“Class 11 Physics Chapter 1: Units and Measurements – Comprehensive Study Material for 2025 Annual Exams

📘 Chapter 1: Units and Measurement (Point-by-Point Detailed Notes) 📚 Based on NCERT Class 11 Physics (2025–26)

🟠 1.1 Introduction

🔹 Measurement of a physical quantity involves comparing it with a basic, arbitrarily chosen, internationally accepted standard called a unit. 🔹 The result of a measurement is a numerical value + unit (e.g., 10 m). 🔹 All physical quantities are interrelated; thus, we only need a limited number of fundamental units to derive others. 🔹

• Fundamental (base) units: Directly measurable (e.g., metre, kilogram, second)
• Derived units: Formed by combinations of base units (e.g., m/s, N, J) 🔹 The full set of base and derived units is called a system of units.

🟡 1.2 The International System of Units (SI)

🔹 Earlier unit systems included CGS, FPS (British), and MKS systems:

• CGS: centimetre, gram, second
• FPS: foot, pound, second
• MKS: metre, kilogram, second

🔹 The SI system (Système Internationale d’Unités) is globally accepted:

• Established by BIPM (Bureau International des Poids et Mesures) in 1971
• Revised in 2018 for global scientific, technical, and commercial use

🧮 SI Base Quantities and Units:

Base Quantity	Unit Name	Symbol	Definition Summary
Length	metre	m	Based on speed of light in vacuum
Mass	kilogram	kg	Defined via Planck constant
Time	second	s	Based on Caesium-133 atom transition frequency
Electric current	ampere	A	Defined using elementary charge
Thermodynamic temperature	kelvin	K	Based on Boltzmann constant
Amount of substance	mole	mol	Number of entities = Avogadro number
Luminous intensity	candela	cd	Defined using luminous efficacy of monochromatic radiation

🔸 Angle Units:

• Plane angle (radian – rad): arc length/radius
• Solid angle (steradian – sr): area on sphere/radius²

🔸 Other Units Still in Use: Some non-SI units are retained for practical convenience (see Table 1.2).

🔸 Derived Units & Prefixes: SI derived units are formed from base units (e.g., 1 N = 1 kg·m/s²). Prefixes denote powers of ten (e.g., kilo, mega, micro).

📍 More to follow in next chunks:

• 1.3 Significant Figures
• 1.4 Dimensions of Physical Quantities
• 1.5 Dimensional Formulae & Equations
• 1.6 Dimensional Analysis & Applications Perfect! Let’s continue with Point 1.3: Significant Figures from Chapter 1 – Units and Measurement (Class 11 Physics, 2025–26 NCERT syllabus). Here’s the detailed and visually formatted explanation:

🔷 1.3 Significant Figures (सार्थक अंक)

📌 What are Significant Figures?

When we measure a physical quantity, the result is never perfectly exact. It includes:

• Certain digits: Measured with certainty.
• One uncertain digit: The first doubtful figure.

🔹 Significant Figures = All certain digits + First uncertain digit

🎯 Purpose of Using Significant Figures:

✅ To indicate precision in measurements
✅ To avoid misleading or false accuracy
✅ To ensure proper representation in calculations

🧠 Examples:

Measurement	Significant Figures	Explanation
1.62 s	3	1 & 6 certain, 2 uncertain
287.5 cm	4	2, 8, 7 certain; 5 uncertain
0.02308 m	4	Only digits 2, 3, 0, 8 are significant (leading zeros ignored)

📋 Rules for Counting Significant Figures:

✔️ Rule 1: All non-zero digits are significant

📝 Example:

• 245 has 3 significant figures

✔️ Rule 2: All zeros between non-zero digits are significant

📝 Example:

• 2008 → 4 significant figures
• 2.005 → 4 significant figures

✔️ Rule 3: Zeros to the left of the first non-zero digit are not significant

📝 Example:

• 0.0023 → 2 significant figures (2 and 3)
• 0.000500 → 3 significant figures (5 and two trailing zeros)

✔️ Rule 4: Trailing zeros in a number without a decimal are not significant

📝 Example:

• 1500 → 2 significant figures
  1. → 4 significant figures (decimal makes trailing zeros significant)

✔️ Rule 5: Trailing zeros in a number with a decimal are significant

📝 Example:

• 2.300 → 4 significant figures
• 0.0600 → 3 significant figures

✔️ Rule 6: Change of unit doesn’t change significant figures

📝 Example:

• 2.308 cm = 0.02308 m = 23.08 mm → All have 4 significant figures

✔️ Rule 7: Use of Scientific Notation removes ambiguity

📌 Format: , where

• has 1 ≤ value < 10
• All digits in are significant

📝 Example:

• 4.700 = 4.700 × 10⁰ → 4 significant figures
• 4700 = 4.7 × 10³ (if written this way, it has only 2 significant figures)

🔍 1.3.1 Arithmetic Operations with Significant Figures

➕ Addition and Subtraction:

• Final answer should have same number of decimal places as the term with least decimal places.

🧮 Example:
436.32 g + 227.2 g + 0.301 g = 663.821 g → round to 663.8 g

✖️➗ Multiplication and Division:

• Final answer should have same number of significant figures as the term with least significant figures

🧮 Example:
Mass = 4.237 g (4 sig. fig.)
Volume = 2.51 cm³ (3 sig. fig.)
Density = 4.237 ÷ 2.51 = 1.69 g/cm³ (3 sig. fig.)

🔘 1.3.2 Rounding Off Uncertain Digits

📝 Rule:

• If digit to be dropped > 5 → Increase preceding digit
• If < 5 → Leave as is
• If = 5 →
  🔹 If preceding digit is even → Leave it
  🔹 If odd → Increase by 1

🧮 Examples:

• 2.746 → 2.75 (last digit > 5)
• 2.743 → 2.74 (last digit < 5)
• 2.745 → 2.74 (4 is even)
• 2.735 → 2.74 (3 is odd → round to 4)

🧮 1.3.3 Uncertainty in Arithmetic Calculations

📌 Example 1:

Length = 16.2 ± 0.1 cm (0.6%)
Breadth = 10.1 ± 0.1 cm (1%)

Area = Length × Breadth
= 163.62 cm² ± (0.6% + 1%) = ±1.6%
⇒ Final result = 164 ± 3 cm²

📌 Example 2:

Mass = 1.02 g ± 0.01 g ⇒ Error = ~1%
Mass = 9.89 g ± 0.01 g ⇒ Error = ~0.1%
🧠 Smaller values have higher relative error

📌 Rule:

In multi-step calculations, retain one extra digit in intermediate steps to avoid rounding errors.

Awesome! Here’s the next section of your detailed and visually rich notes:

🟦 1.4 Dimensions of Physical Quantities

📌 What are Dimensions?

A dimension of a physical quantity tells us the nature of the quantity in terms of the base quantities:

• Length (L)
• Mass (M)
• Time (T)
• Electric current (A)
• Thermodynamic temperature (K)
• Luminous intensity (cd)
• Amount of substance (mol)

These are the 7 fundamental quantities, and each derived quantity can be written in terms of them.

🧠 Why Use Dimensions?

✅ To express physical quantities using base quantities
✅ To understand how quantities depend on one another
✅ To use them in dimensional analysis

📦 Dimensional Notation:

• Written using square brackets: [ ]
• Example:
  • Length = [L]
  • Velocity = [L T⁻¹]
  • Force = [M L T⁻²]

🔍 Example 1: Volume

Volume = Length × Breadth × Height = L × L × L
⟹ Dimensions of Volume = [L³]

✳️ Volume has:

• 3 dimensions in length
• 0 in mass and time ⇒ [M⁰ T⁰ L³]

🔍 Example 2: Force

Force = Mass × Acceleration
= M × (L / T²)
⟹ Dimensions of Force = [M L T⁻²]

✳️ Force has:

• 1 dimension in mass
• 1 in length
• –2 in time

🔍 Example 3: Velocity

Velocity = Distance / Time = L / T
⟹ Dimensions = [L T⁻¹]

🚦 Whether it’s average velocity, final velocity, or speed — all have the same dimensions.

⚠️ Note:

• Magnitudes are not considered, only the nature of the physical quantity.
• Two quantities with the same dimensions may represent different concepts (e.g., torque and energy both have [M L² T⁻²]).

Great! Now let’s move to the next section with full clarity and detail:

🟨 1.5 Dimensional Formulae and Dimensional Equations

📌 What is a Dimensional Formula?

A dimensional formula shows how and which base quantities (like M, L, T) represent a derived physical quantity.

🧠 Structure:

Where:

• = Derived physical quantity
• = Integer powers (positive, negative or zero)

🔹 Examples of Dimensional Formulae:

Physical Quantity	Formula	Dimensional Formula
Volume	length³	[M⁰ L³ T⁰]
Velocity	displacement / time	[M⁰ L¹ T⁻¹]
Acceleration	velocity / time	[M⁰ L¹ T⁻²]
Force	mass × acceleration	[M¹ L¹ T⁻²]
Pressure	force / area	[M¹ L⁻¹ T⁻²]
Energy (Work)	force × distance	[M¹ L² T⁻²]
Power	energy / time	[M¹ L² T⁻³]
Density	mass / volume	[M¹ L⁻³ T⁰]

📌 What is a Dimensional Equation?

When we write an equation equating a physical quantity with its dimensional formula, it’s called a dimensional equation.

📖 Examples:

Quantity	Dimensional Equation
Volume	[V] = [M⁰ L³ T⁰]
Velocity	[v] = [M⁰ L¹ T⁻¹]
Force	[F] = [M¹ L¹ T⁻²]
Density	[ρ] = [M¹ L⁻³ T⁰]

🎯 Purpose of Dimensional Equations:

✅ To check dimensional consistency of physical laws
✅ To derive relationships between quantities
✅ To help in unit conversion

📌 Note: These equations are derived based on physical laws and relationships (like Newton’s second law, etc.).
They’re fundamental tools in dimensional analysis, which we will now explore in the next section.

Awesome! Let’s now dive into the final and very powerful section of this chapter:

🟩 1.6 Dimensional Analysis and Its Applications

📌 What is Dimensional Analysis?

It is a method that uses dimensions of physical quantities to:

🔹 Check the correctness of equations
🔹 Convert units between systems
🔹 Derive relations among physical quantities

📖 Important Principle:
Only quantities having the same dimensions can be added or subtracted.

📌 That means:
Velocity cannot be added to Force,
Current cannot be subtracted from Temperature, etc.

🟢 1.6.1 Checking the Dimensional Consistency of Equations

This method is based on the Principle of Homogeneity of dimensions:

➡️ In a physically correct equation, all terms must have the same dimensions.

🧠 Example:

Equation:

Breakdown:

Term	Dimension
,	[L]
	[L T⁻¹][T] = [L]
	[L T⁻²][T²] = [L]

✅ All terms have [L], so the equation is dimensionally consistent.

⚠️ Important Notes:

✔️ If dimensions on both sides match ⇒ equation may be correct
✖️ If dimensions don’t match ⇒ equation is definitely incorrect

But remember:
✔️ Dimensional correctness ≠ Physically correctness
(e.g., it won’t catch missing constants like ½, 2π, etc.)

🧪 Example 1.3

Check whether the equation is dimensionally correct:

| LHS → [M][L² T⁻²]
| RHS → [M][L T⁻²][L] = [M][L² T⁻²]

✅ So, equation is dimensionally correct.

🧪 Example 1.4

Check which of the following formulas can be ruled out dimensionally (K = Kinetic Energy):

Formula	Dimensional Check (RHS)	Valid?
(a)	[M² L³ T⁻³]	❌ Invalid
(b)	[M L² T⁻²]	✅ Valid
(c)	[M L T⁻²]	❌ Invalid
(d)	[M L² T⁻²]	✅ Valid
(e)	Invalid Addition	❌ Invalid

✳️ Correct dimension ≈ [M L² T⁻²]
Only (b) and (d) are dimensionally valid — though (b) is physically correct.

🟢 1.6.2 Deducing Relations among Physical Quantities

Using method of dimensions, we can derive formulas when we don’t know the constants.

🧪 Example 1.5

Find time period (T) of a simple pendulum, depending on:

• Length (l)
• Mass (m)
• Gravity (g)

Assume:

Use dimensions:

• T → [T¹]
• l → [L¹]
• g → [L T⁻²]
• m → [M¹]

Equating both sides:

Quantity	Exponent
L	x + y = 0
T	–2y = 1 → y = –½
M	z = 0

Solving:
x = ½, y = –½, z = 0

So,

(Note: Constant can’t be derived using dimensional method.)

✅ Uses of Dimensional Analysis:

🔹 Check validity of equations
🔹 Convert units across systems
🔹 Derive possible relations between physical quantities
🔹 Quickly guess approximate formulas in physics

⚠️ Limitations of Dimensional Analysis:

✖️ Cannot find dimensionless constants (e.g., ½, π, e)
✖️ Cannot distinguish between quantities with same dimensions (e.g., energy vs. torque)
✖️ Not applicable to log, sin, cos, exp functions (they must have dimensionless arguments)