Errors in Measurement (त्रुटि)
Definition: Error is the difference between the measured and the true value. Systematic errors (instrument, personal, method) push results one way; random errors scatter results and are reduced by repeating readings. Least count error is ± one least count.
Combination of errors:
Common mistake: subtracting errors in a difference (they always add), or forgetting that the power multiplies the relative error.
CBSE: mean absolute and percentage error numericals. Competitive: multi-variable propagation (density, g from pendulum, resistivity).
Important Definitions (महत्वपूर्ण परिभाषाएँ)
Fundamental unit (मूल मात्रक)
A unit of a base quantity that is independent of all others. हिंदी: वह मात्रक जो किसी अन्य मात्रक पर निर्भर नहीं करता।
SI unit: m, kg, s … Example: metre for length.
Derived unit (व्युत्पन्न मात्रक)
A unit obtained by combining base units. हिंदी: मूल मात्रकों के संयोजन से बना मात्रक।
Example: newton = kg m s⁻².
Dimensions (विमाएँ)
The powers of base quantities in a physical quantity. हिंदी: किसी राशि में मूल राशियों की घातें।
Formula: [Force] = [M L T⁻²]. Example: work has [M L² T⁻²].
Least count (अल्पतमांक)
The smallest value an instrument can measure. हिंदी: यंत्र द्वारा मापा जा सकने वाला सबसे छोटा मान।
Formula: LC (vernier) = 1 MSD − 1 VSD. Example: metre scale LC = 1 mm.
Significant figures (सार्थक अंक)
The reliable digits plus the first doubtful digit. हिंदी: मापन के विश्वसनीय अंक तथा पहला संदिग्ध अंक।
Example: 3.140 has 4.
Absolute error (निरपेक्ष त्रुटि)
Magnitude of the difference between the true and measured value. हिंदी: मापे गए तथा वास्तविक मान का अंतर।
Unit: same as the quantity. Example: 9.80 vs 9.78 gives 0.02 m s⁻².
Relative & percentage error
Relative error = Δa/a; percentage error = (Δa/a)×100%. हिंदी: आपेक्षिक त्रुटि का कोई मात्रक नहीं होता।
Example: 0.02/9.80 ≈ 0.2%.
Principle of homogeneity
Dimensions of every term on both sides of a valid equation must be equal. हिंदी: समीकरण के प्रत्येक पद की विमाएँ समान होती हैं।
Example: v = u + at: all terms are [L T⁻¹].
Formula Sheet (सूत्र)
Symbols: n numeric value, u unit. Use: unit conversion. Mistake: inverting the ratio.
Unit: N. Use: dimension checks. Mistake: forgetting T⁻².
Unit: J. Work, all energies and torque share these dimensions.
Unit: Pa. Pressure, stress and energy density share this.
Unit: N m² kg⁻². From F = Gm₁m₂/r².
Unit: J s. From E = hν (ν = frequency, T⁻¹).
Symbols: N = number of vernier divisions (for a vernier of N divisions matching N−1 main-scale divisions). Use: vernier caliper.
Condition: Z = AᵖBᵠ. Use: maximum error. Mistake: using subtraction.
Condition: small angle. Use: error in g measurement.
Diagram: Reading a Scale
Measuring a rod with a metre scale (LC = 1 mm)
Explanation: each small division is 1 mm. The rod end lies exactly on the 43rd mark. Reading: 4.3 cm = 43 mm, uncertainty ±1 mm (the least count), so the figure 3 is the first doubtful digit.
Key observation: look straight at the mark; viewing at an angle adds parallax error.
Animation: Time Period of a Simple Pendulum
Period T = 2.01 s for l = 1.00 m (g = 9.8 m s⁻²). Key observation: T depends on l and g only, not on mass; this is exactly what dimensional analysis predicted.
Simulation: Error in g from a Pendulum
Change the measured values and watch the percentage error build up.
